Journal of Alloys and Compounds 613 (2014) 55–61

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Journal of Alloys and Compounds

journal homepage: www.elsevier .com/locate / ja lcom

Effects of spin orbital coupling on atomic and electronic structuresin Al2Cu and Al2Au crystal and liquid phases via ab initio moleculardynamics simulations

http://dx.doi.org/10.1016/j.jallcom.2014.06.0050925-8388/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding authors.E-mail addresses: luyh@zju.edu.cn (Y.H. Lu), jiangjz@zju.edu.cn (J.Z. Jiang).

Y. Wang a, Y.H. Lu a,⇑, X.D. Wang a, Q.P. Cao a, D.X. Zhang b, J.Z. Jiang a,⇑a International Center for New-Structured Materials (ICNSM), Laboratory of New-Structured Materials, State Key Laboratory of Silicon Materials, and Department of MaterialsScience and Engineering, Zhejiang University, Hangzhou 310027, People’s Republic of Chinab State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310027, People’s Republic of China

a r t i c l e i n f o

Article history:Received 29 May 2014Received in revised form 31 May 2014Accepted 2 June 2014Available online 11 June 2014

Keywords:Melting pointLiquid structureCohesive energySpin orbit couplingDensity function theory

a b s t r a c t

The origin of different melting points between Al2Cu and Al2Au has been studied using ab initio moleculardynamics simulations. Cohesive energy, electronic structures and structure information of both crystaland liquid phases have been analyzed. It is found that spin orbital coupling (SOC) plays an important roleon the cohesive energy of crystal phase, consistent with the different melting points of these two alloys.Whereas, it seems that SOC has no effect on the formation energy and structure of liquid phase. Possiblemechanism of reduced SOC effect at liquid phase is proposed. Our results are helpful to understand theglass formation ability difference between Al2Cu and Al2Au.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Melting point is considered to be an important parameter foralloys [1,2]. The Al2Cu alloy is composed of Al and Cu elementsand very similar to Al2Au alloy [3–5], as Cu and Au elements belongto the same main group with similar chemical reactivity [6–8].Several studies on their atomic structure evolution in metallicliquids have been recently carried out [9–11], however, the meltingpoint of 1333 K for Al2Au is largely different from 890 K for Al2Cu.Although the atomic structure of Al2Au crystal phase (CaF2-typeface-centered cubic structure with space group of Fm-3m) is muchmore regular than that of Al2Cu (mcm space group), the origin ofthe melting point difference for both alloys is still unsolved. In thispaper we investigated both crystal and liquid phases of Al2Cu andAl2Au alloys. In case of crystal phases, it is revealed almost nodifference in cohesive energy, electronic states and charge transferbetween them without taking spin orbital coupling (SOC) intoconsideration. Whereas, all properties of Al2Au vary largely whenSOC was taken into account, which is largely different from Al2Cu.The SOC is an interaction of a particle’s spin with its motion(mainly between the electron’s spin and the magnetic fieldgenerated by the electron’s orbit around the nucleus). It has an

important influence on heavy metal element atoms [12–14].Distinct local atomic configurations in liquid phase between thesetwo alloys may also play an important role in the origin of meltingpoint difference [15–17]. In case of liquid phases, however, nosignificant changes of geometrical structure was found when SOCwas considered for both Al2Cu and Al2Au melts through the analy-sis of pair-correlation functions, bond angle distributions, Honey-cutt–Anderson (HA) index and Voronoi tessellation methods. TheSOC effect on crystal phase is totally different from that on liquidphase and we confirmed the origin of this could be the randomatomic positions of liquid phase, which reduces the SOC effect onstructural and electronic properties of disordered system.

The paper is organized as following: Section 2, we present thetheoretical background and the details of simulations; Section 3contains calculation results together with their discussions. A briefsummary and the main conclusion are given in Section 4.

2. First-principles methods

The first-principles calculations were performed using densityfunctional theory from Vienna ab initio simulation package [18]with a plane wave basis and we employed Perdew–Burke–Ernzerhof (PBE) for the exchange and correlation functional[19]. The core electrons were represented by the projector-augmented-wave (PAW) potential. A plane wave basis with a

56 Y. Wang et al. / Journal of Alloys and Compounds 613 (2014) 55–61

cutoff of 400 eV was used to converge the structure. For geometricoptimization, a cell containing 8 Al and 4 Au (Cu) atoms with peri-odic boundary conditions was built, the Brillouin zone integrationwas performed with 8 � 8 � 8 k-point sampling in Al2Au(8 � 8 � 10 k-point sampling in Al2Cu). For the calculation of elec-tronic properties, Monkhors–Park 12 � 12 � 12 k-point samplingwas used in the case of Al2Au (12 � 12 � 15 in Al2Cu).

Ab initio molecular dynamics (AIMD) simulations wasperformed for liquid structure with NVT ensemble (fixed number,volume and temperature) [20,21]. The potential was treated thesame as crystal phases calculations. And only the u point was usedto sample the Brillouim zone of supercell with an acceptable accu-racy. An cell containing 96 Al and 48 Au (Cu) atoms was melted atabout 300 K above the melting point (Al2Au in 1600 K and Al2Cu in1200 K). After that an equilibration with 8000 MD steps (time step3 fs) is performed to acquire the final stable configuration. It isnoted that VASP only supports constant volume MD simulationsand we adjusted the cell volume to make the external pressureof the cell to be nearly zero.

3. Results and discussion

Al2Au crystal phase has a CaF2-type structure with Fm-3mspace group (No. 225). It looks like a simple cubic cell consistingof Au atoms occupied face centered cubic (FCC) sites and Al atomsoccupied simple cubic (SC) sites. Its unit cell contains 4 formulaunits as shown in Fig. 1(a). The lattice constants we obtained is6.054 Å with SOC (and 6.065 Å without SOC), in agreement withthose of previous simulations based on plane wave basis [22]. EachAu atom has eight Al neighbors with the Au–Al distance of 2.621 Å,whereas each Al atom has four Au neighbors. On the other hand,Al2Cu has a relatively complicated structure with space group I4/mcm (No. 140), shown in Fig. 1(b). The lattice constants area = 6.043 Å and c = 4.898 Å with SOC (a = 6.060 Å and c = 4.879 Åwithout SOC). Each Cu atom has eight neighboring Al atoms withthe Cu–Al distance of 2.584 Å and Cu–Cu distance of 2.449 Å. Asthe Cu–Cu bond length is shorter than Cu–Al bond length, Cuatoms seem to form the structure of Cu–Cu linear chain, which willbe discussed later.

Cohesive energy is generally associated with the structural sta-bility of crystal phase and a crystal phase with relatively lowercohesive energy is more stable than the one with higher cohesiveenergy [10]. Cohesive energy averaged on each atom are calculatedaccording to equation

Ec ¼1

xþ yðE½AlxMy� � xlAl � ylMÞ ð1Þ

Fig. 1. Atomic crystal structures of Al2Au (a) and Al2Cu (b). The small steel grey balls rep(For interpretation of the references to colour in this figure legend, the reader is referre

where x and y represent the number of Al and M (M = Au or Cu)atom in the cell, and E[AlxMy] is the total energy of the cell of Al2Auor Al2Cu, l is the chemical potential of single atom (Al/Au/Cu),which is calculated in a box of lattice constant more 20 Å with peri-odic condition. The calculated results of Ec are listed in Table 1: allthe values are negative indicating their thermally stability [23,24].It is found that the cohesive energy of Al2Cu with and withoutSOC is almost same. While, the cohesive energy of Al2Au withSOC is about 130 meV/atom lower (3.5%) than that without SOC.Although the cohesive energy difference between them is verysmall without SOC, the cohesive energy of Al2Au is about170 meV/atom lower in energy than Al2Cu, indicating that Al2Aucrystal phase including SOC exhibits a higher structural stability rel-ative to Al2Cu. This means that SOC effect plays an important rolefor crystal stability and it might be helpful to understand theremarkable difference of the melting points between them.

SOC effect has an obvious influence on cohesive energy of crys-tal phases, which relates to the melting points. Does SOC also havean effect on the local structure of liquid phases, which is importantfor melting points. To investigate atomic structures of liquid Al2Cuand Al2Au with and without SOC, their structural evolutions werecalculated by using ab initio molecular dynamics simulations. Theatomic configurations have been analyzed by several statisticsmethods such as pair correlation function g(r), bond angle distribu-tions, Honeycutt–Anderson (HA) index and Voronoi tessellationindex. It is found that the liquid local atomic structure of Al2Auis distinct from Al2Cu. However, no significant change is foundwhen SOC was considered for both systems. The origin of this phe-nomena might be the random atomic positions of liquid phase,which reduce the SOC effect on amorphous system.

3.1. Liquid structures analysis

The pair correlation function (PCF) g(r) is a very significantparameter for structural characterization of liquid or amorphousmaterials [22]. PCF is the statistic average of two body correlationover the system. It interprets the short range structure and chem-ical order. It is well known that the atomic structure of amorphousis similar to liquid structure. The total and partial PCFs for bothsystems are shown in Fig. 2. Comparing with Al2Au system, thefirst peak of PCF becomes narrower and has a higher value in Al2Cusystem. It illustrates that there are more neighboring atoms foreach atom in Al2Cu within first atomic shell. The r value of firstpeak of Al2Au is lower than that of Al2Cu, indicating the averagedlocal atomic distances of Al2Au are shorter than that of Al2Cu. Espe-cially, in partial PCFs for Cu–Cu and Au–Au, the shapes of the mainpeaks are obviously different and the first peak of Cu–Cu has a

resent the Al atoms, the big brown (blue) balls represent the Au atoms (Cu atoms).d to the web version of this article.)

Table 1Lattice parameters, space group, volume, bond length, bader charge and cohesive energy of crystal phase of Al2Au and Al2Cu.

Compound Lattice parameters Volume (Å3/atom) Bond length (Å) Bader charge (e) Cohesion energy (eV/atom)

Al2Au a = b = c = 6.064 Å 18.59 Al–Al 3.032 Al �1.674 �3.691FM-3M (225) a = b = c = 90� Al–Au 2.626 Au 3.347

Au–Au 4.288

Al2Au-SOC a = b = c = 6.054 Å 18.486 Al–Al 3.027 Al �1.700 �3.827FM-3M (225) a = b = c = 90� Al–Au 2.621 Au 3.400

Au–Au 4.280

Al2Cu a = b = 6.060 Å 14.931 Al–Al 2.902 Al �0.836 �3.655I4/MCM (140) c = 4.879 Å Al–Cu 2.588 Cu 1.672

a = b = c = 90� 14.905 Cu–Cu 2.440 �3.656Al2Cu-SOC a = b = 6.043 Å Al–Al 2.906 Al �0.851I4/MCM (140) c = 4.898 Å Al–Cu 2.584 Cu 1.702

a = b = c = 90� Cu–Cu 2.449

Fig. 2. (a) Total pair correlation function, g(r) of liquid Al2Cu and Al2Au systems at 1200 K and 1600 K, respectively. (b) Partial Al–Al g(r). (c) Partial Al–Cu and Al–Au g(r). (d)Partial Cu–Cu and Au–Au g(r). These systems were equilibrated for 8000 MD steps, and the last 4000 configurations were used for statistical average.

Fig. 3. Fractions of various HA indices in Al2Cu and Al2Au alloys. All the data wereacquired by averaging over the last 4000 configurations.

Y. Wang et al. / Journal of Alloys and Compounds 613 (2014) 55–61 57

higher g(r) value and a lower r value than those of Au–Au.However, PCFs including SOC do not exhibit any significant changein both systems. Thus, there are more neighboring Cu atoms foreach Cu within a shorter distance in Al2Cu system than the coun-terpart Au atoms in Al2Au.

Another way to identify the atomic structure is the Honeycuttand Andersen (HA) index [25], which has been widely used tocharacterize local atomic configuration. Here we employ the HAindex proposed by Ganesh and Widom [26,27] to obtain moredetailed information about the atomic structure of Al2Cu and Al2Ausystems. HA index is characterized by three integer parameters(i jk). The integer parameter indicates one atomic pair (in our casewe consider only one atom pair, i.e., i = 1) are close enough to forma bond if they are within a specified cutoff distance, which is aspherical shell of a radius corresponding to the position of the firstminimum in the corresponding PCFs. And j describes the numberof common neighbors of the atom pair. Meanwhile, k illustratesthe number of bonds in the common neighbors. Fig. 3 shows thefraction of HA indices in both systems in the studied temperature.It is known that 144 and 166 types are BCC crystal like indices and142 type is the characteristic type of FCC or HCP crystal like indices[28]. The 155 atomic pair type is signature of icosahedral structure,while the 153 and 143 types are distorted icosahedral structures orrelated to defective icosahedral structures [29]. By means of HA

index analysis, we found that 131, 143 and 142 types have a highpopulation in Al2Au. In contrast, it is mainly 143, 154 and 142 typestructure in Al2Cu. Ideal and distorted icosahedral clusters (155and 154) exist in both systems but do not dominate in liquid struc-tures. 130, 120, 121, 131 types differ from commonly known local

Fig. 4. Fractions of main Voronoi indices in both systems at each temperature(1200 K and 1600 K).

58 Y. Wang et al. / Journal of Alloys and Compounds 613 (2014) 55–61

atomic structures, e.g., icosahedral, FCC or BCC. Therefore, we cometo the conclusion that local atomic structures difference does existbetween Al2Au and Al2Cu liquid systems. However SOC effects arelargely reduced.

Voronoi tessellation method is a another helpful way to uncoverlocal atomic environment around each atom in liquid structure andto provide more direct information on atomic clusters [30,31]. Thismethod could obtain the atomic configurations of the central atomwith the nearest-neighbor atoms by using a cutoff distance whichis determined by the first valley in the corresponding PCFs. TheVoronoi polyhedral index is design as hn3,n4,n5,n6,. . .i, where ni

denotes the number of i-edged faces of Voronoi polyhedron. Wechoose four indices hn3,n4,n5,n6i (trilateral, quadrilateral, pentago-nal and hexagonal) to mark the polyhedron which are consideredto be well enough to present polyhedral structures. The idealicosahedra have the Voronoi index h0,0, 12,0i that are usually

Fig. 5. Bond angle distributions in Al2Cu and Al2Au. (a) Al–Al–Al (b) Al–Al–Cu and Al–Al–AAu (f) Cu–Cu–Cu and Au–Au–Au.

dominant in amorphous alloys and liquid structures. Fig. 4 showsthe evolutions of the fifteen most prevailing types of polyhedralindices during the equilibration processes. Configurations of 8000MD simulation steps are statistically analyzed and averaged. It isfound that h0,4, 4,2i, h0,3,6,1i, h0,3,6,2i, h0,4,4,3i, h0,2, 8,1i havethe relative higher populations corresponding coordinate number(CNs) of 10 and 11 respectively in Al2Au system. Whereas,h0,2, 8,1i, h0,3, 6,3i and h0,2,8,2i is dominant in Al2Cu system. Itis in agreement with those of previously reported structures thathigh coordinated Voronoi cells are dominant in metallic liquidand bulk metallic glasses. Contents of distorted icosahedral struc-tures h0,2,8,1i, h0,2,8,2i are more abundant in Al2Cu than in Al2-

Au, indicating that local atomic environment in Al2Cu is moredisordered. In contrast, Al2Au system is more ordered, implyingthe potential connection existence between crystal and liquidstructure.

Bond angle distribution function is often used to examine thehigher order correlations in the liquid and amorphous systems.Fig. 5 illustrates the first-nearest partial bond angle distribution(BAD) functions of liquid phases using the cut-off distance, whichcorresponds to the radial distance of the first coordination sphereof pair function. Here, the nearest neighbor atom is set as 0�. ForAl–Al–Al bond angle distributions in Fig. 5a, Al–Al bonds aroundAl atom has a peak near 54� and a broad second peak around100� in Al2Au system. In contrast, Al atom has a peak at about55� and 102� in Al2Cu system. This suggests the Al–Al bondsaround Al atoms in both systems have the nearly same BAD. More-over, similar phenomenon is observed in Al–Al–Au and Al–Al–Cupartial BADs. Furthermore, it is the same in Al–Cu–Al and Al–Au–Al partial BADs. However, the broad second peak in Al–Au–Albecomes rather flat. Nevertheless, significant differences presentin the rest bond angle distributions. Particularly in Cu–Cu–Cuand Au–Au–Au, it can be seen that the prominent peak and broadmaximumare largely different between these two systems. Ourresults indicate the local atomic structures between two systemsare distinct. There is no observable difference when SOC wasconsidered for both systems at liquid state.

u (c) Al–Cu–Al and Al–Au–Al (d) Cu–Cu–Al and Au–Au–Al (e) Cu–Al–Cu and Au–Al–

Table 2The cohesive energy differences (DE) between considering SOC and non-SOC for 11selected liquid structures of Al2Au.

Number in MD steps DE cohesion energy (meV)

0 (crystal phase) 1364000 36.54400 36.04800 36.45200 36.95600 36.26000 37.16400 36.16800 36.27200 36.17600 36.58000 36.6

Y. Wang et al. / Journal of Alloys and Compounds 613 (2014) 55–61 59

3.2. Electronic property and cohesive energy difference for liquid phase

SOC effect was assumed to be significant in metallic liquid inour expectations, but no change was found in both Al2Cu and Al2Ausystems. How did the liquid structures reduce the effect of SOC?We investigated cohesive energy and PDOS in Al2Au with andwithout SOC to see the difference. We estimated the cohesiveenergy differences (DE) between considering SOC and non-SOC atdifferent time step (selected randomly from the 4000th, 4400th,4800th, 5200th, 5600th, 6000th, 6400th, 6800th, 7200th, 7600thand 8000th MD steps). Results are listed in Table 2. DE for selectedliquid structures is all around 36.5 meV, which is about 100 meVlower in energy than that of crystal phase structures (136 meV).Since the liquid configurations were chosen randomly from theMD steps, these results reveal that cohesive energy in liquid waspretty smaller than that in crystal. Hence, the random atomicpositions in liquid phase played an important role in reducingthe cohesive energy. In other words, complex liquid atomicconfigurations eliminate SOC effects. In addition, we further studythe question: how about the electronic property difference inliquid with and without SOC? To answer this question, we plotd-projected local density of states (LDOS) projected onto Au/Cuin both systems with and without SOC effect to provide electronic

Fig. 6. Local density of states (LDOS) ofAl2Au and Al2Cu liquid phase projected onto Au/SOC. (d) LDOS on Cu with SOC.

level insights into the origin of the problem. For example, we selectone liquid structure at the 4000th MD step. Due to muchlower intensities of s and p orbitals, here we only analyzed thed-projected LDOS onto Au/Cu. Fig. 6 shows the projected DOS forthe d states of the Au and Cu atoms. The 5d states of the Auatom with SOC express an obvious splitting comparing withnon-SOC Au atom. Meanwhile, the 3d states of Cu atom withSOC and without-SOC present the similar occupation. Consideringthis fact, SOC effect results in a strong interaction among d orbitalsof Au atoms. This is in agreement with that in crystal phases,which will be discussed in next section. It is noted that SOC effectleads to a great difference in cohesive energy for crystal phases,whereas no obvious change was found in that of liquid phase.Consequently, liquid local atomic configurations do influencethe cohesive energy because of the random packing. Conversely,SOC effect does not exhibit any remarkable contribution to struc-tural factors although SOC has a significant effect on electronicproperties.

3.3. Electronic property for crystal phase

Next, we investigate the electronic structure of Al2Au and Al2Cucrystal phases to gain more insight into the origin of different melt-ing points. As we discussed above, cohesion energy of Al2Auincluding SOC is larger than that of Al2Cu, it is very crucial forthe explanation of different melting points. Fig. 7 illustrates thecharge density difference between Al2Au and Al2Cu crystal phases,which are calculated through subtracting non-self-consistentcharge density (a simple superposition of atomic charge densities)from self-consistent charge density. The charge densities have nosignificant change between Al2Au with and without SOC (notshown here), as well as Al2Cu. It is found that the charge transferbetween Al and Au is larger than that between Al and Cu, indicat-ing stronger Al–Au interaction in Al2Au than Al–Cu interaction inAl2Cu since we have used the same level of isosurface. The mostsignificant interaction in Al2Au is Al–Au bond whereas in Al2Cu itis Cu–Cu bond instead of Al–Cu bond. Fig. 8 shows the total densityof states (TDOS) for Al2Au and Al2Cu, as well as the d-projectedlocal density of states (LDOS) projected onto Au/Cu, respectively.The TDOS of Al2Au with SOCis totally different from that without

Cu. (a) LDOS on Au without SOC. (b) LDOS on Au with SOC. (c) LDOS on Cu without

Fig. 7. Geometric structures of Al2Au (a) and Al2Cu (b) with charge-density isosurface. The yellow isosurfaces represent positive charge transfer, the cyan isosurfacesrepresent negative charge transfer. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

60 Y. Wang et al. / Journal of Alloys and Compounds 613 (2014) 55–61

SOC, mainly due to electronic structure localization on Au asshown in LDOS projected onto Au. In line with our expectations,no obvious difference of TDOS between Al2Cu with and withoutSOC is observed. In TDOS of Al2Au with SOC, the weight center ofTDOS shifts downward the energy axis and forms a gap mainlydue to Au, which can be seen from the d-projected LDOS of Au. Thisgap agrees well with former d-orbital splitting in Al2Au liquid

Fig. 8. Density of states of Al2Au and Al2Cu. (a and e) Total density of states (TDOS)of Al2Au and Al2Cu without considering spin orbital coupling (SOC), respectively. (band f) TDOS of Al2Au and Al2Cu considering SOC, respectively. (c and g) Localdensity of states of states (LDOS) projected on 5d (3d) orbitals of Au (Cu) withoutconsidering SOC, respectively. (d and h) LDOS projected on 5d (3d) orbitals of Au(Cu) considering SOC, respectively. The insets are partial charge density isosurface(a and b) in energy range from �8 eV to �6.5 eV without and with considering SOC.

structure with SOC (Fig. 6d). It is noted that this SOC effect doesnot affect geometrical structure in liquid phases and onlystrengthens the interaction between atoms, which enhances thesplitting of d-orbital of Au. Distinctionin DOS between SOC andnon-SOC in Al2Au crystal system isconfirmed by cohesive energydifference shown in Table 1. Inorder to view d-orbital influenceon the interaction between Al and Au atoms in real space, we plot-ted partial charge density isosurface (insets of Fig. 8a and b) inenergy range from �8 eV to �6.5 eV, which is around the splittingd-orbital energy gap. The charge density isosurface of Al2Au withSOC is much more localized than that without SOC, implying SOCeffect pushes electrons between Al and Au atoms close to Aunucleiand lowers the total energy of the whole system. As a result, SOCeffect has an obvious effect on electronic properties and stabilityof Al2Au crystal phase.

4. Conclusions

In summary, we have studied the origin of different meltingpoints between Al2Au and Al2Cu using ab initio calculation. We cal-culated the cohesive energy, electronic states and charge transferfor crystal phase, as well as pair correlation functions, bond angledistributions, Honeycutt–Anderson index, Voronoi tessellationindex, cohesive energy and electronic properties for liquid phase.It reveals that both SOC effect and local atomic structures areimportant for melting point difference of these two systems,although they have similar valence electrons. The SOC effect affectsthe cohesive energy of crystal phase, which is believed to be themain reason for the large difference of melting points betweenthem. However, our investigations also reveal that the effect ofSOC was reduced due to the random local atomic structures inliquids. On the other hand, the local geometrical structures are alsovery important for the melting points. Further studies are desiredto understand the melting point difference between Al2Cu andAl2Au.

Acknowledgements

Financial supports from the National Key Basic ResearchProgram of China (2012CB825700), National Natural ScienceFoundation of China (Grants 51371157, 11374009, 11179026 and51071141), Natural Science Foundation of Zhejiang Province(Grants Z1110196 and Y4110192), and the Fundamental ResearchFunds for the Central Universities are gratefully acknowledged. Thecomputer resources at the Shanghai Supercomputer Center aregratefully acknowledged.

Y. Wang et al. / Journal of Alloys and Compounds 613 (2014) 55–61 61

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