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Geometric, electronic properties and the thermodynamics of pure and Al--doped Li clusters

Geometric, electronic properties and the thermodynamics of pure and Al–doped Li clusters
Mal–Soon Lee1 , S. Gowtham2 , Haiying He2 , Kah–Chun Lau2 , Lin Pan2 , and D. G. Kanhere1
1

arXiv:cond-mat/0609215v2 [cond-mat.mes-hall] 23 Oct 2006

Centre for Modeling and Simulation, Department of Physics, University of Pune, Ganeshkhind, Pune 411 007, India.

2

Physics Department, Michigan Technological University, Houghton, MI 49931-1295, USA. (Dated: February 6, 2008)

The ?rst–principles density functional molecular dynamics simulations have been carried out to investigate the geometric, the electronic, and the ?nite temperature properties of pure Li clusters (Li10 , Li12 ) and Al–doped Li clusters (Li10 Al, Li10 Al2 ). We ?nd that addition of two Al impurities in Li10 results in a substantial structural change, while the addition of one Al impurity causes a rearrangement of atoms. Introduction of Al–impurities in Li10 establishes a polar bond between Li and nearby Al atom(s), leading to a multicentered bonding, which weakens the Li–Li metallic bonds in the system. These weakened Li–Li bonds lead to a premelting feature to occur at lower temperatures in Al–doped clusters. In Li10 Al2 , Al atoms also form a weak covalent bond, resulting into their dimer like behavior. This causes Al atoms not to ‘melt’ till 800 K, in contrast to the Li atoms which show a complete di?usive behavior above 400 K. Thus, although one Al impurity in Li10 cluster does not change its melting characteristics signi?cantly, two impurities results in ‘surface melting’ of Li atoms whose motions are con?ned around Al dimer.
PACS numbers: 61.46.Bc, 36.40.–c, 36.40.Cg, 36.40.Ei

I.

INTRODUCTION

clusters of assembled materials, e.g. Aln Li5n (n > 1). This idea was further explored by Akola and Manni-

There have been considerable experimental and theoretical studies to understand the physical and chemical properties of cluster which include structural and electronic properties, the nature of bonding, thermodynamics, and spectroscopic properties. Since most of these studies have been carried out on the homogeneous clusters, the physics of mixed clusters remains less explored. It is well known that the dilute impurities alter the electronic structure and geometries of the bulk system. A similar phenomenon is also observed in impurity– doped clusters.4,5 Cheng et al. have investigated the energetics and the electronic structure of small Li-Al clusters.6 They suggested a special role of AlLi5 unit as a building block for
3 2 1

nen.7 They investigated small Li–rich AlN Li5N (N =16,10) clusters using ?rst–principle calculations. They reported that Al ions form a compact inner core embedded in Li atoms. However, they did not ?nd AlLi5 to be a favorable candidate as a building block for larger clusters. Further, they observed a signi?cant charge transfer from Li to nearby Al atoms, strengthening ionic bonds between Li and Al, as well as a formation of Al–Al covalent bonds. Similar ?ndings on Li10 Al8 cluster have been reported by Kumar.8 Another issue of considerable interest is the ?nite temperature behavior of homogeneous as well as impurity– doped clusters. Joshi et al. have investigated the ?nite temperature behavior of impurity–doped cluster, Li6 Sn.5

2 Their work indicates that the addition of one impurity results in lowering of melting temperature by about 125 K. A similar observation has made be Aguado et al. for the case of LiNa54 and CsNa54 clusters.9 Recently, an interesting study on the e?ect of a single impurity in the icosahedral clusters of silver has been reported by Mottet et al.10 In contrast to the previous studies,5,9 they showed that a single impurity of Ni or Cu can lead to an increase in the melting temperature of the host. A recent study by Zorriassatein et al. on the melting of Si16 Ti also reveals that a single impurity like Ti can change the ?nite temperature behavior of the host cluster signi?cantly.11 Therefore, it is of considerable interest to investigate the impurity induced e?ects in Li clusters. In the present work, we have carried out the ?rst–principles density functional molecular dynamics simulations on Li10 Al and Li10 Al2 clusters. The results have been compared with those of pure Li clusters. In particular, we have investigated the equilibrium geometries, the nature of bonding, and the ?nite temperature behavior of these clusters. In sec. II, we brie?y describe the computational procedure. We present our results and discussion in sec. III. A brief summary of results is given in sec. IV.
A. II. COMPUTATIONAL DETAILS The Geometries and the Electronic Structures III. RESULTS AND DISCUSSION

at least 60 ps. Then, resulting trajectories are used to choose several initial con?gurations for geometry optimizations. In this way, we have obtained various equilibrium geometries. The geometries are considered to be converged when the force on each ion is less than 0.005 eV/? with a convergence in the total energy to A be the order of 10?4 eV. To investigate the nature of the bonding, we have examined the total charge density and the molecular orbitals (MO’s). To examine the ?nite temperature behavior of clusters, molecular dynamics simulations have been carried out at 16 temperatures for Li10 , and at 13 temperatures for Li12 and Li10 Alm (m=1,2) within the range of 100 K ≤ T ≤ 800 K. The simulation time for each temperature is at least 150 ps. We have discarded the ?rst 30 ps for each temperature to allow the system to be thermalized. The resulting trajectories have been used to calculate standard thermodynamic indicators as well as the ionic speci?c heat via multiple histogram technique. The details can be found in Ref. 15.

The lowest energy structures and various isomers of We have employed Bohn–Oppenheimer molecular dynamics12 using Vanderbilt’s ultrasoft pseudopotentials13 within the local density approximation, as implemented in the VASP package.14 A cubic supercell of length 20 ? A with energy cuto? of 9.5 Ry was used for the total–energy convergence. We have obtained the lowest energy structure and other equilibrium geometries by a two step process. In the ?rst step, starting from random con?gurations, the clusters are heated to a few representative temperatures, below and above expected melting points. The clusters are maintained at these temperatures for Li10 , Li12 , Li10 Al, and Li10 Al2 are shown in Figs. 1 and 2. The ground state (GS) geometry of Li10 consists of two interconnected pentagonal rings with planes perpendicular to each other (Fig. 1–a(i)), which agrees with that found by Fournier et al.16 as well as with our earlier study.4 The structure with an atom at the center (Fig. 1–a(ii)) is nearly degenerate with the ground state (Fig. 1–a(i)). This is the structure reported by Jones et al. as their ground state.17 A high energy structure shows an interconnection of dodecahedron and octahedron (Fig. 1– a(iii)). We have found ?ve nearly degenerate equilibrium

3

Li10

Li10Al

a(i)

a(ii)
?E = 0.03 eV

a(iii)
?E = 0.10 eV

c(i)
Li Li Li

c(ii)
?E = 0.03 eV

c(iii)
?E = 0.04 eV

Li12
Li

Al Li

Li Li Li

Li Li

c(iv)
b(i) b(ii)
?E = 0.004 eV

c(v)
?E = 0.16 eV

c(vi)
?E = 0.26 eV

b(iii)
?E = 0.006 eV

?E = 0.11 eV

Li10Al2

b(iv)
?E = 0.07 eV

b(v)
?E = 0.07 eV

b(vi)
?E = 0.11 eV

d(i)

d(ii)
?E = 0.002 eV

d(iii)
?E = 0.03 eV

FIG. 1: The ground–state geometry and some isomers of Li10 and Li12 . The label (i) represents the ground–state geometry. The energy di?erence ?E is given in eV with respect to the ground–state energy.

d(iv)
?E = 0.09 eV

d(v)
?E = 0.13 eV

d(vi)
?E = 0.19 eV

geometries of Li12 within the energy range of 0.007 eV. We examined the stability of these structures by vibrational analysis. Three of these are shown in Figs. 1–b(i) ? 1–b(iii), where Fig. 1–b(ii) have been reported as the lowest–energy structure by Fournier et al.16 All the structures consist of two units: a pentagonal bipyramid and a distorted octahedron connecting to each other with different angles. As we shall see, the existence of these nearly degenerate structures have a bearing on the shape of the speci?c heat curve at low temperatures. In high energy structures, the octahedron is destroyed ?rst (Figs. 1–b(iv) and 1–b(v)) and then the destruction of pentag-

FIG. 2: The ground–state geometry and some isomers of Li10 Al and Li10 Al2 . The label (i) represents the ground–state geometry. The blue circle represents the Li atoms and the yellow circle represents the Al atoms. The energy di?erence ?E is given in eV with respect to the ground–state energy.

onal bipyramid follows (Fig. 1–b(vi)). Now, we discuss the geometries of impurity–doped systems. So far as Li10 is concerned, a single Al atom replaces one of the Li atoms in one unit of the pentagonal bipyramids in the GS geometry of Li10 Al (Fig. 2–c(i)). This causes the Li atoms in another unit to rearrange in

4 the form of antiprism, with Al atom at the center. As shown in Figs. 2–c(ii) to 2–c(v), the low energy geometries are dominated by the presence of pentagonal bipyramid, whereas the high energy structures are dominated by antiprism. The addition of one more Al atom in Li10 Al changes the lowest energy structure signi?cantly. The
Li10Al2 Li10Al2 Li10 Li12 Li12 Li10Al Li10Al

(a)
3.0

εdef (b)
Eb (eV/atom)

2.0

equilibrium structures of Li10 Al2 do not show the pentagonal bipyramidal structure any more. Instead, there is an octahedron consisting of four Li atoms and two Al atoms with Li atoms forming a central plane. This core octahedron is common to all the isomers. Di?erent isomers represent di?erent ways of capping this core by the remaining Li atoms. Symmetric arrangements of Li atoms give rise to low energy isomers (Fig. 2–d(i) ? 2– d(iii)). The lowest energy structure (Fig. 2–d(i)) is the most symmetric structure among all the clusters studied. This is because two Al atoms share Li atoms equally to ?ll their unoccupied p–orbitals. The nearly degenerate structure shown in Fig. 2–d(ii) has been found as the lowest energy structure by Cheng et al.6 The stability of two nearly degenerate structures has been veri?ed by carrying out a vibrational analysis. The structures with broken symmetry have higher energies (Figs. 2–d(iv) ? 2–d(vi)). We used the deformation parameter εdef to examine the shape of the clusters. εdef is de?ned as, εdef = 2Q1 Q2 + Q3

1.0

-1.2 -1.4 -1.6 -1.8 -2.0 Li10
eV/atom.

FIG. 3: The properties of the ground state geometry. (a) The deformation parameter (εdef ); (b) The binding energy Eb in

doping reduces the deformation considerably. We also show the binding energies of these clusters in Fig. 3(b). Clearly the increase in the binding energy by adding Al atom (0.2 eV/atom) is much higher than that by adding Li atom (0.04 eV/atom). Thus, we conclude that the increase of the binding energy in the mixed clusters is mainly due to the formation of strong Li–Al bonds. We examine the change in the nature of the bonding

where Q1 ≥ Q2 ≥ Q3 are eigenvalues of the quadrupole tensor Qij =
I

RIi RIj

due to the presence of Al atoms via the total electron charge density, and the molecular orbitals (MO’s). In addition, we have also calculated the di?erence charge density between electron charge densities of mixed cluster Li10 Alm (m=1,2) and separated units of Li10 and Alm (m=1,2), by keeping the atomic positions the same as those in the mixed cluster. We ?rst examine the charge

with RIi being ith coordinate of ion I relative to the center of mass of the cluster. A spherical system has εdef = 1 (Q1 = Q2 = Q3 ), while εdef > 1 indicates a deformation. The calculated εdef for the GS geometry of the clusters are shown in Fig. 3(a). It can be seen that

5

FIG. 4: The isovalued surfaces of the total charge density at 1/3 of its maximum values, where the maximum charge density of Li10 Al is 0.233 and that of Li10 Al2 is 0.239.

distribution in Al2 dimer and Li–Al diatomic cluster. Evidently, there is a covalent bond in Al2 dimer as expected, while Li–Al diatomic cluster shows a polar bond. This can be explained on the basis of electronegativity di?erence between Li(0.98) and Al(1.61), which is not large enough to establish an ionic bond. We note that the bond lengths of these clusters are 2.68 ? for Al–Al and A 2.88 ? for Li–Al. The isovalued surface of total charge A density in Li10 Al and Li10 Al2 are shown in Figs. 4(a) and 4(b), respectively. In Figs. 5(a) to 5(d), we show the isosurface of di?erence charge density for these Al– doped clusters. When we add one Al atom to the Li10 cluster, the total charge distribution shows a mixed character of localization near Al atom and delocalization on the pentagonal bipyramid in Li10 Al (Fig. 4(a)). It can be seen that the charge, localized around Al atom in the antiprism part, is nearly spherical. This is due to a signi?cant charge transfer from nearby Li atoms to the Al atom to ?ll its unoccupied p–orbitals However, there is an insigni?cant charge transfer from the Li atoms to the Al atom, forming the pentagonal bipyramidal unit. This leads to a delocalized charge distribution in this unit. This observation is con?rmed by the examination of Figs. 5(a) and 5(b), depicting the constant density contour of the di?erence charge density. It may be noted that the values of the maxima in the di?erence charge density are 0.042 and 0.051, and those of the minima are -0.037 and -0.084 in Li10 Al and Li10 Al2 , respectively. Figs. 5(a) to 5(d) show the isosurfaces for the value of 0.017 for the charge gain, and -0.008 for the charge loss. It can be seen that a ring shape region in between Li and Al (around Al) gains charge in Li10 Al (Fig. 5(a)). Further, most of the charge is lost by Li atoms nearer to the Al atom (Fig. 5(b)). This charge transfer from Li to Al causes Li atoms to be positively charged. These positively charged Li atoms pull the charge distribution around Al atom
FIG. 5: The di?erence charge density between the mixed cluster and separated units of Li and Al clusters at the same positions. (a) and (c) show the region where the charge is gained and (b) and (d) show the region where the charge is lost. The ?gures are shown the isosurfaces of di?erence charge density at the value of charge gain of 0.017 and loss of 0.008, respectively.

6 to polarize it. This leads to a multicentered bonding between Al atom and nearby Li atoms. The addition of one more Al atom changes the charge density distribution from a mixed localized and delocalized one to a mainly localized one. In Li10 Al2 , the charge distribution (Fig. 4(b)) is mainly around two Al atoms polarized by

1.4

Li10
1.2

Normalized Specific Heat

nearby Li atoms. The di?erence charge density shows a very symmetric charge gain region in between Li and Al atoms (Fig. 5(c)) and loss of charge at each atomic site (Fig. 5(d)). These di?erence in the bonding between Li10 Al and Li10 Al2 are also seen in molecular orbitals (?gures not shown). For example, in Li10 Al three highest occupied molecular orbital (HOMO) show a delocalized charge distribution, while only the HOMO shows a delocalization in Li10 Al2 . As noted earlier, since electronegativity di?erence of Li and Al is not very high, the Li atoms donate their charge to Al atom(s) partially. We expect this charge transfer to result in weakening of Li– Li metallic bond in the system. This feature has also been noted in Li6 Sn cluster. Further, this partial charge transfer also results in the sharing of charge between two Al atoms to ?ll their unoccupied p–orbitals in Li10 Al2 . As we shall see, this di?erence in the bonding leads to a di?erent thermodynamic behavior in two clusters.
5

1.0

Li12
1.2

1.0

Li10Al
1.2

1.0

Li10Al2
1.2

1.0 100 300 500 700

Temperature (K)
B. Thermodynamics FIG. 6: The normalized speci?c–heat as a function of temper1. Pure Li clusters ature. C0 = (3N ? 9/2)kB is the zero–temperature classical limit of the rotational plus vibrational canonical speci?c–heat.

To investigate the melting behavior of the clusters, we calculate the canonical speci?c heat using multiple histogram technique. The calculated ionic speci?c heats for all the clusters are shown in Fig. 6. It is well known that a small cluster exibits a broad melting transition. Our observatios are consistent with this observation. In addition, in Li10 the ionic speci?c heat also shows a remarkable premelting feature between 150 K and 225 K, whose maximum value is close to that of the main peak around 575 K. The examination of the ionic motion reveals that the two structures shown in Figs. 1–a(i) and 1–a(ii) are observed in the temperature range of 150 K ?225 K. The system visits these two isomers in the time spent of a few pico seconds at 175 K, indicating that

7 this premelting feature is due to the isomerization. At higher temperature of 260 K, we observe the structure shown in Fig. 1–a(iii). At still higher temperatures, the cluster visits these three isomers frequently. It is di?cult to identify the melting temperature for this cluster because liquidlike behavior develops over a wide range of temperature (300 K to 700 K). The root–mean–square bond length ?uctuation (δrms ) is another indicator for studying a melting transition. According to the Lindemann criteria (for bulk) solid–liquid transition is signi?ed when the value of δrms exceeds 0.1. However, it is generally observed that for clusters a liquidlike behavior is seen when δrms exceeds 0.25?0.3. As shown in Fig. 7, δrms increases in two steps. At 175 K it exceeds the value of 0.1, which corresponds to maximum speci?c heat of the shoulder. However, as discussed earlier, the cluster is not in the liquidlike state. δrms increases from 200 K again until it saturates about 0.3 at 575 K where we observe the maximum value of the speci?c heat. In can be seen from Fig. 6 that Li10 Al shows a premelting feature in the low temperature region around 150 K. This may be caused by a weakening of the Li–Li metallic bond due to the charge transfer from Li atoms to the Al atom in antiprism unit. This e?ect has also been observed in Li6 Sn.5 The Li atoms in this unit break the Li–Li bonds at low temperature and one of them moves towards the pentagonal bipyramidal unit. This is the way the cluster visits the other low energy isomers shown in The speci?c heat of Li12 shows a weak shoulder around 125 K before it increases gradually. This shoulder is related to the existence of ?ve degenerate isomers. At low temperatures up to 150 K, the motion is dominated by vibrations and the cluster visits the low energy isomers. As temperature increases, the liquidlike behavior evolves over a very wide range of temperature. At 175 K, we observe high energy isomers where the octahedral structure is absent (two of them are shown in Figs. 1–b(iv) and 1– b(v)). With further increase of temperature to 375 K, we observe that the pentagonal bipyramidal structure is destroyed (one of these structures is shown in Fig. 1–b(vi)). It can be seen that δrms also increases continuously from 125 K to 500 K (Fig. 7). It is interesting to note that even though the isomerization is seen in both the clusters Li10 and Li12 , evidently the isomerization in Li10 leads to a prominent shoulder in its speci?c heat. This is related δrms = 1 N
i<j

to the nature of the process by which their ground state visits the isomer(s). In the case of Li10 , there is a barrier at about 175 K. The system overcomes this barrier with sudden increase in the accessible density of states. For Li12 , since the di?erent isomers di?er only in the angle between two constituent units, the ground state visits them almost continuously, aided by the vibrational motion. This results in di?erent speci?c heat curves.

2.

Thermodynamics of Al-doped Li clusters

Figs. 2–c(ii) and 2–c(iii). For the temperatures above 175 K, we observe the motion of Li atoms around the Al atom without destroying the overall shape of the structure. However, the four Li atoms consisting of pentagonal bipyramidal unit do not interchange their position with others till about 575 K. At this temperature, the system shows a di?usive liquidlike behavior. We calculate the root–mean–square bond length ?uctuation (δrms ) for Li–Li, Li–Al, and Al–Al separately, to see the di?erence between their ?uctuations. δrms is de?ned as,
2 Rij t

? Rij
t

2 t

Rij

where Rij is the distance between ith and j th ions with i and j for relevant bonds, and N is the number of bonds. ...
t

denotes a time average over the entire trajectory. It

can be seen in Fig. 7 that in the low temperature region

8

0.4

lar orbitals with those of pure Al dimer, they looked alike

Li10
0.2

even though their bond lengths are di?erent. The bond length of Al–Al in Li10 Al2 is 2.97?, while that of pure A Al dimer is 2.68?. Thus, there is a weak covalent bond A between Al atoms. This covalent bond between them re-

0.0

stricts the motion of Al atoms. We observe that Al-Al

Li12
0.2

bond does not break at least up to the temperature of about 800 K. As shown in Fig. 7, δrms of Li–Li and Li–Al saturate to a value of about 2.5 at 500 K, while that of Al–Al reaches a value of 0.1 at 800 K. The saturation values of δrms for Li–Li and Li–Al are lower than corre-

δrms

0.0

Li10Al
0.2

sponding δrms of other clusters, indicating the motion of atoms in Li10 Al2 is more restricted than other systems at comparable temperature.

IV.

SUMMARY

0.0

Li10Al2
0.2

We have employed ab initio molecular dynamics to study the equilibrium geometries, the electronic structure, and the ?nite temperature properties of pure lithium clusters Li10 and Li12 , and Al–doped lithium clus-

0.0 0 400 800

ters Li10 Al and Li10 Al2 . We ?nd that there is a substantial structural change upon doping with two Al atoms, while the addition of one Al atom results in a rearrangement. The analysis of the total charge density and the molecular orbitals reveal that there is a partial charge transfer from Li atoms to Al atom(s) in Al–doped clusters, forming a polar bond between them. This leads to a multicentered bonding and weakens the Li–Li metallic bonds in these clusters. In Li10 Al2 , Al atoms also form

Temperature (K)
Li-Li Li-Al
FIG. 7:

Al-Al

The root–mean–square bond length ?uctuation

(δrms ) of Li–Li, Li–Al, Al–Al as a function of temperature.

δrms of Li–Al is much lower than that of Li–Li. This is because of relatively stronger Li–Al bond. Interestingly, the speci?c heat of Li10 Al2 at low temperature also shows a shoulder which is very similar to the speci?c heat of Li10 . After this premelting feature, the speci?c heat remains ?at. This is due to the presence of Al dimer in the cluster. When we compare its molecu-

a weak covalent bond. These changes in the nature of bonding upon doping a?ect the ?nite temperature properties of pure host cluster. We observe that the presence of dimer–like Al atoms with weak covalent bond con?nes the motion of Li atoms around them. Thus, a substitution of two Li atoms by Al atoms in Li12 leads to a surface melting only, showing a continuous phase change

9 over a very broad range of temperature. DGK acknowledges with pleasure, the hospitality of the Department of Physics, MTU, USA during the course of
V. ACKNOWLEDGMENTS

this work.

We acknowledge partial assistance from the Indo– French Center for providing the computational support.

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S.

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