First Principle Calculation on AunAg2 (n = 1 - 4) Clusters

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First Principle——第一性原理

Introduction to Computer Simulation: Edinburgh, May 2010
Set up the problem
Let’s start defining various quantities Assume that the nuclei (Mass Mi) are at: R1, R2, …, RN Assume that the electrons (mass me) are at: r1, r2, …, rm Now let’s put some details in the SE
ˆ R , R ,..., R ,r ,r ,...,r E R , R ,..., R ,r ,r ,...,r H 1 2 N 1 2 m 1 2 N 1 2 m
Introduction to Computer Simulation: Edinburgh, May 2010
Quantum Mechanics: Density Functional Theory and Practical Application to Alloys
Stewart Clark Condensed Matter Section Department of Physics University of Durham
Introduction to Computer Simulation: Edinburgh, May 2010
From first principles
The equipment
Application
Scientific problemsolving “Base Theory” (DFT) Implementation (the algorithms and program)

NPr共掺杂二氧化钛的第一性原理计算

N:Pr共掺杂二氧化钛的第一性原理计算高攀1，吴晶1，张学军2，周文芳1，柳清菊1（1.云南省高校纳米材料与技术重点实验室，云南大学，昆明 650091;2. 湖南城市学院物理与电信工程系，湖南益阳，413000）摘要：本文采用基于密度泛函理论的平面波超软赝势方法研究了N:Pr共掺杂二氧化钛的晶体结构、电子结构和光学性质，计算结果表明：共掺杂同时在禁带中和价带顶形成了杂质能级，一方面，这有利于光生电子-空穴对的分离，可提高TiO2的光催化活性，另一方面，掺杂后禁带宽度有一定的减小，从而使得TiO2基本吸收带边红移到可见光区。

关键词：TiO2；共掺杂；第一性原理First-principles study on Nitrogen and Praseodymiumcodoped anatase TiO2GAO Pan1, WU Jing1, ZHANG Xuejun2, ZHOU Wenfang1, LIU Qingju1(1. Yunnan Key laboratory of nanomaterials & technology, Yunnan University, Kunming 650091;2. Department of physics and Electric Information Engineering, Hunan City University, Hunan,Yiyang 413000, )Abstract: The crystal structures, electronic structures and optical properties of Nitrogen and / or Praseodymium doping anatase TiO2 were calculated by first principles with the plane-wave ultrasoft pseudopotential method based on density functional theory. The calculated results show that the formation of impurity energy levels of N+Pr-doped TiO2 are located at the centre of the band gap and on the top of the VB. On the one hand, it is very effective for the separation of photoexcited electron-hole pairs and the improvement of the photocatalytic activity of TiO2. On the other hand, the band gap of N+Pr-doped TiO2 is narrowed, therefore, the optical absorption edge of N+Pr-doped TiO2 can be red-shift.Key words: TiO2; codoping; first-principle0 引言作为光催化环境净化材料，TiO2由于具有无毒、成本低、稳定性好等诸多优点而成为最具研发潜力的光催化剂。

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First-principles calculations of electronic

First-principles calculations of electronic structure and optical properties of Boron-doped ZnO with intrinsicdefectsYen-Chun Peng,Chieh-Cheng Chen,Hsuan-Chung Wu ⇑,Jong-Hong LuDepartment of Materials Engineering,Ming Chi University of Technology,New Taipei 24301,Taiwana r t i c l e i n f o Article history:Received 11August 2014Received in revised form 27October 2014Accepted 27October 2014Available online 15November 2014Keywords:First principles B-doped ZnO Intrinsic defectElectronic structure Optical propertya b s t r a c tThis study adopted ﬁrst-principles calculations to evaluate the effects of intrinsic defects on the elec-tronic structure and optical properties of Boron-doped ZnO (BZO).Four types of defect were considered:non-defective (B Zn ),Zn vacancies (V Zn ),O vacancies (V O ),and interstitial Zn (Zn i ).Calculations of forma-tion energy illustrate that O-rich conditions tend to induce V Zn ,while O-poor conditions tend to induce V O and Zn i .With respect to electric properties,V Zn defects in BZO decrease carrier concentration as well as mobility,which consequently decreases the conductivity of BZO.The existence of V O or Zn i defects in BZO leads to n-type conductive characteristics and increases the optical band gap.The existence of Zn i defects in BZO also increases the effective mass,which decreases the mobility and conductivity of BZO.As for the optical properties,the introduction of V Zn to BZO leads to an increase in transmittance in the visible light region,but a decrease in the UV region.The introduction of intrinsic V O and Zn i defects to BZO leads to a signiﬁcant decrease in transmittance in the visible as well as UV regions.The calculated results were also compared with experimental data from the literature.Ó2014Elsevier B.V.All rights reserved.1.IntroductionZnO is an abundant,non-toxic material with a wide band gap (3.37eV)and transparent properties under visible light.ZnO has recently attracted considerable attention as an alternative for Tin-doped In 2O 3(ITO),which is currently the most common choice of transparent conductive oxide for a variety of applications [1,2].The resistivity of pure ZnO is on the order of 10À2X -cm,which is far higher than that of ITO (10À4X -cm order).A great deal of research has gone into enhancing the conductivity of ZnO through the addition of various dopants,which can mainly be divided into metals [3–5]and non-metals [6,7].B-doped ZnO (BZO)thin ﬁlm shows considerable promise for its superior photoelectric proper-ties and stability [8,9].Many groups have investigated the effects of process parameters on the electric and optical properties of BZO thin ﬁlm,with the aim of optimizing performance [7–13].Miyata et al.[7]indicated that the transmittance of BZO thin ﬁlm could be improved through the introduction of O 2gas from 0sccm to 10sccm.David et al.[10]reported that annealing temperature and atmosphere strongly affect the conductivity of BZO.Yang et al.[11]concluded that the low oxygen partial pressure during deposition increases the carrier density of oxygen vacancies,which leads to a strong decline in resistivity.However,resistivity in sam-ples produced under the high oxygen partial pressure is far higher than in samples deposited under low oxygen partial pressure,which suggests the existence of p-type carriers of Zinc vacancies in ﬁlms grown under high oxygen partial pressure.Patil et al.[12]synthesized B-doped ZnO powders using a mechanochemical method.The photoluminescence (PL)spectra at room temperature is an indication that a greater number of oxygen vacancies exist in nonmetal-doped ZnO,compared to pure ZnO.In the fabrication of BZO microrods,Yılmaz et al.[13]investigated the inﬂuence of B diffusion doping on optical emission and defect formation.PL spec-tra results revealed that the intensity of the deep level visible band emission increases with an increase in annealing time,which implies a signiﬁcant increase in the concentration of intrinsic defects.As outlined above,various process conditions inﬂuence the type and number of intrinsic defects with a subsequent inﬂuence on the electric and optical properties of BZO.Gaining a comprehensive understanding of the electric and optical characteristics of BZO would require in-depth study into the effects of intrinsic defects on the properties of BZO.First-principles calculations can provide information concerning materials at the microscopic scale to eluci-date the connection between structure and properties.It is well known that the use of conventional density functional theory/10.1016/j.optmat.2014.10.0580925-3467/Ó2014Elsevier B.V.All rights reserved.⇑Corresponding author at:Department of Materials Engineering,Ming Chi University of Technology,84Gungjuan Road,Taishan,New Taipei 24301,Taiwan.Tel.:+8862290898994675;fax:+886229084091.E-mail address:hcwu@.tw (H.-C.Wu).(DFT)leads to a considerable underestimation of the calculated band gap in ZnO [14–16].In our previous study [17],we used the DFT plus Hubbard U (DFT +U)method to avoid underestimat-ing the band gap.This approach reduced the differences in calcu-lated band gap and lattice constant to within 1%of the experimental values.The current study extended the utilization of the DFT +U method to calculate and analyze the effects of intrin-sic defects (V Zn ,V O ,and Zn i )on the formation energy,crystal struc-ture,electronic structure,and optical properties of BZO.These results clarify the connections among the fabrication process,structure,and properties of BZO,for use in determining the criteria for future material designs.2.Calculation methodsThis study considered a 2Â2Â2supercell of a Wurtzite ZnO,including 16Zn atoms and 16O atoms,as shown in Fig.1.A B-monodoping model was constructed by substituting one Zn atom (number 1site)with one B atom (B Zn model),which correspond to the B concentrations of 6.25at.%.We also considered three intrinsic defects in the B Zn structure,in which Zn vacancies(B Zn V Zn ),O vacancies (B Zn V O ),and interstitial Zn (B Zn Zn i )are repre-sented as 2,3,and 4,respectively.The V Zn ,V O ,and Zn i concentra-tions corresponds to doping levels of 6.25, 6.25,and 5.88at.%,respectively.The defect concentration could be reduced using a larger supercell for the real systems;however,this study was lim-ited with regard to computer resources.Therefore,the properties of the defects calculated from a 2Â2Â2ZnO supercell such as this could be used as qualitative analysis.1432ZnO BTable 1Formation energy and optimized structure of BZO with varying intrinsic defects.Formation energy (eV)Optimized structure O-richO-poor Zn–O (Å)B–O (Å)4V (%)ZnO –– 1.981––B Zn3.750.39 1.996 1.526À3.1B Zn V Zn 5.68 5.81 1.993 1.530À3.3B Zn V O 7.550.70 1.995 1.521À5.3B Zn Zn i10.513.662.0031.5174.74.5 eV2.15 eV3.25 eV4.68 eV4.41 eV(a)(b)(c)(d)Band structures of B-doped ZnO for (a)B Zn ,(b)B Zn V Zn ,(c)B Zn V O models.Y.-C.Peng et al./Optical Materials 39(2015)34–3935All models presented in this study were developed using CASTEP software [18].Structural optimization was performed on each model before calculating properties.The Monkhorst–Pack scheme [19]K-points grid sampling in the supercells was set at 4Â4Â2.Electron–ion interactions were modeled using the ultrasoft pseudo-potential method [20].The valence conﬁgurations of the atoms were 4s 23d 10for Zn,2s 22p 4for O,and 2s 22p 1for B.The elec-tron wave functions were expanded in plane wave with an energy cutoff of 380eV.In the structural optimization process,the change in energy,maximum force,maximum stress,and maximum displacement tolerances were set at 10À5eV/atom,0.03eV/Å,0.05GPa,and 0.001Å,respectively.The energy convergence crite-rion for the self-consistent ﬁeld was set at 10À6eV.To describe the electronic structures more accurately,we adopted the DFT +U d +U p method [21],in which the U d value for Zn-3d and the U p value for O-2p orbitals were set at 10and 7eV,respectively.The band structures,band gaps,and Zn-3d orbital locations of pure ZnO,which were used for the selection of U d and U p values,can be referenced in our previous research [17,22].3.Results and discussion 3.1.Optimized structureThe average bond lengths and volume difference ratio,as obtained from geometric optimization,are summarized in Table 1.In pure ZnO,each Zn atom is bonded to its three horizontal and one vertical oxygen neighbors.The average bond length of Zn-O is 1.981Åand optimized lattice constants are a =b =3.249Åandc =5.232Å,which are in agreement with the experimental values of a =b =3.249,c =5.206Å[23].Following the substitution of one B atom for one Zn atom (B Zn model),the Zn–O bond length is longer than that of B–O (1.526Å).This is because the B 3+radius (0.27Å)is smaller than that of Zn 2+(0.74Å)[24].Therefore,the cell volume of B Zn model shrinks,which is consistent with the experi-mental results [25].Clearly,the presence of Zn or O vacancies in BZO also leads to a shrinkage in volume.Conversely,the presence of interstitial Zn leads to a longer Zn–O length and expansion in volume.3.2.Formation energyTo examine the relative stability of BZO with intrinsic defects in neutral charge state,the defect formation energy can be expressed as follows:[26,27]E f ðD Þ¼E tot ðD ÞÀE tot ðZnO ÞþXn i l ið1Þwhere E tot (ZnO)and E tot (D )are the total energy in pure ZnO and in the defective systems,respectively.n i is the number of i atoms removed from or added to the supercell.If an atom is removed from the supercell,n i is positive,otherwise is negative.l i is the chemical potential of atom i .Formation energy depends on the growth envi-ronment during the preparation process,which can be O-rich or O-poor (Zn-rich).In thermo-dynamic equilibrium,D l Zn +D l O =D H f (ZnO),where D H f (ZnO)represents the formation enthalpy of ZnO.For the chemical potential of B,this study adopted the relation of 2D l B +3D l O 6D H f (B 2O 3)under O-rich conditions and l B =l B(bulk)under O-poor conditions,where D H f (B 2O 3)represents the(a)(b)(c)(d)V ZnV OZn iBZnOO density difference for (a)B Zn ,(b)B Zn V Zn ,(c)B Zn V O ,and (d)B Zn Zn i models.The red,orange,yellow,green,and interpretation of the references to color in this ﬁgure legend,the reader is referred to the web version 36Y.-C.Peng et al./Optical Materials 39(2015)34–39formation enthalpy of B2O3.D l i represents the chemical potential of atom i referred to as the elemental solid/gas of l i(bulk/molecule).It is well known that a defective structure with lower formation energy forms more readily and denotes an increased occurrence of defects.Table1presents a summary of the calculated formation energy of BZO with various intrinsic defects,based on the neutral charge state.With the existence of B Zn,E f(B Zn V Zn)<E f(B Zn V O)<E f (B Zn Zn i)under O-rich conditions,implying that O-rich conditions are more likely to induce the formation of V Zn,followed by V O and Zn i.Under O-poor conditions,E f(B Zn V O)<E f(B Zn Zn i)<E f (B Zn V Zn),which implies that O-poor conditions are more likely to induce the formation of V O.As a result,process conditions,such as O2gasﬂow rate and substrate temperature,largely determine the type of intrinsic defects that form in BZO during preparation. The occurrence of V O is far more likely under a low-O atmosphere, and V Zn is more likely to occur under a high-O atmosphere.For the sake of comparison,we also calculated the formation energy of a single intrinsic defect(V Zn and V O)in pure ZnO.The calculated values of E f(V Zn)and E f(V O)are3.09and4.33eV under O-rich conditions and6.58and0.84eV under O-poor conditions. Thus,we can see that the formation energy of a Zn vacancy from pure ZnO(E f(V Zn))is greater than that obtained from BZO(E f(B Zn V Zn)ÀE f(B Zn)=1.93eV under O-rich conditions and5.42eV under O-poor conditions).This demonstrates that Zn vacancies form more easily in BZO than in ZnO.These results are similar to those calculated for O vacancies,which implies that B-doping facilitates the formation of V Zn and V O.Previous studies [12]obtained similar results,indicating that a greater number of oxygen vacancies or defects exist in BZO than in pure ZnO.3.3.Electronic structureTo clarify the inﬂuence of intrinsic defects on the electronic structure of BZO,we calculated the band structure,difference in charge density,and density of states(DOS),as shown in Figs.2–4,respectively.In our previous study[17,22],the calculated band structure revealed a band gap of3.37eV in pure ZnO,which is in excellent agreement with values obtained in experiments.In the present study,we focus on the properties of BZO with intrinsic defects.Fig.2presents the band structures for BZO with various intrin-sic defect models.The Fermi level indicated by the dotted line was set to zero.Fig.2(a)shows the situation in which a Zn atom in pure ZnO is replaced by a B atom,in which the Fermi level shifts from the valence band(VB)maximum to the bottom of the conduction band(CB),resulting in a shallow donor level at the bottom of the CB.The shallow donor level at B doping causes an increase in the optical band gap to4.5eV at B concentration of6.25at.%,which is well known as the Burstein-Moss effect[28].The deﬁnitionof Fig.4.Density of states for(a)B Zn,(b)B Zn V Zn,(c)B Zn V O,and(d)B Zn Zn i models.optical band gap is from the top offor n-type semiconducting materials and the bottom of conduction band for materials.Similar tendencies were experiment-based studies[29,30].Asin the vicinity of B impurities appears atoms.The calculated Mulliken bond and B–O bonds are0.371and0.658, that a B–O bond is more covalent than Mulliken bond population represents characteristics).As shown in Fig.4(a),to the bottom of CB are the Zn-4s anda few O-2s and O-2p orbitals.The main extra electron toﬁll up the CBM.According to the results calculated regarded as an intrinsic defect under the B Zn V Zn model(Fig.2(b)),when donor levels coexist,the empty states produced trons from the B Zn donor level,resulting level as well as the formation of p-type band gap of B Zn V Zn can be narrowed to eration of conduction electrons requires energy from the Fermi level to the CB, be required in the B Zn model.Thus,in may lead to a decrease in the carrier known that mobility is related to the time.The relaxation time could not be software and was assumed as a defects in BZO.The followingeffects of the effective mass on thenear the Fermi level appear nearlyof carriers with a smaller curvature The larger effective mass is related toTherefore,V Zn defects in BZO reduce both carrier concentration as well as mobility,which consequently increases the resistivity of BZO.Fig.3(b)shows that the O atoms surrounding a Zn vacancy gain fewer electrons(green color),implying the occurrence of a number of empty states of O atoms.These empty states are O-2p orbitals near the Fermi level,as shown in Fig.4(b).V O and Zn i can be regarded as intrinsic defects in an O-poor environment.Fig.2(c)and(d)show the band structures in B Zn V O and B Zn Zn i models,in which n-type conductive characteristics appear and the optical band gap increases to4.68eV and4.41eV, respectively.One shallow donor state and one deep donor state occur in these two models.In the B Zn V O model,the deep donor level is probably the charge remaining in the oxygen vacancy (Fig.3(c));in the B Zn Zn i model,it is probably the covalence charge in the vicinity of the interstitial Zn atom(Zn i)(Fig.3(d)).Fig.4(c) and(d)show that the shallow donor level in both models origi-nated from B doping,whereas the deep donor level in the B Zn V O and B Zn Zn i models originated from the addition of V O and Zn i, respectively.The deep donor level in the B Zn V O model comprises mainly Zn and O atoms;however,in the B Zn Zn i model,it also includes B atom(B-2s and B-2p states).The shallow donor states provide conduction electrons;however,the deep donor states may contribute less to the increase in carrier concentration.Qual-itatively,the curvature of the energy band near the Fermi level in the B Zn Zn i model is smaller than that in the B Zn V O model.There-fore,Zn i defects present in BZO increase the effective mass,which may consequently decrease the mobility and conductivity of BZO.3.4.Optical propertiesThe optical properties can be described via the dielectric func-tion e(x)=e1(x)+i e2(x)[31].The imaginary part of the dielectric function e2(x)is calculated as follows:e2¼2e2pX e0Xk;v;cu ckuÁrj j u v k2d E ckÀE vkÀxÀÁð2Þwhere e is the electronic charge;X is the unit cell volume;u is the vector deﬁning the polarization of the incident electricﬁeld;x is the frequency of light;and u c k and u v k are the wave functions of the conduction and valence bands,respectively.Fig.5(a)shows the e2(x)of BZO with various intrinsic defects. In the B Zn model,a blue-shift in the intrinsic absorption edge occurred due to an enlarged optical band gap as compared with ZnO.The shallow donor levels mentioned in Section3.3resulted in an absorption peak at1.2eV.The absorption peaks in the intrin-sic defect models were as follows:B Zn V Zn(0.3eV),B Zn V O(1.5eV), and B Zn Zn i(0.9eV).These peaks resulted in enhanced absorption in the visible range.The peak of B Zn V Zn is the lowest,which can probably be attributed to the transition between occupied states and unoccupied states near the Fermi level.In the B Zn V O,and B Zn Zn i models,the absorption in the visible range may be the result of a shift from the shallow and deep donor occupied states to the unoccupied states of the conduction band.Fig.5(b)presents the transmittance of BZO under various defec-tive models.Table2presents the calculated values for average transmittance associated with each model under UV and visible light.It is should be noted that the calculated results were based on the doping levels of B(6.25at.%),V Zn(6.25at.%)V O(6.25at.%), and Zn i(5.88at.%).The average transmittance of pure ZnO is89.2 %in the visible region and65.6%in the UV region.Fig.5(b)shows that the incorporation of B into ZnO decreased transmittance in the range of800–1200nm(infrared region)and400–800nm(visible light region),but increased transmittance in the range of200–400nm(UV region),compared with pure ZnO.When V Zn was introduced to BZO,transmittance in the visible light region was Optical properties of BZO with varying intrinsic defects.(a)Imaginary dielectric function,(b)Transmittance.38increased,but transmittance in the UV region decreased,compared with BZO.When V O or Zn i was introduced to BZO,the transmit-tance of visible as well as UV light declined signiﬁcantly,which implies that transmittance could be reduced by employing a low-O environment for fabrication.This may explain why trans-mittance is enhanced by an increase in oxygen pressure during processing[7].4.ConclusionsThis study used the DFT+U method to investigate the inﬂuence of intrinsic defects on the formation energy,crystal structure,elec-tronic structure,and optical properties of BZO.Our results revealed that the formation energy of V Zn is lowest under O-rich conditions and the formation energy of V O is lowest under O-poor conditions. V Zn defects in BZO may decrease carrier concentration as well as mobility,which increases transmittance in the visible light region but decreases transmittance in the UV region.V O or Zn i defects in BZO lead to the appearance of n-type conductive characteristics, increasing the optical band gap,and decreasing transmittance in the visible light and UV regions.In addition,Zn i defects increase the effective mass,which may consequently decrease the mobility and conductivity of BZO.Conﬂict of interestsThe authors declare that there is no conﬂict of interests regard-ing the publication of this article.AcknowledgementsThis work was supported by the National Science Council in Taiwan(NSC102-2221-E-131-008),for which the authors are grateful.We also acknowledge the National Center for High-performance Computing for computer time and the use of its facilities.Reference[1]T.Minami,Semicond.Sci.Technol.20(2005)S35–S44.[2]L.Zhao,G.Shao,S.Song,X.Qin,S.Han,Rare Metals30(2011)175–182.[3]V.Bhavanasi,C.B.Singh,D.Datta,V.Singh,K.Shahi,S.Kumar,Opt.Mater.35(2013)1352–1359.[4]C.Huang,M.Wang,Z.Deng,Y.Cao,Q.Liu,Z.Huang,Y.Liu,W.Guo,Q.Huang,J.Mater.Sci.–Mater.Electron21(2010)1221–1227.[5]J.L.Zhao,X.W.Sun,H.Ryu,Y.B.Moon,Opt.Mater.33(2011)768–772.[6]L.Cao,L.P.Zhu,W.F.Chen,Z.Z.Ye,Opt.Mater.35(2013)1293–1296.[7]T.Miyata,Y.Honma,T.Minami,J.Vac.Sci.Technol.A25(2007)1193–1197.[8]X.L.Chen,B.H.Xu,J.M.Xue,Y.Zhao,C.C.Wei,J.Sun,Y.Wang,X.D.Zhang,X.H.Geng,Thin Solid Films515(2007)3753–3759.[9]D.W.Kang,J.Y.Kwon,D.J.Lee,M.K.Han,J.Electrochem.Soc.159(2012)H61–H65.[10]C.David,T.Girardeau, F.Paumier, D.Eyidi, croix,N.Papathanasiou,B.P.Tinkham,P.Gu´erin,M.Marteau,J.Phys.:Condens.Matter23(2011)334209.[11]H.Yang,X.Xu,X.Zhou,Y.Ma,J.Dong,T.Wang,J.Miao,Y.Jiang,J.Mater.Sci.47(2012)6513–6516.[12]A.B.Patil,K.R.Patil,S.K.Pardeshi,J.Solid State Chem.184(2011)3273–3279.[13]S.Yılmaz,J.Nisar,Y.Atasoy,E.McGlynn,R.Ahuja,M.Parlak,E.Bacaksız,Ceram.Int.39(2013)4609–4617.[14]L.Li,W.Wang,H.Liu,X.Liu,Q.Song,S.Ren,J.Phys.Chem.C113(2009)8460–8464.[15]X.Qu,W.Wang,S.Lv,D.Jia,Solid State Commun.151(2011)332–336.[16]G.Ji,Z.Gu,M.Lu,J.Zhou,S.Zhang,Y.Chen,Physica B405(2010)4948–4950.[17]H.C.Wu,Y.C.Peng,C.C.Chen,Opt.Mater.35(2013)509–515.[18]M.D.Segall,P.J.D.Lindan,M.J.Probert,C.J.Pickard,P.J.Hasnip,S.J.Clark,M.C.Payne,J.Phys.:Condens.Matter.14(2002)2717–2744.[19]H.J.Monkhorst,J.D.Pack,Phys.Rev.B13(1976)5188–5192.[20]D.Vanderbilt,Phys.Rev.B41(1990)7892–7895.[21]X.Ma,B.Lu,D.Li,R.Shi,C.Pan,Y.Zhu,J.Phys.Chem.C115(2011)4680–4687.[22]H.C.Wu,Y.C.Peng,T.P.Shen,Materials5(2012)2088–2100.[23]R.D.Vispute,V.Talyansky,S.Choopun,R.P.Sharma,T.Venkatesan,Appl.Phys.Lett.73(1998)348–350.[24]D.R.Lide,CRC Handbook of Chemistry and Physics,87th ed.,Taylor andFrancis,Philadelphia,2006.[25]X.G.Xu,H.L.Yang,Y.Wu,D.L.Zhang,S.Z.Wu,Appl.Phys.Lett.97(2010)232502.[26]A.Janotti, C.G.Van de Walle,Phys.Rev.B:Condens.Matter.76(2007)165202.[27]T.Guo,G.Dong,Q.Chen,X.Diao,F.Gao,J.Phys.Chem.Solids75(2014)42–47.[28]P.V.Kamat,N.M.Dimitrijevic, A.J.Nozik,J.Phys.Chem.93(1989)2873–2875.[29]L.Gao,Y.Zhang,J.M.Zhang,K.W.Xu,Appl.Surf.Sci.257(2011)2498–2502.[30]B.N.Pawar,S.R.Jadkar,M.G.Takwale,J.Phys.Chem.Solids66(2005)1779–1782.[31]R.Chowdhury,S.Adhikari,P.Rees,Physica B405(2010)4763–4767.Table2Average transmittance of BZO with varying intrinsic defects.UV region(%)Visible region(%)ZnO65.689.2B Zn91.175.6B Zn V Zn68.186.8B Zn V O52.053.2B Zn Zn i59.256.9Y.-C.Peng et al./Optical Materials39(2015)34–3939。

TiB2电子结构、光学和晶格动力学性质的第一性原理计算

Applied Physics 应用物理, 2019, 9(5), 259-268Published Online May 2019 in Hans. /journal/apphttps:///10.12677/app.2019.95031First Principles Calculation of the Electronic Structure, Optical and Lattice KineticProperties of TiB2Liang Yu, Yajian Zhang, Xiaoan Kang, Yanli Jiang*Key Laboratory of New Processing Technology for Nonferrous Metals & Materials, Ministry of Education,School of Materials Science and Engineering, Guilin University of Technology, Guilin GuangxiReceived: May 8th, 2019; accepted: May 21st, 2019; published: May 28th, 2019AbstractThe electronic structure, optical and lattice dynamics properties of TiB2 are calculated by using the first principles of density functional theory. The calculation results show that the TiB2 has me-tallic characteristics, and the conduction band is mainly composed of the 2p orbital antibonding state of B atom and the 3d electron of Ti atom. The σ bonds of B atoms are formed from 2s and 2p hybridized orbitals, and the p z orbitals form a single continuous π bond. The transitions between these states are reflected in the absorption spectrum. The electron cloud orbital distribution in-dicates that the 3d orbital of Ti atom transfer to the 2p orbital of B atom in TiB2, is similar to the charge transfer of MgB2. Lattice dynamics calculations show that the splitting between LA (LO) and TA (TO) in TiB2 is significant, and the higher part of the frequency comes from the vibration of the B atom in the hexagonal plane. All of the nine eigen values at Γ point are summarized and labeled according to their frequency. The prediction of TiB2 superconducting temperature should be 0 K.KeywordsTiB2, Electrical Structure, l Energy Band, Optical Property, Phonon DispersionTiB2电子结构、光学和晶格动力学性质的第一性原理计算喻亮，张亚坚，康晓安，姜艳丽**通讯作者。

会计学原理约翰·J·怀尔德版上海交通大学

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计算凝聚态物理研究

where
VH [n(r)]
n(r)n(r') d 3rd 3r' | r r'|
History of Density Functional Theory
Variational principle: E = min{ E[n(r)] }
We have single electron equation (Kohn-Sham):
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Doping of graphene hydrogen fuel cell Pt catalyst carbon alloy to replace Pt which kind of structure is the key how to realize it
About the VASP code
1. Written by Georg Kresse, Martijn Marsman, and Jurgen Furthmuller Computational Physics, Faculty of Physics Universitat Wien Sensengasse 8, A-1130 Wien, Austria
1 2
n(r )n(r ' )drdr ' | r r'|
Approximation for Exc[n(r)]
The local density approximation (LDA)
ExLcDA[n(r)] exc[n(r)]n(r)dr
For a uniform electron gas:
exc (n)
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Simple but the results are surprisingly useful

Electronic structure of LaFe1-xCoxAsO from first principle calculations

Electronic structure of LaFe1-x Co x AsO from first principle calculationsHaiming Li1, Jiong Li1, Shuo Zhang1,2, Wangsheng Chu1, Dongliang Chen1 and Ziyu Wu1,2,3,*1Beijing Synchrotron Radiation Facility, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, P. R. China2National Synchrotron Radiation Lab, University of Science and Technology of China, Hefei 230026, P. R. China3Theoretical Physics Center for Science Facilities (TPCSF), Chinese Academy of Sciences, Beijing 100049, P. R. ChinaBased on the first-principles calculations, we have investigated the geometry, binding properties, density of states and band structures of the novel superconductor LaFe1-x Co x AsO and its parent compounds with the ZrCuSiAs structure. We demonstrate that La–O bond and TM-As (TM=Fe or Co) bond are both strongly covalent, while the LaO and TMAs layers have an almost ionic interaction through the Bader charge analysis. Partial substitution of iron with cobalt modify the Fermi level from a steep edge to a flat slope, which explains why in this system Co doping suppresses the spin density wave (SDW) transition.PACS numbers: 74.25.Jb, 74.70.-b, 71.20.-b* Electronic address: wuzy@IntroductionFollowing the discovery of high-T c copper oxides 1 and MgB 22, the appearance of superconductivity in quaternary pnictide-oxides triggered new researches and stimulated the entire scientific community. Of this new class of compounds, iron- and nickel-based layered compounds LaFePO and LaNiPO were the first systems that showed superconductivity although with a low transition temperature T 34c : 5 K and 3 K, respectively. In the last months, the superconductivity transition temperature raised first 26 K in the LaFeAsO by a partial substitution of O with F atoms (LaFeAsO 1-x F x ), and 43 K at high pressure . The progress encouraged researchers to look for superconductivity in other FeAs-based materials and other families of FeAs quaternary pnictide-oxides with Ce, Pr, Nd, Sm, Gd replacing La were identified with superconducting temperature higher than 50 K . The quaternary pnictide-oxides LaFeAsO belong to a tetragonal family with the ZrCuSiAs type structure and the space group of P4/nmm . They consist of alternative stacking of FeAs tetrahedral layers and LaO tetrahedral layers along the c-axis. These two layers are supposed to be positively and negatively charged, respectively so that the LaO layer mainly acts as a donor and superconducting pairing is supposed to occur in the FeAs layers. The parent materials without doping undergo a weakly first order structural phase transition from tetragonal (P4/nmm) to orthorhombic (Cmma), and then followed by antiferromagnetic spin density wave (SDW) transition at about 140 K . More recently, it has been reported that other free oxygen FeAs-based compounds: Ba 567-1213141-x K x Fe 2As 215-17, Sr 1-x K x Fe 2As 218, Eu 1-x K x Fe 2As 219, Ca 1-x Na x Fe 2As 220, Li 0.6FeAs may exhibit a similar superconducting behavior as the LaFeAsO 211-x F x , with T c of 38 K, 38 K, 32 K, 20 K and 18 K, respectively. The similar SDW instablity is also observed in these systems. The maximum T c achieved up today is 56 K by with the partial substitution of Gd by Th in the GdFeAsO . At the moment it is not possible to predict if T 22c in FeAs based compounds will continue to grow as rapidly as happened in cuprate superconductors, nevertheless this unexpected discovery will be certainly useful to better understand the physics of high-T c superconductor materials. The suppression of SDW transition is actually believed to be an important prerequisite for the appearance of superconductivity in these doped FeAs-based materials .8, 14, 23, 24In general, superconductors are doped in the donor layers by nonmagnetic atom, as in the case of La 2-x Ba x CuO 41 and Na x CoO 2·yH 2O 25, etc.. Indeed, it was well established that doping by magnetic atoms in a conducting layer generally destroys Cooper pairs and induces large distortion in the layer. However, a new type of FeAs based superconductor: the LaFe 1-x Co x AsO with T c ~ 10 K is just theresult of a real doping by a magnetic atom (cobalt) in the superconducting-active FeAs layer, and the doping system suppress the SDW transition26, 27. In addition, Co-doped SrFe2As228 and BaFe2As229 show also superconductivity at T c ~ 20 K and 22 K, respectively. The evidence of Co-doping inducing superconductivity has challenged our knowledge between superconductivity and magnetic interactions. Additional accurate investigations are then mandatory.Theoretical investigations in the frames of the density functional theory successfully reproduced the slope of the density of states (DOS) near the Fermi level, and gave many reasonable explanations on the interaction of FeAs-based superconductors24, 30-35. In this study, we focus through first principle calculations, to both geometry and electronic structure near the Fermi level of the new iron-pnictide system LaFe1-x Co x AsO, and explain why in this system Co doping suppresses the spin density wave (SDW) transition.Calculating methodThe present calculations have been performed using the first-principles plane-wave Vienna ab initio simulation package (V ASP)36, 37 while the exchange-correlation is described by the Perdew-Burke-Ernzerhof general gradient approximations (GGA)38. The projector augmented wave (PAW) method in its implementation of Kresse and Joubert was used to describe the electron-ion interaction39, 40.The La (5s25p65d16s2), Fe (3d64s2), Co (3d74s2), As (3s23p3), O (2s22p4) are treated as valence states. To ensure an enough convergence, the energy cutoff was chosen to be 600 eV, while the Brillouin zone was sampled with a mesh of 16×16×8 k points generated by the Monkhorst–Pack41 scheme for the pure LaFeAsO and LaCoAsO. Doping was modeled with a supercell of the parent material with Co atoms, with well converged grid k points in calculations. A first-order Methfessel–Paxton method with σ=0.2 eV has been considered for the relaxation42. The crystal cell and the internal parameters were optimized using the conjugate gradient method until the total forces on each ion was less than 0.02 eV/Ǻ. The density of states (DOS) calculations were performed using the tetrahedron method with the Blöchl corrections43.Results and discussionI Pure LaFeAsO and LaCoAsO systemsA. Crystal structureLaFeAsO and LaCoAsO crystallize in a tetragonal structure with the space group P4/nmm. Corresponding Wyckoff positions of the space group for different atoms are La(2c) (0.25, 0.25, z La), Fe or Co(2b) (0.75, 0.25, 0.5), As(2c) (0.25, 0.25, z As), and O(2a) (0.75, 0.25, 0), where z La and z As are both internal coordinates. Partial replacement of Fe with Co atoms in the LaFeAsO structure generates the LaFe1-x Co x AsO whose crystal structure is displayed in Fig. 1.Table 1 reports the calculated lattice constants and the internal coordinates of both LaFeAsO and LaCoAsO together with available experimental data. The experimental determinations are consistent and in good agreement with our calculations. The existing differences among experimental data and calculations may be addressed to the poor description of the exchange-correlation interaction by both LDA and GGA approaches in density functional theory. Both LaFeAsO and LaCoAsO systems are layered structure with alternating stack of LaO and FeAs (CoAs) layers. FeAs (CoAs) layers are conducting layers formed by a square lattice sheet of Fe (Co) ions coordinated by As above and below the plane to form face sharing FeAs4 (CoAs4) tetrahedra24. The Fe (Co) atoms coordinate tetrahedrally with four As atoms with a bond length of 2.34 Ǻ, forming a distorted tetrahedra with two different As-Fe-As (As-Co-As) angles of 118.86˚ and 104.99˚ (119.58˚ and 104.67˚), in agreement with results of Refs. 5, 14, 44. Every Fe (Co) has also four neighboring Fe (Co) atoms within the same layer with a bond length of about 2.85 Ǻ (2.86 Ǻ).B. Electronic structureThe LaTMPnO (Pn=P, As) systems crystallize in quasi two-dimensional structure and the LaO and TMPn layers interact through an ionic interaction. For a deeper understanding of the framework of LaFeAsO and LaCoAsO systems, we performed also a Bader analysis of the charge density45, 46. Table 2 compares the charges of each atom using the Bader analysis with the pure ionic picture. It addresses that the large charge transfer of both La-O bonds and TM-As bonds, which is quite different from the ionic description, are strong chemical bonding. Moreover, the charge transfer between LaO and TMAs layers is considerably smaller, which implies an ionic bonding between layers. Similar conclusions have also addressed in both LaFePO47 and LaNiPO31.The density of states of LaFeAsO and LaCoAsO compound was calculated. Fig. 2 shows that the DOS lineshape of these compounds is quite similar and in the case of LaFeAsO our DOS is in goodagreement with previous reports24, 30, 48.It is well recognized that electrons near the Fermi surface contribute to the superconductivity. From the analysis of Fig. 2 around the Fermi level in the range -3 eV to 2 eV we found Fe 3d states while As 4p and O 2p states appear at lower energies, from -2 eV down to -5 eV. Actually As 4p states may hybridize with Fe 3d near the Fermi surface while the O 2p contribution can be neglected. In the Co doped system, Co2+ (3d7) ions contribute with one more electron respect to Fe2+ (3d6) and the Fermi level is pushed up of about 0.7 eV. We can see that the DOS of LaFeAsO near the Fermi energy is monotonous with the energy, while in the LaCoAsO the Fermi level locates near the t2 peak, and the Fermi level both locate at the steep edge. We may address that the changes near the Fermi level are correlated to the spin density wave transition in the LaFeAsO near 150 K5, 14, 49 and in the LaCoAsO at 66 K44 which make these parent materials out from superconductivity.Both Fe and Co 3d states split by the exchange interaction and the crystal field and, in the ideal TMAs tetragonal structure, the crystal field due to the four nearest neighbor As atoms splits the fivefold degenerate d states of a free TM atom into a doubly degenerate e band (d z2 and d x2-y2) and a triply degenerate t2 band (d xy, d xz and d yz). Similarly, in TM-doped GaN diluted magnetic semiconductor systems, the large interaction among TM t2 states and N 2p states pushes up the t2 band above the e band50-52. However, from the orbital resolved DOS of Fe and Co 3d states in the parent compounds shown in Fig. 3, all the fivefold degenerate d states contributes around the Fermi energy indicating that the hybridization and the crystal field are relatively small in the distorted tetrahedral TMAs system. Moreover, the complex distributions of TM 3d states are influenced by the TM-TM direct interactions with bond length of 2.85 Ǻ in the same layers. The complicate distribution of Fe or Co 3d states is different from CuO2 planes. In a comparison with the copper-oxide superconductor systems, a Cu2+ occupies a planar fourfold square site and the DOS at the Fermi level are mainly correlated with the d x2-y2 orbital, suggesting a different superconductive mechanism between cuprate and new FeAs based superconductors.The band structures of LaFeAsO and LaCoAsO are showed in Fig 4. The band structure of LaFeAsO is in good agreement with previous data24, 48, and the lack of dispersion along both Г-Z and A-M directions suggests a quasi 2-D structure, in consistent with the Bader charge analysis. The band shape of these compounds is quite similar, except at the Fermi energy where it is pushed up about 0.7 eV in the LaCoAsO, in agreement with the DOS data. To clarify the role of electrons at the Fermi surface wecompare in Fig 5 the band structures of LaFeAsO and LaCoAsO in the range -0.6 eV to 0.6 eV. The dispersion near Fermi level between these two parent compounds is greatly different as a result of the Fermi level shift, which may result in different superconductivity mechanics.II LaCo x Fe1-x AsO systemIn the experiment reported by Wang et al.27, Co replaces Fe in a single crystalline phase as shown by XRD. According to our lattice and internal parameter relaxation steps, we claim that the doped system is energetically stable for the partial substitution of cobalt for iron. We find that the two kinds of As-Co-As angels change to 107.41˚ and 113.68˚, trend to less distorted tetrahedral than the parent compounds. The Bader analysis data are very close to the pure phase.As discussed for the parent materials, the Fermi level will push up about 0.7 eV for the replacement of Co for Fe in LaFeAsO. Thus, it is expected that Fermi level would locate at the range in the doping system. In Fig. 6 we compare the total DOS of both LaCo0.125Fe0.875AsO and LaCo0.25Fe0.75AsO and those of the pure LaFeAsO and LaCoAsO compounds. Like for other calculations on Fe-based superconductors such as LaFeAsO24, 30, 48 and LaFePO47, the Fermi energy lies just above a peak in the DOS. Actually, their DOS have a very steep and negative slope near the Fermi level, which drives the system close to a magnetic instability and at the moment no Fe-based parent materials exhibit superconductivity. At higher Co concentration the occupation at Fermi level decreases, i.e., 4.85 for the pure LaFeAsO, 3.25 for LaCo0.125Fe0.875AsO and 1.5 for LaCo0.25Fe0.75AsO, and the DOS near the Fermi level become flat pushing the doped system away from the magnetic instability. Many experiments in other superconductor systems revealed that the appearance of spin density wave transition may destroy superconductivity in Fe-based compounds; however nothing is known for these systems and it is fundamental to explore the relationship between SDW and superconductivity in these Fe-based compounds.ConclusionsWe have investigated the geometry structure, binding properties and electronic structure of LaFe1-x Co x AsO and its parent materials through first principle calculations. We find that FeAs-based ZrCuSiAs structure compounds LaFeAsO, LaCoAsO and the cobalt doping systems have quasi two-dimensional character, with ionic layer-layer interaction. The Fe or Co 3d states mainly contributenear the Fermi level, and the orbit resolved components of 3d states distribute complicate, which reveals an underlying mechanics different from cuprate superconductors. We further compare the DOS near the Fermi energy of LaFeAsO, LaCoAsO and the cobalt doping compounds, and demonstrate that cobalt doping push up the Fermi level of the LaFeAsO from a steep and negative edge towards the flat distribution of doping system, which suppresses the spin density wave transition in LaFe1-x Co x AsO doping systems. It would be interesting to explore the relationship between spin fluctuations and superconductivity for future work.AcknowledgementsThis work is partially supported by the National Outstanding Youth Fund (Project No. 10125523 to Z.W.), by the Knowledge Innovation Program of the Chinese Academy of Sciences (KJCX2-SW-N11), and by Supercomputing Center, CNIC, CAS. We are grateful for discussions with A. Marcelli.Table 1. Comparison of experimental and calculated crystal lattice and internal parameters of LaFeAsO and LaCoAsO compounds.(Ǻ) ( Ǻ) a c La z As z LaFeAsO Calculated 4.026 8.611 0.1451 0.6381 Experiment 5 4.035 8.741 0.1415 0.6512 Experiment 13 4.038 8.754 - - Experiment 14 4.030 8.737 0.1418 0.6507 LaCoAsO Calculated 4.045 8.518 0.1454 0.6382 Experiment 13 4.054 8.472 - -Table 2. Comparison of the electronic charges to different species obtained by the Bader analysis and their pure ionic picture.La TM As O LaO TMAs Bader (LaFeAsO) 9.1009 7.7103 5.8928 7.2960 16.3969 13.6031 Ionic picture (LaFeAsO) 8(La3+) 6(Fe2+) 8(As3-) 8(O2-) 16 14 Bader (LaCoAsO) 9.0957 8.8937 5.7158 7.2947 16.3904 14.6095 Ionic picture (LaCoAsO) 8(La3+) 7(Co2+) 8(As3-) 8(O2-) 16 15Figure 1. Cystal structure of the LaFe1-x Co x AsO. Elements are labeled inside the spheres.Figure 2. Total density of states of LaFeAsO (top) and LaCoAsO (bottom) and their partial density of states within the GGA approximation. The character of Fe (Co), O and As is shown separately. All energies are relative to the Fermi energy.-6-4-2021020D O S (s t a t e s /e V )Energy (eV)LaFeAsO Fe 3d As 4p O 2p-6-4-2021020D O S (s t a t e s /e V )Energy (eV)LaCoAsO Co 3d As 4p O 2pFigure 3. Orbital resolved DOS of Fe 3d states in the LaFeAsO (top) and of Co 3d states in LaCoAsO (bottom).-6-4-20251015D O S (s t a t e s /e V )Energy (eV)51015D O S (s t a t e s /e V )Energy (eV)Figure 4. Calculated band structures of LaFeAsO (top) and LaCoAsO (bottom). All energies are relative to the Fermi energy.-6-4-22ΓE n e r g y (e V )ΓX MZ R A-6-4-22E n e r g y (e V )ΓX M ΓZ R A MFigure 5. Band structures of LaFeAsO (top) and LaCoAsO (bottom) near the Fermi energy from -0.6 eV to +0.6 eV . All energies are relative to the Fermi level.-0.6-0.4-0.20.00.20.40.6ΓE n e r g y (e V )ΓX MZ R AM-0.6-0.4-0.20.00.20.40.6E n e r g y (e V )ΓX M ΓZ R A MFigure 6. Comparison of the total density of states of LaCo x Fe 1-x AsO, LaFeAsO and LaCoAsO. All energies are relative to the Fermi level.-2-10125101520D O S (s t a t e s /e V )Energy (eV)LaFe 0.875Co 0.125AsO LaFe 0.75Co 0.25AsOpure LaFeAsO pure LaCoAsOReferences1 J. G. Bednorz and K. A. 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First-principles calculation of electronic structures and optical

Author's personal copy
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Q.Y. Chen et al. / Physica B 403 (2008) 1666–1672
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density functional theory (DFT) in CASTEP code. The calculated electron structures are found to be in good agreement with the experimental results. On the other hand, the interesting optical properties are calculated and discussed.
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Author's personal copy
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First-principles calculation of electronic structures and optical properties of wurtzite InxAl1ÀxN alloys
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution

第一性原理整理总结英文版适合组会

theorem Kohn-Sham equation Local -Density Approximation(LDA)

Pseudopotentials
2
※What
are first principles calculations?
A material is simply a collection of atoms that are bound by chemical reactions. Chemical reactions, in turn, are simply interactions between electrons.
11
2.Kohn-Sham equation
Within the framework of KohnSham DFT (KS DFT), the intractable manybody problem of interacting electrons in a static external potential is reduced to a tractable problem of non-interacting electrons moving in an effective potential .
10
E V drv r r
The original H–K theorems held only for non-degenerate ground states in the absence of a magnetic field, although they have since been generalized to encompass these.
4
※How do we do first principles calculations? Solving the Schrö dinger equation by brute mathematical force is extremely demanding computationally and not practical for all but tiny systems.

First-principles prediction of the glass-forming ability in Zr–Ni binary metallic glasses

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 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RuC高压相变的第一性原理计算外文翻译及原文

本科毕业设计（论文）外文翻译译文学生姓名：院（系）：材料科学与工程专业班级：材料1101指导教师：完成日期：2015年3月1日要求1、外文翻译是毕业设计（论文）的主要内容之一，必须学生独立完成。

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3.外文翻译译文用A4纸打印。

文章标题用3号宋体，章节标题用4号宋体，正文用小4号宋体，20磅行距；页边距上、下、左、右均为2.5cm，左侧装订，装订线0.5cm。

按中文翻译在上，外文原文在下的顺序装订。

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RuC高压相变的第一性原理计算First-principle calculations of high-pressure phasetransformations in RuC作者：Jian Hao, Xiao Tang, Wenjing Li, Yinwei Li起止页码：46004-p1~p5出版日期（期刊号）：EPL, 105 (2014) 46004,2014年2月27日出版单位：IOP, EPL (Europhysics Letters)摘要- 使用第一原理计算在高压下RuC的结构稳定性。

结果表明，在9.3GPa的压力下，RuC从ZB型（闪锌矿型）结构转变为空间群为I4mm的四面体结构。

通过RuC5金字塔构造的I4mm结构的稳定性达26GPa，在更高压力下，则更有利成为WC型结构。

观察到伴随ZB型→ I4mm → WC型的相序，配位数增加从4至5，然后至6。

能带结构的计算表明，ZB型相是半导体，而I4mm和WC型相是金属。

此外，对所有三个阶段的RuC的机械特性进行了讨论。

简介-经压缩，由于原子间相互作用的变化和电子密度的再分配，化合物通常经历若干次相变。

结构的变化也因此可以引起物理性质的剧烈变化[1]。

热力学性质第一原理计算

INPHON ATOM_NAME =Al LSUPER=.TRUE. NDIM= 4 4 4 POSCAR SYSTEM Al 4.047 0.5 0.5 0.0 0.0 0.5 0.5 0.5 0.0 0.5 1 Direct 0.0 0.0 0.0
The second step: Use super-cell to calculate force by VASP code
The all freedom degree of vibrations are independently and have the same frequency.
Debye model
The key word of Debye model: Continuum Sound velocity for an isotropic crystal as
ON
ATOM_NAME =Al #LSUPER=.TRUE. NDIM= 4 4 4 #ND=5 #NPOINTS=61 #QI= 0.000 0.000 0.000 0.500 0.000 0.500 0.500 0.250 0.750 0.500 0.500 1.000 0.000 0.000 0.000 #QF= 0.500 0.000 0.500 0.500 0.250 0.750 0.500 0.500 1.000 0.000 0.000 0.000 0.500 0.500 0.500 MP=51 51 51
The forth step: Calculate the phonon and related properties 1. Phonon
Modified the file INPHON
ATOM_NAME =Al #LSUPER=.TRUE. NDIM= 4 4 4 ND=5 NPOINTS=61 QI= 0.000 0.000 0.000 0.500 0.000 0.500 0.500 0.250 0.750 0.500 0.500 1.000 0.000 0.000 0.000 QF= 0.500 0.000 0.500 0.500 0.250 0.750 0.500 0.500 1.000 0.000 0.000 0.000 0.500 0.500 0.500

First-principles Calculations of the Electronic Structure and Spectra of Strongly Correlate

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 optical absorption properties of CaCO3______ Vaterite

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 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First Principles Calculations of Fe on GaAs (100)

a r X i v :c o n d -m a t /0301563v 1 [c o n d -m a t .m t r l -s c i ] 29 J a n 2003First Principles Calculations of Fe on GaAs (100).S.Mirbt,B.Sanyal,C.Isheden,and B.JohanssonDepartment of Physics,Uppsala University,Uppsala,Sweden(Dated:February 2,2008)We have calculated from ﬁrst principles the electronic structure of 0.5monolayer upto 5monolayer thick Fe layers on top of a GaAs (100)surface.We ﬁnd the Fe magnetic moment to be determined by the Fe-As distance.As segregates to the top of the Fe ﬁlm,whereas Ga most likely is found within the Fe ﬁlm.Moreover,we ﬁnd an asymmetric in-plane contraction of our unit-cell along with an expansion perpendicular to the surface.We predict the number of Fe 3d -holes to increase with increasing Fe thickness on p -doped GaAs.PACS numbers:75.70.-i,82.65.+r,75.50.Bb,72.25.MkI.INTRODUCTIONIn the context of magnetoelectronics it is important to understand the structural and electronic properties of the interface between Fe and GaAs [1,2].Several experimen-tal investigations have been performed on this interface [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17].Under As-rich conditions the Fe ﬁlm growing on top of GaAs is found to have a reduced magnetic moment,or even a zero magnetic moment close to the interface [3,4,5].Moreover,structural investigations of the Fe ﬁlm lead to the conclusion that between a bcc Fe ﬁlm and the GaAs substrate there exists an intermediate phase,F e x Ga y As z [6,7].Under Ga-rich conditions no traces of an interme-diate phase are found and the magnetic moment is found to be bulk-like even close to the interface [8,9].Other theoretical investigations of the Fe/GaAs(100)interface [18,19,20,21]have been performed.Except of the calculations by Erwin et al [20],all other calcula-tions considered only an ideal Fe/GaAs interface.Erwin focused on Fe adatom growth and magnetic properties of the interface and allowed for ionic relaxations only per-pendicular to the surface (i.e along the z-direction).In general their results agree with our calculations.In this paper we present an investigation of the (completely)re-laxed Fe/GaAs interface structure.We study the Fe mag-netic moment as a function of the Fe ﬁlm thickness and the relaxed structure of the Fe ﬁlm.In addition we study As and Ga segregation.We discuss the origin of the so called magnetically dead layers on top of GaAs and why Fe growth diﬀers on As terminated and Ga terminated interfaces.II.COMPUTATIONAL DETAILSWe have performed self-consistent ﬁrst-principles den-sity functional calculations employing a plane wave pseu-dopotential code (VASP )[22].PAW pseudopotentials [23]with an energy cutoﬀof 24.61Ryd were used.Ex-change correlation was treated within the generalized gradient approximation (GGA)[24].We used a (4x4x4)folding of special k-points [25].We used a unit cell hav-ing 6semiconducting layers (3Ga and 3As)as the sub-strate and 1to 10Fe atoms (0.5monolayer to 5mono-layers)on top of the GaAs substrate.A vacuum of 10˚A thickness or more was kept to avoid interactions between neighboring unit cells in the (001)direction.To simulate a bulk semiconductor,the lowermost Ga/As layer was passivated with pseudohydrogen atoms [26].Volume and shape relaxations were allowed along with the relaxation of the atoms.All atoms were relaxed except the 2bulk semiconducting atoms.It is worth mentioning that the bcc Fe is perfectly lattice matched to the GaAs substrate (zinc-blende lattice).The GaAs lattice constant (5.65˚A )is almost twice of the lattice constant of bcc Fe (2.87˚A ).Thus the GaAs unit cell is equivalent to two bcc unit cells.We used a lattice constant of 5.735˚A which cor-responds to the calculated GaAs bulk equilibrium lattice constant within GGA.The lattice constant of bulk Fe is calculated to be only 0.05%smaller.The energies were converged with an accuracy of 10−4eV.The Pulay stress we ﬁnd to be negligible,being of the order of 0.1kB.Local magnetic moments were obtained by projecting wave functions onto spherical harmonics within spheres centered on the atoms [27].We used 1.302˚A 1.402˚A and 1.355˚A for Fe,Ga,and As spheres,respectively.In order to compare energies of diﬀerent interface geometries with a non-equivalent number of atoms,we used the following deﬁnition of the formation energy:E form =E total −iN i µi .(1)N i is the number of non equivalent atoms of species i ,andµi is the chemical potential.We estimated the chemical potential by calculating the total energy of species i in its bulk form.III.GAAS SURF ACEThe GaAs (100)surface structure depends on the growth environment.Under As rich conditions the GaAs (100)surface shows a c(4x4)[28]reconstruction,whereas under Ga rich conditions the GaAs (100)surface shows an ǫ(4x2)[29]reconstruction.All these reconstructions diﬀer from an ideally terminated surface -where either22 2.22.4 2.6 2.8D [Å]−13−12−11−10−9−8C o h e s i v e e n e r g y [e V ]FeGaFeAsFIG.1:Cohesive energy as a function of the Fe-X distance (X=As,Ga)for a bulk zincblende FeX structure.The up-per curve (circles)corresponds to zincblende FeGa,the lower curve (squares)to zincblende FeAs.only Ga or only As exists at the surface-by the existence of Ga-Ga and (or)As-As dimers.In a ﬁrst attempt to understand the interface between Fe and GaAs we neglect any surface reconstructions and assume an ideal Ga(As)terminated surface.In order to justify this approximation,let us assume we would grow Fe on top of some reconstructed GaAs surface.The interaction of Fe with the dangling bonds of As or Ga is already included in our calculation assuming an ideal cut.The question is now what happens with the As-As (Ga-Ga)dimers,when Fe ispresent at the surface?In a recent paper Erwin et al.[20]calculate that Ga and As surface dimers become unstable under Fe adsorption and Fe-Ga resp.Fe-As bonds form instead.This implies that our calculation assuming an ideal cut surface covers most of the physics at the Fe-GaAs interface.Moreover,experimental results by Kneedler et al.[16]warrant our approximation to ignore the surface reconstruction de-tails.They investigated the inﬂuence of the GaAs sur-face reconstructions on the properties of the growing Fe ﬁlm.They compared two As-rich terminations,the 2x4and the c(4x4).In summary they ﬁnd,although the Fe island morphology is diﬀerent,the growth mode,inter-facial structure,magnetic behaviour,and the uniaxial anisotropy to be independent on the chosen As-rich sur-face reconstruction.The diﬀerence in Fe growth between a Ga-terminated and an As-terminated surface can be explained largely by kinetic arguments.In Fig.1we show the cohesive energy as a function of the Fe-Ga (Fe-As)distance for bulk FeGa (FeAs)in a zincblende structure.(These artiﬁcial struc-tures only serve as a tool to understand the FeAs and FeGa interaction.)The cohesive energy of zincblende FeAs is about 4eV lower than of FeGa.Thus the in-teraction between Ga and Fe is much weaker than be-tween As and Fe,because As has three p -electrons and Ga only one p -electron,which gives rise to a larger pd -hybridization between As and Fe.According to Eq.2,energy is gained,when Fe replaces a Ga atom.There-Fe sFe i Fe s +Ga i<1−10>BULK GaAsV A C U U M<001>FIG.2:Geometrical structure of 0.5ML Fe on top of GaAs(100).Circles (diamonds)represent As (Ga)and tri-angles represent Fe.The four columns correspond to four dif-ferent interface conﬁgurations as discussed in the text.The atomic positions are shown along the <1¯10>direction,bonds are indicated by the lines connecting the atoms.fore,if it is kinetically possible,Fe replaces a Ga atom in the interface region independent of the surface termina-tion.In addition,energy is gained,-independently of the termination-,if As segregates to the surface (see section VI).The experimentally observable physical properties of the Fe on GaAs(100)system are thus governed by ex-trinsic eﬀects like growth conditions and in addition the surface termination governs the kinetic conditions for the Fe growth,i.e the height of involved barriers,diﬀusion probability,probability distribution of As atoms versus Ga atoms within the Fe ﬁlm.In this paper we discuss the physical properties of an Fe ﬁlm on an As-terminated surface,but our conclusions are valid independent of the termination,since we do not calculate any kinetic properties.IV.STRUCTURAL PROPERTIESIn GaAs bulk an Fe impurity is most likely found in a Ga-substitutional position,i.e Fe replaces a Ga atom.For 0.5ML Fe on top of a (100)GaAs surface,we compare here four diﬀerent Fe conﬁgurations (Fig.2).We show the four diﬀerent Fe conﬁgurations in the order of decreasing formation energy:F e ideal scorresponds to one Fe atom (i.e.0.5ML)sitting on top of GaAs at the position of a Ga atom,that is Fe sits substitutionally (s)following the ideal stacking (no lattice relaxations).F e s corresponds to one Fe atom having replaced the Ga atom closest to the surface,that is the Fe atom becomes buried under the surface.F e i corresponds to one Fe atom sitting in a buried position but at an interstitial (i)position of the ideal GaAs lattice.F e s +Ga i ﬁnally corresponds to one Fe atom sitting in a substitutionally buried position,but in addition a Ga atom is now sitting in an interstitial0500100015002000E f o r m [m e V ]0.511.522.53M F e [µB]FIG.3:Formation energy,E f orm ,of 0.5ML Fe conﬁgurations (see Fig.2)relative to the lowest conﬁguration,F e s +Ga i .Also indicated is the magnetic moment of the Fe atom (right scale and diamonds).position.In Fig.3we show the formation energy relative to the formation energy of the F e s +Ga i conﬁguration.Let us now discuss these results:The four conﬁgurations dif-fer by the number of vacant neighbours that each Fe atom has:In the F e ideal sconﬁguration,Fe has four va-cant neighbours,in the F e s ,Fe has three vacant neigh-bours,and in the other two conﬁgurations,Fe has only two vacant neighbours.The more neighbours the Fe atom has,the more energy is gained due to increased wavefunction overlap.Thus the F e ideal sconﬁguration is highest in energy,because the wavefunction overlap be-tween Fe and its surrounding is minimal.Next we con-sider the F e s conﬁguration,which is already 1eV lower than F e ideal s.Besides having one more neighbour than F e ideals ,in the F e s conﬁguration two Fe-As bonds have formed.For this conﬁguration the Fe magnetic moment is totally quenched.This will be further discussed in the next section.The F e i and F e s +Ga i conﬁgurations are lower than F e s because they only have two vacant neigh-bours.These two conﬁgurations are similar except that the topmost Ga and Fe positions have been interchanged.We ﬁnd from our calculationE Ga −As +EF e i >E F e −As +E Ga i ,(2)where E Ga −As (E F e −As )is the formation energy of aGa-As (Fe-As)surface bond and E F e i (E Ga i )is the for-mation energy of an interstitial surface defect F e i (Ga i ).We compared the charge density and performed an (unrelaxed)calculation without the respective interstitial atom.In both conﬁgurations,the top Ga atom forms to-gether with the Fe atom a metallic layer.The diﬀerence is the bonding:For F e i we have mainly two Ga-As bonds and one Ga-Fe metallic bond,whereas for F e s +Ga i we have two Fe-As bonds and again one Ga-Fe metallic bond,but in addition the interstitial Ga atom bonds to it’s As neighbours.This Ga i -As interaction reduces the Fe-As interaction,whereby the Fe magnetic moment is close to it’s bulk value.It is thus the Ga i that lowers the energy of the F e s +Ga i conﬁguration.In summary,aFe thickness [ML]−5005001000E (F e i)−E (F e s +G a i) [m e V ]FIG.4:Energy diﬀerence between an interface conﬁguration where Fe sits interstitial (F e i )and where Fe sits substitutional together with an interstitial Ga (F e s +Ga i )as a function of the Fe ﬁlm thickness.TABLE I:Calculated c/<a >ratio and in-plane lattice contraction of the unit cell along the <1¯10>and <110>direction for the lowest energy conﬁguration of the speciﬁed Fe ﬁlm.c/<a ><110>1.07-6.92%1ML Fe +0.5ML As -0.51%1.02-2.79%2ML Fe +0.5ML As -0.81%1.05-2.79%5ML Fe +0.5ML As -1.15%1.03-1.83%TABLE II:Total magnetic moment and atomic distances for the respective closest topmost atoms for1ML of Fe for the three interface conﬁgurations shown in Fig.6.Ga-As(h) (GaAs(v))indicates the horizontal(vertical)distance.Fe i Fe s+Ga istartFe-As 2.42˚A2.96˚A 2.51˚AGa-As(h) 2.87˚A- 3.61˚AGa-Ga 2.44˚A4.63µB0.01µB52 2.2 2.4 2.6 2.83D [Å]−1123L o c a l M F e [µB ]M BulkFIG.7:Local magnetic moment of Fe,M F e ,as a function of the Fe-As distance ,D .Solid (dashed)horizontal line indi-cates the bulk Fe magnetic moment (zero-line)and the dashed vertical line at 2.37˚A indicates the Fe-As distance at which the Fe magnetic moment becomes zero.2.12.3 2.5 2.7D [Å]−101234L o c a l M F e [µB ]M BULKFIG.8:Bulk magnetic moment per Fe atom for zincblende FeAs (circles),zincblende FeSe (diamonds),and zincblende FeTe (triangles)as a function of the Fe-anion distance,D .The dashed vertical line is the same as in Fig.7shown here for comparison.caused by pd -hybridization:The Fe spin-polarization is driven by the on-site exchange interaction.Therefore the Fe magnetic moment decreases with increasing delocal-ization of the Fe d -states.Because,the delocalization of the Fe d -states increases with increased overlap between the Fe d -states and As p -states,we ﬁnd that the Fe mag-netic moment is quenched for small Fe-As bond distances.The spread in magnetic moments for bond lengths above 2.36˚A is due to the other factors inﬂuencing the Fe mag-netic moment,like number of Fe neighbours and number of vacant neighbours.Our interpretation is supported by the behaviour of the Fe magnetic moment in the bulk FeAs zincblend com-pound (circles in Fig.8).We show the Fe magnetic mo-ment as a function of the Fe-anion distance (D )in the zincblend lattice.The same trend as before is observed,namely the Fe magnetic moment becomes smaller at acritical D .In contrast to Fe on GaAs,the Fe mag-netic moment is not quenched on ZnSe [31].In order to understand the diﬀerence between GaAs and ZnSe we also calculated zincblende FeSe and for comparison zincblende FeTe.In all three cases,i.e.FeAs,FeSe,and FeTe,the Fe d -electrons hybridize with the respective an-ion’s p -electrons.Therefore there exists a bond distance at which the Fe magnetic moment becomes quenched (Fig.8).The exact value of the critical D depends on the system in question.For a given lattice constant (set by for example the semiconductor host),the anion will occupy a certain fraction of the unit cell volume that de-pends on its atomic radius.GaAs and ZnSe have about the same lattice constant.The As atomic radius (1.14˚A )is larger than the Se radius (1.03˚A ).The pd -overlap between Fe and Se is therefore (for the same lattice con-stant)smaller than between Fe and As and the Fe mag-netic moment starts to become quenched for a smaller D .On the other hand the Te radius (1.23˚A )is larger and the magnetic moment becomes quenched already for a larger D .As clearly visible from Fig.8there exist high-spin,low spin,and zero moment phases as a function of D .The energy diﬀerence between these phases becomes rather small for a given D .This will be further investigated in the future.We conclude that the magnetic conﬁguration of Fe on GaAs will be determined by pd -hybridization,but that the speciﬁc magnetic phase of the Fe-As com-plex depends on small variations of the Fe-As bonding.For example,in earlier publications [19,20]an antifer-romagnetic (AFM)solution of the Fe ﬁlm has been dis-cussed and calculated.Especially for the 1ML thick Fe ﬁlm an AFM solution was found.We also investigated the possibility of an AFM solution.For a 1ML Fe ﬁlm covered with a complete ML of As we ﬁnd an AFM solu-tion for a F e s interface conﬁguration (0.22µB ,−0.22µB ).The involved magnetic moments are so small because of the Fe-As pd -hybridization as discussed above.The total energy is 3meV higher than a solution with a zero mag-netic moment on both Fe sites,i.e the AFM solution is almost degenerate with the zero moment per Fe atom so-lution.For the F e s +Ga i interface conﬁguration we again ﬁnd an AFM solution (0.05µB ,-0.05µB )being 10meV higher than a solution having a total magnetic moment of 0.5µB .Another example we ﬁnd from our calculations is a ferrimagnetic solution.The structure consists of a 5ML thick Fe ﬁlm on top of GaAs covered with 0.5ML As and an additional 0.5ML As within the Fe ﬁlm.The top Fe atom has a magnetic moment of −2.1µB ,whereas the two Fe atoms/cell in the next layer beneath have a mag-netic moment of 0.95µB and 1.2µB ,respectively.Below this layer the additional 0.5ML As is located.The rest of the Fe atoms have a magnetic moment close to or larger than the bulk value.The average magnetic moment for this ferrimagnetic solution is 1.55µB per Fe atom.Since the magnetic phase diagram of the Fe-As inter-action is rather complex,it very much depends on small details of the Fe ﬁlm conﬁguration which magnetic mo-6123l o c a l M F e [µB /a t o m ]As Fe GaFe Fe Fe Fe FeAs<1−10><001>FIG.9:Local Fe magnetic moment,M F e ,proﬁle for 6MLFe (10Fe atoms/cell)on top of GaAs covered with 0.5ML As and 0.5ML Ga within the Fe ﬁlm.The horizontal line indicates the calculated bulk Fe magnetic moment.The ﬁlled (open)circles correspond to the average (individual)Fe mag-netic moment of the respective layer.On the x-axis the atomic structure is sketched along the <001>direction projected on the <1¯10>direction.123L o c a l M F e [µB /a t o m ]FIG.10:Same as Fig.9,but for 5ML Fe (9Fe atoms/cell)covered with 1ML As.ment will be measured.But the physical mechanism be-hind the reduction of the Fe magnetic moment on top of GaAs is without any doubt the Fe-As pd -hybridization.As can be seen from Fig.5,the Fe magnetic moment is larger for F e i than for F e s +Ga i .In general the F e s +Ga i conﬁguration has a smaller Fe-As bond distance than F e i ,which explains the smaller magnetic moment for the F e s +Ga i conﬁguration.For 1.5and 2ML of F e ,the en-ergy gain due to increased Fe-As binding is smaller than the energy gain due to the increased spin-polarization of the Fe atoms.This explains why F e i is lower than F e s +Ga i for 1.5and 2ML of F e (see discussion above and Fig.4)In Figs.9and 10we show the structure and magnetic proﬁle of a 5ML Fe ﬁlm on top of GaAs(100)with one Ga atom within the Fe ﬁlm for two segregation proﬁles:In Fig.9for 0.5ML As (i.e one As surface atom per cell)on top of the Fe ﬁlm;in Fig.10for 1ML As (i.e two Asatoms per cell)on top of the Fe ﬁlm.The Ga atom does not inﬂuence the Fe magnetic moment,i.e the Ga-Fe in-teraction is very weak.On the other hand,the Fe-As interaction quenches the magnetic moment (as discussed before).For the 1ML As coverage,the Fe magnetic mo-ment of the top Fe layer is thus almost zero (Fig.10).If Fe is covered with only 0.5ML As,the Fe-Fe interaction is stronger than the Fe-As interaction and the Fe magnetic moment is not reduced (Fig.9).From our calculations we can conclude:E xcF e >E F e −As .(3)E F e −As >E xcF e +E F e −F e .(4)E xcF e +E F e −F e >E As −F e −As .(5)Here E F e −As is the Fe-As surface bond formation energy,E xc F e is the energy gain due to the spin-polarization of the Fe atom,E F e −F e is the Fe-Fe surface bond formation en-ergy,and E As −F e −As is the bond formation energy be-tween a Fe surface atom bonded to two As atoms.These three equations explain the calculated magnetic proper-ties.Equation 3:This equation states that the Fe magnetic moment will be quenched if Fe only has one or more As neighbours.For example,the Fe magnetic moment becomes zero for 0.5ML Fe with a F e s IC (Fig.3).Here the Fe atom has no other Fe around it but an As atom.Equation 4:This equation states that the Fe magnetic moment is not quenched if the Fe atom is close to one As atom and at least one Fe atom.This is the case for the F e i IC of 1ML Fe on top of GaAs (Fig.6and tab.2).Equation 5:The Fe magnetic moment becomes quenched again,if Fe is bonded to two As atoms (Fig.10,Fig.6),independent on the number of Fe neighbours.For completeness,we show in Fig.11the spin-polarization of the GaAs host.The Fe magnetic mo-ment is not shown here (see Fig.10).The induced spin-polarization is mostly antiparallel and rather small.In a recent X-ray absorption study [12],the number of Fe 3d -holes was determined as a function of the Fe ﬁlm thickness on n -doped GaAs.Freeland et al ﬁnd the num-ber of holes to increase with decreasing Fe ﬁlm thickness.They explain this due to charge transfer from Fe to As which they also believe to be the cause for the reduced Fe magnetic moment.Our calculation describes the charge transfer between Fe and p -doped GaAs,because due to numerical reasons the Fermi energy of bulk GaAs is ﬁxed at the top of the valence band edge.From our calculations we directly get the number of Fe 3d -holes as a function of the Fe ﬁlm thickness (Fig.12).Our absoult number of the Fe7−0.1−0.050.05L o c a l M [µB /a t o m ]AsAsGaAsGaAs Ga<001>F e −f i l mV A C U U MB U L KFIG.11:Local magnetic moment,M ,proﬁle of Ga and As for the geometrical structure of 5ML Fe on top of GaAs covered with 1ML As and 0.5ML Ga within the Fe ﬁlm.The Fe magnetic moment is not shown here.Fe thickness [ML]33.23.43.63.84N u m b e r o f F e 3d −h o l e sFIG.12:Calculated number of Fe 3d -holes as a function of the Fe ﬁlm thickness.The horizontal line indicates the number of Fe 3d -holes for Fe bulk.3d -holes depends of course on the chosen Wigner-Seitz radius for Fe,but the trend we ﬁnd to be independent on the chosen Wigner-Seitz radius.In contrast to exper-iment we ﬁnd an increase of the holes with increasing Fe ﬁlm thickness.Our explanation is the following:The Fe-As pd -hybridization determines the Fe magnetic moment,not a charge transfer between Fe and As.The electron transfer between Fe and GaAs is diﬀerent for n -type and p -type conditions,because for n -type (p -type)GaAs,the Fermi level of Fe has to align with the conduction (valence)band of GaAs.Under p -type conditions,a consequence of the hybridization is that on average there are slightly more delocalized electrons on the Fe site,i.e.the num-ber of 3d -holes decreases.The thicker the Fe ﬁlm be-comes,the more Fe atoms are without an As neighbour ,which is why the number of d -electrons (holes)decreases (increases)with increasing Fe thickness.Under n -type conditions it is plausible to assume vice versa that on av-erage there are slightly less delocalized electrons on the Fe site.We therefore predict that the experimentally ob-G a −S e g r g a t i o n e n e r g y [m e V ]FIG.13:Ga segregation energy for four diﬀerent Gaconﬁgu-rations as discussed in the text for 5ML Fe on top of GaAs.1000200030004000A s s e g r e g a t i o n e n e r g y [m e V ]FIG.14:As segregation energy for four diﬀerent As conﬁgu-rations as discussed in the text for 5ML Fe on top of GaAs.served charge transfer diﬀers between p -doped GaAs and n -doped GaAs and that similar experiments performed instead on p -doped GaAs should ﬁnd an increase of the 3d -holes with increasing thickness in agreement with our calculations.VI.SURF ACE SEGREGATIONIt is well known that at metal-semiconductor inter-faces the semiconductor constitutents segregate towards the surface.For example,for Fe on GaAs it is known that As segregates to the surface,whereas Ga is not found at the surface but within the metal ﬁlm [14,15,16].It is found that the segregation of As is independent of tem-perature,but the segregation of Ga is dependent on tem-perature.A quantitative understanding of segregation is obtained with a simple model put forward by Weaver et al.[32].The free energy of segregation is determined by the strain energy and surface energy,the latter of which they estimated with the cohesive energy.There model predicts for Fe on GaAs both Ga and As to segregate to the surface,whereas in experiment only As surface segregation is observed.In the following we resolve this discrepancy.In Figs.13and 14,we show the Ga (respective As)seg-8regation energy relative to the energetically lowest conﬁg-uration.For Ga we have to consider two cases separately: (i)no As has segregated to the surface and(ii)As has segregated to the surface.In case(i)(insert of Fig.13), Ga prefers to segregate to the surface(1ML Ga Surface). The segregation energy of1ML Ga amounts to2.4eV.In case(ii)where As already has segregated to the surface (Fig.13),Ga prefers to leave the interface(Ga Interface) and stay somewhere within the Feﬁlm(Ga inF e−film).It is more costly(280meV)for the Ga to be in the surface layer together with an As atom(Ga Surface).Therefore, if As has segregated to the surface,no Ga will be found at the surface.The As segregation energy(Fig.14)shows the same trend as the Ga segregation energy(inset of Fig.13),but the segregation energy of one As atom to the surface is 1.5eV larger than for one Ga atom.Notice,that both Ga and As prefer to be within the Feﬁlm rather than at the interface.In summary,this suggests the following:On top of the Feﬁlm always As will be found independent of the GaAs surface termination.The segregation proﬁles should more or less be independent on the GaAs sur-face termination,because the Fe-As interaction is much stronger than the Fe-Ga interaction.The As segregation path is a result of the lattice relax-ation due to the chemical interaction between the Fe and As,which implies that As segregates already at T=0 K.The segregation is thus not diﬀusion controlled,but only controlled by chemical bonding,which is in agree-ment with experiment.Weﬁnd Ga to leave the interface, but only to segregate to the surface,if no As already has segregated.In contrast to As,the segregation of Ga,does not take place at T=0K.There exists an activation bar-rier for the segregation,i.e.the segregation is diﬀusion controlled.The amount of Ga on top of or within the Feﬁlm is therefore strongly dependent on temperature, whereas the amount of As on top of the Feﬁlm is only weakly dependent on temperature.This is in agreement with an experimental study of the Fe/GaAs interface in-terdiﬀusion[6].Regarding the Weaver model,weﬁnd that Ga seg-regates to the surface(in agreement with the Weaver model),if no As has segregated.If As has segregated, Ga will stay within the Fe-ﬁlm(in agreement with ex-periment).VII.DISCUSSION AND SUMMARYWeﬁnd As to segregate to the surface on top of the Fe ﬁlm independent of termination,which is in agreement with experiments.The Fe/GaAs interface is not stable against further segregation.An As atom within the Fe ﬁlm has a lower energy than at the interface.In an ex-periment one will thus alwaysﬁnd As within the Feﬁlm, where the Fe-As pd-hybridization will quench the Fe mag-netic moment.Since Ga also has a lower energy within the Feﬁlm than at the interface,one will also alwaysﬁnd Ga within the Feﬁlm.Ga itself does not inﬂuence the magnetic moment(Fig.9),but probably Ga in the Feﬁlm will prevent further As segregation.This would explain the much lower thickness of the magnetically quenched Fe layers for Ga terminated samples.The thickness of the magnetically quenched Fe layers is then determined by the probability ofﬁnding As and Ga within the Feﬁlm. This probability depends on the termination and the Fe growth conditions.Our results presented here for Fe on GaAs are most likely also valid for other semiconductor substrates.For example,weﬁnd more or less the same structures for Fe on ZnSe[31]and the segregation be-haviour is the same,but the Fe-Se hybridization is much weaker.AcknowledgmentsWe are grateful to the Swedish foundation for strate-gic research(SFS),the Swedish Research Council(VR), the G¨o ran Gustafsson foundation,and the European RTN network on”computational spintronics”forﬁnan-cial support.We thank the Swedish supercomputer cen-ter(SNAC)for providing us with supercomputer time.[1]G.A.Prinz,Science282,1660(1998).[2]H.J.Zhu,M.Ramsteiner,H.Kostial,M.Wassermeier,H.-P.Sch¨o nherr,and K.H.Ploog,Phys.Rev.Lett.87,016601 (2001).[3]J.J.Krebs,B.T.Jonker,and G.A.Prinz,J.Appl.Phys.61,2596(1987).[4]A.Filipe,A.Schuhl,and P.Galtier,Appl.Phys.Lett.70,129(1997).[5]M.Gester,C.Daboo,R.J.Hickens,S.J.Gray,A.Ercole,and J.A.C.Bland,J.Appl.Phys.80,347(1996).[6]M.Rahmoune,J.P.Eymery,and M.F.Denanot,J.Magn.Magn.Mater.175,219(1997).[7]llaizon, B.Lepine,S.Ababou, A.Guivarch,S.Deputier,F.Abel,and C.Cohen,J.Appl.Phys.86,5515(1999).[8]F.Bensch,G.Garreau,R.Moosb¨u hler,E.Beaurepaire,and G.Bayreuther,J.Appl.Phys.89,7133(2001). [9]Y.B.Xu,M.Tselepi, C.M.Guertler, C.A.F.Vaz,G.Wastlbauer,and J.A.C.Bland,J.Appl.Phys.89,7156 (2001).[10]F.P.Zhang,P.S.Xu,E.D.Lu,H.Z.Guo,F.Q.Xu,andX.Y.Zhang,J.Appl.Phys.86,1621(1999).[11]R.A.Gordon,E.D.Crozier,D.-T.Jiang,T.L.Monch-esky,and B.Heinrich,Phys.Rev.B62,2151(2000). [12]J.W.Freeland,I.Coulthard,A.P.J.Stampﬂ,and W.J.Antel,Phys.Rev.B63,193301(2001).[13]G.Wedler, B.Wassermann,R.N¨o tzel,and R.Koch,Appl.Phys.Lett.78,1270(2001).。

First-principles study of the structural, vibrational, phonon and thermodynamic

1. Introduction Ultra-high temperature ceramics (UHTCs) with melting temperatures in excess of 3000 K are usually composed by the refractory borides, carbides and nitrides of early transition metals [1–7]. Among the UHTCs, transition metal carbides (TMC) such as TiC, ZrC and HfC are metallic compounds with unique physical and chemical properties including an extremely high melting point and hardness, chemical stability, corrosion resistance combined with metallic electrical and thermal conductivities [5–10]. These features give transition metal carbides the capability to withstand high temperatures in oxidizing environments, making them candidates for applications in the atmosphere of extreme thermal and chemical environments [6,7]. The structural, vibrational, phonon and thermodynamic properties of IVb group transition metal carbides have been investigated experimentally [10–17] and theoretically [13,18–28] in the earlier reports. In the 1970s, the phonon dispersion relations of TiC, ZrC and HfC were measured using inelastic neutron scattering by Pintschovius et al. [10] and Smith et al. [15–17]. Lattice dynamics calculation and the phonon dispersion relations of transition metal carbides such as ZrC and HfC were reported using a phenomenological ‘‘double-shell’’ model theory [18] where long-range interatomic interactions were taken into account in order to get a

A first principles investigation of the mechanical properties of g-ZnO

A ﬁrst principles investigation of the mechanical properties of g -ZnO:The graphene-like hexagonal zinc oxide monolayerQing Peng ⇑,Chao Liang,Wei Ji,Suvranu DeDepartment of Mechanical,Aerospace and Nuclear Engineering,Rensselaer Polytechnic Institute,Troy,NY 12180,USAa r t i c l e i n f o Article history:Received 5October 2012Accepted 16October 2012Available online 8December 2012Keywords:Mechanical propertiesHigh order elastic constants Density functional theory 2D materialsHoneycomb structure g -ZnOa b s t r a c tWe investigate the mechanical properties,including high order elastic constants,of the graphene-like hexagonal zinc oxide monolayer (g -ZnO)using ﬁrst-principles calculations based on density-functional pared to the graphene-like hexagonal boron nitride monolayer (g -BN),g -ZnO is much softer,with 17%in-plane stiffness and 36%,33%,and 33%ultimate strengths in armchair ,zigzag ,and biaxial strains respectively.However,g -ZnO has a larger Poisson’s ratio,0.667,about three times that of g -BN.It was found that the g -ZnO also sustains much smaller strains before the rupture.We obtained the sec-ond,third,fourth,and ﬁfth order elastic constants for a rigorous continuum description of the elastic response of g -ZnO.The second order elastic constants,including in-plane stiffness,are predicted to monotonically increase with pressure while the Poisson’s ratio monotonically decreases with increasing pressure.Ó2012Elsevier B.V.All rights reserved.1.IntroductionWith a wide bandgap (3.37eV),a large exciton biding energy (60meV),piezoelectricity,chemical stability,and biocompatibility,zinc oxide (ZnO)is a verstile functional material and is widely used in optics,photonics,and electronics.The applications includes light-emitting diodes,solar cells,ﬁeld effect transistors,mechani-cal actuators,piezoelectric sensors,surface acoustic wave devices,waveguides,photocatalysts,and nanogenerators [1–3].A hexago-nal zinc oxide monolayer (g -ZnO)is a graphene-like 2D material,which is only monoatomically thick (Fig.1)and chemically stable from both ab initio DFT calculations [4–7]and experiment [8].The electronic and magnetic properties were extensively studied [9–16],as well as the elasticity and piezoelectricity [5,7].The band-gap of g -ZnO is predicted to be 3.57–5.64eV [7,9],larger than that in bulk.The in-plane Young’s modulus was reported to be 24%of graphene [7].The Poisson ratio is 0.71.Two-dimentional (2D)nanomaterials are the basic ‘‘building blocks’’for all other structures:buckyballs (0D)by wrapping,nanotubes (1D)by rolling,bulk (3D)by stacking [17].However,due to the quantum conﬁnement resulting from the reduction of the third dimention,2D nanomaterials present very different prop-erties from bulk.For example,the mechanical strengths are an or-der of magnitude larger than those in bulk.Mechanical properties are critical in designing parts or structures with g -ZnO regarding the practical applications.Strain engineering is a common and important approach tailoring the functional and structural proper-ties of nanomaterials [18,9].One can expect that the properties of g -ZnO will be affected by applied strain too.In addition,g -ZnO is vulnerable to be strained with or without intent because of its monatomic thickness.For example,there are strains because of the mismatch of lattices constants or surface corrugation with sub-strates [19,20].Therefore,the knowledge of the mechanical prop-erties of g -ZnO is highly desired.Depending on the loading,the mechanical properties are di-vided into four strain domains:linear elastic,nonlinear elastic,plastic,and fracture.Materials in the ﬁrst two strain domains are reversible,i.e.,they can restore to equilibrium status after the re-lease of the loads.On the contrary,the last two domains are non-reversible.Defects are nucleated and accumulated with the increase of the strain,until rupture.As in graphene,the nonlinear mechanical properties are prominent since it remained elastic until the intrinsic strength was reached [21,22].Thus it is of great inter-est to examine the nonlinear elastic properties of g -ZnO,which is necessary to understand the strength and reliability of structures and devices made of g -ZnO.Several previous studies have shown that 2D monolayers pres-ent a large nonlinear elastic deformation during the tensile strain up to the ultimate strength of the material,followed by a strain softening until fracture [23–26,22].We expect that the g -ZnO be-haves in a similar manner.Under large deformation,the strain en-ergy density needs to be expanded as a function of strain in a Taylor series to include quadratic and higher order terms.The higher order terms account for both nonlinearity and strain0927-0256/$-see front matter Ó2012Elsevier B.V.All rights reserved./10.1016/matsci.2012.10.019Corresponding author.Tel.:+15182796669;fax:+15182766025.E-mail address:pengq3@ (Q.Peng).URL: (Q.Peng).softening of the elastic deformation.They can also express other anharmonic properties of 2D nanostructures including phenomena such as thermal expansion and phonon–phonon interaction [21].Due to a large bandgap,ZnO is an alternative material to the ni-tride semiconductors.Thus it is interesting to compare the mechanical properties of g -ZnO with g -BN,the graphene analogue of BN [27].The goal of this paper is to study the mechanical behav-iors of g -ZnO at large strains and ﬁnd an accurate continuum description of the elastic properties from ab initio density func-tional theory calculations.The total energies of the system,forces on each atom,and stresses on the simulation boxes are directly ob-tained from DFT calculations.The response of g -ZnO under the nonlinear deformation and fracture is studied,including ultimate strength and ultimate strain.The high order elastic constants are obtained by ﬁtting the stress–strain curves to analytical stress–strain relationships that belong to the continuum formulation [24].We compared g -ZnO with the well known 2D materials such as g -BN,graphene,and graphyne [28].Based on our result of the high order elastic constants,the pressure dependence properties,such as sound velocities and the second order elastic constants,including the in-plane stiffness,are predicted.Our results for the continuum formulation could also be useful in ﬁnite element mod-eling of the multiscale calculations for mechanical properties of g -ZnO at the continuum level.The remainder of the paper is organized as follows.Section 2presents the computational details of DFT calculations.The results and analysis are in Section 3,followed by conclusions in Section 4.2.Density functional theory calculationsWe consider a conventional unit cell containing six atoms (3Zinc atoms and 3oxygen atoms)with periodic boundary condi-The total energies of the system,forces on each atom,stresses,and stress–strain relationships of g -ZnO under the desired deforma-tion conﬁgurations are characterized via ﬁrst-principles calcula-tions with density-functional theory (DFT).DFT calculations were carried out with the Vienna ab-initio Simulation Package (VASP)[30–33]which is based on the Kohn–Sham Density Functional The-ory (KS-DFT)[34,35]with the generalized gradient approximations as parameterized by Perdew,Burke,and Ernzerhof (PBE)for ex-change–correlation functions [36].The electrons explicitly included in the calculations are the 3d 104s 2electrons for zinc atoms and 2s 22p 6for oxygen atoms.The core electrons are replaced by the pro-jector augmented wave (PAW)and pseudo-potential approach [37,38].A plane-wave cutoff of 500eV is used in all the calculations.The calculations are performed at zero temperature.The criterion to stop the relaxation of the electronic degrees of freedom is set by total energy change to be smaller than 0.000001eV.The optimized atomic geometry was achieved through minimizing Hellmann–Feynman forces acting on each atom until the maximum forces on the ions were smaller than 0.001eV/Å.The atomic structures of all the deformed and undeformed con-ﬁgurations were obtained by fully relaxing a 6-atom-unit cell where all atoms were placed in one plane.The simulation invokes periodic boundary conditions for the two in-plane directions while the displacement to out-of-plane direction is forbidden.The irreducible Brillouin Zone was sampled with a Gamma-centered 21Â21Â1k -mesh.Such a large k -mesh was used to re-duce the numerical errors caused by the strain of the systems.The initial charge densities were taken as a superposition of atomic charge densities.There was a 15Åthick vacuum region to reduce the inter-layer interaction to model the single layer system.To eliminate the artiﬁcial effect of the out-of-plane thickness of the simulation box on the stress,we used the second Piola–Kirchhoff stress [25]to express the 2D forces per length with units of N /m .For a general deformation state,the number of independent com-ponents of the second,third,fourth,and ﬁfth order elastic tensors are 21,56,126,and 252respectively.However,there are only fourteen independent elastic constants that need to be explicitly considered due to the symmetries of the atomic lattice point group D 6h which consists of a sixfold rotational axis and six mirror planes [22].The fourteen independent elastic constants of g -ZnO were determined by a least-squares ﬁt to the stress–strain results from DFT based ﬁrst-principles studies in two steps,detailed in our pre-vious work [25].In the ﬁrst step,we used a least-squares ﬁt of ﬁve stress–strain responses.Five relationships between stress and strain are necessary because there are ﬁve independent ﬁfth-order elastic constants (FFOECs).We obtain the stress–strain relation-ships by simulating the following deformation states:uni-axial strain in the zigzag direction (zigzag );uni-axial strain in the arm-chair direction (armchair );and equibiaxial strain (biaxial ).From the ﬁrst step,the components of the second-order elastic constants (SOECs),the third-order elastic constants (TOECs),and the fourth-order elastic constants (FOECs)are over -determined (i.e,the num-ber of linearly independent variables is greater than the number of constraints),and the ﬁfth-order elastic constants (FFOECs)are well -determined (the number of linearly independent variables is equal to the number of constraints).Under such circumstances,a second step is needed:a least-square solution to these over-and well-determined linear equations.3.Results and analysis 3.1.Atomic structureWe ﬁrst optimize the equilibrium lattice constant for g -ZnO.The total energy as a function of lattice spacing is obtained by spec-ifying nine lattice constants varying from 3.0Åto 3.6Å,with full2.Atomic structure of g -ZnO in the conventional unit cell (six atoms)in undeformed reference conﬁguration.Fig.1.Left:g -ZnO plane.Right:geomoetry of a conventional g -ZnO cell.Science 68(2013)320–324321relaxations of all the atoms.A least-squareﬁt of the energies ver-sus lattice constants with a fourth-order polynomial function yields the equilibrium lattice constant as a=3.291Å.The most energetically favorable structure is set as the strain-free structure in this study and the atomic structure,as well as the conventional cell,is shown in Fig.2.Speciﬁcally,the bond length of Zn–O bond is 1.90Å.The Zn–O–Zn and O–Zn–O angles are120°and all atoms are within one plane.Our results of atomic structure is in good agree-ment with previous DFT calculations[4–7,9–16]and experiment (1.92Å)[8].3.2.Strain energyWhen the strains are applied,all the atoms are allowed full free-dom of motion within their plane.A quasi-Newton algorithm is used to relax all atoms into equilibrium positions within the de-formed unit cell that yields the minimum total energy for the im-posed strain state of the super cell.Both compression and tension are considered with Lagrangian strains ranging fromÀ0.1to0.4with an increment of0.01in each step for all three deformation modes.We deﬁne strain energy per atom E s=(E totÀE0)/n,where E tot is the total energy of the strained system,E0is the total energy of the strain-free system,and n=6is the number of atoms in the unit cell.This size-independent quan-tity is used for the comparison between different systems.Fig.3 shows the E s of g-ZnO as a function of strain in uniaxial armchair, uniaxial zigzag,and biaxial deformation.E s is seen to be aniso-tropic with strain direction.E s is nonsymmetrical for compression (g<0)and tension(g>0)for all three modes.This nonsymmetry indicates the anharmonicity of the g-ZnO structures.The harmonic region where the E s is a quadratic function of applied strain can be taken betweenÀ0.02<g<0.02.The stresses,derivatives of the strain energies,are linearly increasing with the increase of the ap-plied strains in the harmonic region.The anharmonic region is the range of strain where the linear stress–strain relationship is invalid and higher order terms are not negligible.With even larger loading of strains,the systems will undergo irreversible structural changes, and the systems are in a plastic region where they may fail.The maximum strain in the anharmonic region is the critical strain. The critical strain is0.18under armchair deformation.However, for the other two directions,the critical strains are not observed. The ultimate strains are determined as the corresponding strain of the ultimate stress,which is the maxima of the stress–strain curve,as discussed in the following section.3.3.Stress–strain curvesThe second P–K stress versus Lagrangian strain relationship for uniaxial strains along the armchair and zigzag directions,as well as biaxial strains,are shown in Fig.4.The stresses are thederivatives322Q.Peng et al./Computational Materials Science68(2013)320–324of the strain energies with respect to the strains.The ultimate strength is the maximum stress that a material can withstand while being stretched,and the corresponding strain is the ultimate strain.Under ideal conditions,the critical strain is larger than the ultimate strain.The systems of perfect g-ZnO under strains beyond the ultimate strains are in a metastable state,which can be easily destroyed by long wavelength perturbations and vacancy defects, as well as high temperature effects[39].The ultimate strain isdetermined by the intrinsic bonding strengths and acts as a lower limit of the critical strain.Thus it has a practical meaning in consid-eration for its applications.The ultimate strengths and strains corresponding to the differ-ent strain conditions are in Table1,compared with that of g-BN, graphene,and graphyne.The material behaves in an asymmetric manner with respect to compressive and tensile strains.With increasing strains,the Zn–O bonds are stretched and eventually rupture.When the strain is applied in the armchair direction,the bonds of those parallel with this direction are more severely stretched than those in other directions.The ultimate strain in armchair deformation is0.17,smaller than that of g-BN,graphene, and graphyne.The critical strain is0.18under armchair deforma-tion,where there is a signiﬁcant drop of the stresses thereafter (Fig.4top panel),indicating the failure of the system.However, for other two directions,the critical strains are not spotted.Under the zigzag deformation,in which the strain is applied perpendicu-lar to the armchair,there is no bond parallel to this direction.The bonds that are at an incline to the zigzag direction with an angle of 30°are more severely stretched than those in the armchair direc-tion.The ultimate strain in this zigzag deformation is0.24,smaller than that of g-BN,while bigger than that of graphene and graph-yne.At this ultimate strain,the bonds that are at an incline to the armchair direction appear to be ruptured(Fig.4middle panel). Under the biaxial deformation,the ultimate strain is g b m¼0:20, which is smaller than that of g-BN and graphene,but larger than graphyne.At this applied ultimate strain,all the Zn–O bonds are observed to be ruptured(Fig.4bottom).It should be noted that the softening of the perfect g-ZnO under strains beyond the ultimate strains only occur for ideal conditions. The systems under this circumstance are in a metastable state, which can be easily destroyed by long wavelength perturbations and vacancy defects,as well as high temperature effects,and enter a plastic state[39].Thus only the data within the ultimate strain has physical meaning and was used in determining the high order elastic constants in the following subsection.3.4.Elastic constantsThe elastic constants are critical parameters inﬁnite element analysis models for mechanical properties of materials.Our results of these elastic constants provide an accurate continuum descrip-tion of the elastic properties of g-ZnO from ab initio density func-tional theory calculations.They are suitable for incorporation into numerical methods such as theﬁnite element technique.The second order elastic constants model the linear elastic re-sponse.The higher(>2)order elastic constants are important to characterize the nonlinear elastic response of g-ZnO using a contin-uum description.These can be obtained using a least squaresﬁt of the DFT data and are reported in Table2.Corresponding values for graphene are also shown.The in-plane Young’s modulus Y s and Poison’s ratio m may be obtained from the following relationships:Y s¼C211ÀC212=C11 and m=C12/C11.We have Y s=47.8(N/m)and m=0.667.The in-plane stiffness of g-ZnO is very small compared to g-BN(17%), graphene(14%),and graphyne(29%).Our obtained Poisson ratio is in good agreement with a previous DFT calculation(0.71)[7].Knowledge of higher order elastic constants is very useful in understanding the ing the higher order elastic continuum description,one can calculate the stress and deforma-tion state under uniaxial stress,rather than uniaxial strain[22]. Explicitly,when pressure is applied,the pressure dependent sec-ond-order elastic moduli can be obtained from the high order elas-tic continuum description[41–43,26].The third-order elastic constants are important in understanding the nonlinear elasticity of materials such as changes in acoustic velocities due toﬁnite strain.As a consequence,nano devices such as nano surface acous-tic wave sensors and nano waveguides could be synthesized by introducing local strain[25,26].Stress–strain curves in the previous section show that they will soften when the strain is larger than the ultimate strain.From the view of electron bonding,this is due to the bond weakening and breaking.This softening behavior is determined by the TOECs and FFOECs in the continuum aspect.The negative values of TOECs and FFOECs ensure the softening of g-ZnO monolayer under large strain.The hydrostatic terms(C11,C22,C111,C222,and so on)of g-ZnO monolayers are smaller than those of g-BN and graphene,consis-tent with the conclusion that the g-ZnO is‘‘softer’’.The shear terms (C12,C112,C1122,etc.)in general are smaller than those of g-BN and graphene,which contributes to its high pared to graphene,graphyne,and g-BN,one can conclude that the mechanical behavior of g-ZnO is similar to graphyne,and much softer than graphene and g-BN.3.5.Pressure effect on the elastic moduliWith third-order elastic moduli,we can study the effect of the second-order elastic moduli on the pressure p acting in the planeTable1Ultimate strengths R am ;R zm;R bmin units of N/m and ultimate strains g a m;g z m;g b mÀÁunder uniaxial strain(armchair and zigzag)and biaxial from DFT calculations, compared with g-BN,graphene,and graphyne.g-ZnO g-BN[24]Graphene[23]Graphyne[26]R a m 8.423.628.617.8g am0.170.180.190.20R z m 8.726.330.418.8g zm0.240.260.230.20R bm9.327.832.120.64g b m 0.200.240.230.18Table2Nonzero independent components for the SOEC,TOEC,FOEC,and FFOEC tensorcomponents,Poisson’s ratio m and in-plane stiffness Y s of g-ZnO from DFT calculations,compared with g-BN,graphene,and graphyne.g-ZnO g-BN[24]Graphene[40]Graphyne[26]a 3.291 2.512 2.468 6.889Y s47.8278.3340.8162.1m0.6670.2250.1780.429C1186.0293.2352.0198.7C1257.366.162.685.3C111À525.4À2513.6À3089.7À890.9C112À456.3À425.0À453.8À872.6C222À452.7À2284.2À2928.1À1264.2C1111139016,54721,927À7966C11122284260927314395C11224728221538888662C22225612,28818,7791154C11111À3296À65,265À118,79189,000C111127685À8454À19,173À10,393C11122À20,648À28,556À15,863À26,725C12222À17,926À36,955À27,463À15,495C222221607À10,0469À134,752À14,262Q.Peng et al./Computational Materials Science68(2013)320–324323of g-ZnO.Explicitly,when pressure is applied,the pressure depen-dent second-order elastic moduliðe C11;e C12;e C22Þcan be obtained from C11,C12,C22,C111,C112,C222,Y s,and m as:e C11¼C11ÀðC111þC112Þ1ÀmY sPð1Þe C22¼C11ÀC2221ÀmY sPð2Þe C12¼C12ÀC1121ÀmsPð3ÞThe second-order elastic moduli of g-ZnO are seen to increase linearly with the applied pressure(Fig.5).However,Poisson’s ratio decrease monotonically with the increase of pressure.e C11is asym-metrical to e C22unlike the zero pressure case.e C11¼e C22¼C11only occurs when the pressure is zero.This anisotropy could be the out-come of anharmonicity.4.ConclusionsIn summary,we studied the mechanical response of g-ZnO un-der various strains using DFT basedﬁrst-principles calculations.It is observed that g-ZnO exhibits a nonlinear elastic deformation up to an ultimate strain,which is0.17,0.24,and0.20for armchair, zigzag,and biaxial directions,respectively.The deformation and failure behavior and the ultimate strength are anisotropic.It has a low in-plane stiffness(47.8N/m)and a large Poisson ratio com-pared to g-BN and pared to g-BN,g-ZnO has17% in-plane stiffness,36%,33%,and33%ultimate strengths in arm-chair,zigzag,and biaxial strains respectively,and a three times of Poisson’s ratio.We also found that the g-ZnO can sustain much smaller strains before the rupture.The nonlinear elasticity of g-ZnO was investigated.We found an accurate continuum description of the elastic properties of g-ZnO by explicitly determining the fourteen independent components of high order(up toﬁfth order)elastic constants from theﬁtting of stress–strain curves obtained from DFT calculations.This data is useful to develop a continuum description which is suitable for incorporation into aﬁnite element analysis model for its applica-tions at large scale.The second order elastic constants including in-plane stiffness are predicted to monotonically increase with pressure while Poisson’s ratio monotonically decreases with increasing pressure.5.AcknowledgementsThe authors would like to acknowledge the generousﬁnancial support from the Defense Threat Reduction Agency(DTRA)Grant #BRBAA08-C-2-0130,the U.S.Nuclear Regulatory Commission Faculty Development Program under contract#NRC-38-08-950, and U.S.Department of Energy(DOE)Nuclear Energy University 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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

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。

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