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PHYSICAL REVIEW B 88, 161402(R) (2013)

Geometrically induced melting variation in gallium clusters from first principles

K. G. Steenbergen and N. Gaston*

MacDiarmid Institute for Advanced Materials and Nanotechnology, Victoria University of Wellington, P.O. Box 600,6140 Wellington, New Zealand

(Received 19 May 2013; published 11 October 2013)

First-principles Born-Oppenheimer molecular dynamics simulations of gallium clusters reproduce experimen-tal specific heats to a high degree of accuracy, with no experimental input or fitting. Remaining systematic shiftsof the melting temperature highlight important limitations of density-functional approximations. An analysis ofthe structural changes observed as a function of finite temperature provides evidence for the assignment of amelting temperature to a specific peak in the specific heat. The structural basis for features in the specific heatcurves, in particular, the existence of multiple peaks, is demonstrated.

DOI: 10.1103/PhysRevB.88.161402 PACS number(s): 36.40.Ei, 31.15.xv, 68.35.Rh, 71.15.Mb

Experimental measurements of the melting temperatures ofsmall gallium clusters demonstrated that, at the size of tens ofatoms, gallium clusters melt at temperatures up to 2.5 timesthe bulk melting temperature.1–3 The addition of a single atomwas also shown to dramatically change the cluster meltingtemperature and latent heat behavior. Both observations definea new size regime for melting where the transition temperatureno longer scales with the cluster radius. Greater-than-bulkmelting temperatures have also been observed in isolated sizesof aluminum and tin clusters,4–6 but gallium clusters exhibitthe behavior more systematically.

Gallium is also one of the most unusual elemental metals.The lattice structure is orthorhombic, based upon covalentlybound pairs of gallium atoms,7 and it exhibits propertiescharacteristic of both a metallic and molecular solid. Inparticular, the low melting temperature of gallium has beenascribed to the molecular nature of the solid;8 this observationcan itself provide some explanation of the greater-than-bulkmelting behavior of the clusters.9,10 Differences between themelting behavior and structures of gallium and aluminumclusters give insight into the origin of the different bulkphases.

Numerous theoretical studies have addressed the globalminimum (GM) structure of small gallium clusters over anextended size range.9,11–15 However, to date, these structureshave not been verified by comparison to experimental data.Previous molecular dynamics simulations have covered in-dividual sizes,16 with only two neutral clusters simulatedconsecutively.17 An explanation of the melting features basedon first-principles calculations has yet to be completed.Previous efforts suggested that covalent bonding in the clustersmight be responsible,17 but recent work has presented evidencerefuting this theory.9,10,13,18

Our Rapid Communication includes first-principles molec-ular dynamics (MD) simulations of five consecutive sizes ofcationic gallium clusters. Illustrated in Figure 1, we makean explicit comparison to experiment, in order to ascertainthe accuracy of density-functional-based MD in capturing thethermodynamic behavior of these small systems. We showthat although a melting temperature shift is observed in allclusters, these shifts are broadly systematic and we relate thisbehavior to known deficiencies of standard density functionals.The nature of the experimental curves is accurately replicated,demonstrating that first-principles MD captures fine details

FIG. 1. (Color online) Simulated canonical specific heat curves(thick black) for the 20-atom and 32–36 atom cations, with groundstate structures displayed to the right (from two angles for 32–36).To compare directly with experiment (Refs. 2 and 3) (red squares),all curves are shifted vertically by −0.2CV(T )/C0, with temperatureshifts +90 K (blue dashed) for the Ga20

+, Ga32+, Ga34

+, and Ga35+,

while the shifts for Ga33+ and Ga36

+ are given to the left of thosecurves.

such as a narrow double peak. The finite temperature dynamicsreveals these features can be attributed to adatom-inducedstructural reorganization prior to melting. Finally, we observethe influence of geometric shell closure for Ga33

+ and Ga36+.

Born-Oppenheimer (BO) molecular dynamics simulationswere completed for five cationic clusters sized 32–36 atoms.

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All calculations utilized the Vienna Ab Initio Software Package(VASP) 5.2,19–22 with the Perdew-Wang 1991 generalizedgradient approximation (GGA-PW91) functional23,24 coupledwith projector-augmented wave (PAW) pseudopotentials25,26

treating 3s23p1 as valence electrons. The effect of angularmomentum on configuration space was considered to benegligible, as randomly selected structures from each constantenergy trajectory demonstrated only small variations in theprinciple moments of inertia.27,28

Initial structures for the clusters with 32–34 atoms wereobtained from a set of putative global minimum (GM) alu-minum structures,29 while Ga35

+ and Ga36+ were identified by

Nunez et al.9 Parallel tempering (PT) allowed for configurationswapping between different energy trajectories based on aMonte Carlo random walk algorithm,30 enabling effectiveexploration of the extensive structural landscape known forgallium.13,14,18,31 The lowest energy structures identified by PTmatch the GM structures identified by Nunez et al.,9 with smallbond length differences arising from the different functional.

While the structural progression has been previously de-scribed in detail,9 we will re-iterate that the GM configurationsfor the 33- and 36-atom cations are geometric closed-shellstructures, where Ga36

+ has a stable tetrahedral core. TheGM for Ga32

+ is similar to that of Ga33+, but small atomic

rearrangements arise from the single-atom vacancy in oneof the surface planes. The GM structure for the 34-atomcation adds a single atom to that of Ga33

+, creating afourth plane, while Ga35

+ undergoes a more notable structuralreorganization, leaving three atoms in the fourth plane. Eachof these vacancies or adatoms has a significant destabilizingeffect on surface structure.

MD convergence was determined by measuring changesin the specific heat peak temperature and overall nature,as detailed in our previous work,18 and required simulationtimes of 207, 400, 543, 278, and 240 ps, for the 32–36atom clusters, respectively. The converged peak temperaturesvaried by less than 8 K in 10 ps; however, the relatively shortsimulation times feasible for first-principles MD could leadto larger errors in the simulated melting temperatures, sinceergodicity is nontrivial to establish.

Figure 1 illustrates each simulated canonical specific heatcurve calculated by the multiple histogram (MH) method,32

compared with experimental data.2,3 For three of the fiveclusters, we note a simulated melting temperature ∼90 Klower than the corresponding experimental peaks, with a−0.2CV(T )/C0 vertical shift. This same shift was noted inprevious results for Ga20

+,18 as shown.The temperature and specific heat shifts for Ga33

+ andGa36

+ differ in magnitude and direction. As illustrated inFig. 1, the simulated melting temperatures for the 33- and36-atom clusters are higher than experiment. A shift of −25 Kand −0.2CV(T )/C0 was applied to the simulated specific heatof the 33-atom cation, while the 36-atom result was shifted−110 K and −0.2CV(T )/C0.

Each shifted curve is included in Figure 1, demonstratingthe excellent agreement in the overall nature of the specificheat curves compared to experimental results for the 20-, 33-,34-, 35-, and 36-atom clusters. As illustrated in Table I, eventhe temperature spacing between the double peaks agrees wellwith experiment, measuring within 10 K of the experimental

TABLE I. Temperatures (T ), given in Kelvin, at which peaks areobserved in the specific heat curves with corresponding specific heatvalues (C), given in CV(T )/C0. Subscripts indicate where there aremultiple peaks.

Simul. Expt.

Cluster Tp1 Cp1 Tp2 Cp2 Tp1 Cp1 Tp2 Cp2

Ga20+ 615 1.60 705 1.35

Ga32+ 510 2.10 600 1.35

Ga33+ 565 2.00 540 1.95

Ga34+ 400 1.45 615 1.35 490 1.40 695 1.10

Ga35+ 400 1.50 535 1.50 500 1.30 600 1.30

Ga36+ 635 1.75 550 1.50

difference for Ga34+ and 35 K for Ga35

+. Only the nature of thelatent heat behavior for Ga32

+ differs notably from experiment.Although it has vertical and horizontal shifts comparable to the20-, 34-, and 35-atom simulations, the simulation measuresa finely peaked melting transition compared with a truncatedexperimental peak. The 32-atom structure shown here matchesthat discovered by an extensive GM search,9 and we do notidentify a lower energy isomer in our simulations.

The Ga34+ and Ga35

+ specific heat curves display a distinctdouble peak. The question naturally arises: Which of thesepeaks corresponds to the melting transition? Analysis ofthe root-mean-square bond length deviation18 (δrms) yieldsadditional evidence of the nature of each peak. This measuresignals a melting transition by a sharp rise in bond lengthvariance. Although the sharp rise is broadened due to finite sizeeffects in clusters33 as well as the use of PT,34 δrms provides agood measure of the structural response at finite temperature.

As illustrated in Fig. 2, all clusters exhibit a classic δrms

melting signature, with low bond length variance at lowtemperatures followed by a steep rise and smoothly increasingδrms values at high temperatures, characteristic of liquidbehavior. For both Ga34

+ and Ga35+ where a double peak

is observed, the onset of converged δrms correlates well withthe higher-temperature peak. This is even true for the 34-atomcluster, where the peak at Tp2 is notably lower than at Tp1 . Itcan be clearly seen from the ground state structures (Fig. 1)that both the 34- and 35-atom clusters correspond to sizeswhere the geometric shell closure at Ga33

+ is still evident,

FIG. 2. (Color online) Root-mean-square bond length deviations(δrms) for each cluster as a function of temperature. The simulated Tm

are indicated by the vertical dashed-dotted lines and correspondingsymbols along the x axis.

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FIG. 3. (Color online) Each row gives the MSD results for a givencluster size, distinguishing the atomic mobilities of the central atoms(black dashed) from surface atoms (red solid). The left plots are theMSDs of the lowest-temperature simulations, while the middle andright plots correspond to Tp1 and Tp2 or Tmax, as annotated. Due to thedistinctive MSD for Ga36

+ at the melting temperature (middle, Tm),we also include the MSD at the highest temperature (∼1070 K).

and the additional atoms are loosely attached to the 33-atomcore. We thus see a structural cause for the double peak in theheat capacities, revealing that the first peak can be attributedto structural rearrangement at finite temperature.

Figure 3 offers additional insight into the nature of meltingin these small cations with a comparison of surface and cen-tral atomic mobilities. Atomic mean-square displacements18

(MSDs) were averaged separately for surface and centralatoms, and between configuration swaps, eliminating anyeffect of PT. For all cluster sizes, the left column represents theMSD at the lowest temperature, while the second representsthe temperature just above Tp1 . For Ga34

+ and Ga35+, the third

column illustrates the atomic mobilities just above Tp2 .At low temperatures, central atoms exhibit significantly

lower mobilities than surface atoms. All atoms become equallymobile at temperatures above Tp1 for the clusters of 32, 33,and 36 atoms. For Ga34

+ and Ga35+, however, the surface and

central atoms do not become equally mobile until Tp2 , furthersupporting a melting transition at the higher-temperature peakfor these sizes.

The MSD for Ga36+ is distinctive. The coupled mobilities

of the surface and central atoms signify latticelike vibra-tions, indicating a highly symmetric, rigid structure at finite

temperature. Although the surface and central atoms meet ourcriterion of being equally mobile at the melting temperature,periodic motion is still evident. Thus, an additional MSDis included to illustrate that decoupled mobility occurs athigher temperatures. This behavior explains the bulklike δrms

melting signature of Ga36+, where the steep rise in bond length

variance correlates exactly with the simulated Tm. With itsclosed geometric shell and stable tetrahedral core, we againobserve finite temperature behavior with a strong geometricdependence.

It is interesting to note that this geometric dependencewas not observed for aluminum clusters in the same sizerange.35 For example, although the GM for Ga35

+ and Ga36+

are nearly identical to their size analogs in aluminum, thespecific heat characteristics are entirely distinct between thetwo metals. There is no adatom-induced double specific heatpeak observed for either the experimental4 or simulated35

Al35+ cluster. Additionally, although the GM structures for

the 36-atom clusters both have closed geometric shells, Al36+

exhibits a microcanonical double specific heat peak, whileGa36

+ melts with a fine, single-peak in both the canonical andmicrocanonical measures. These results suggest that for atleast these two cluster sizes, the thermodynamic response ingallium is demonstrably more sensitive to geometric structure.This increased influence of geometric structure is consistentwith the more complex phase diagram of bulk galliumcompared to its lighter congener, providing an indication ofthe origin of their very different bulk properties.

Both δrms and MSD analyses support the assignment of amelting phase transition at 510, 565, 615, 535, and 635 Kfor the 32–36 atom cations, respectively. Where there aretwo peaks in the specific heat, the higher-temperature peakcorresponds to the melting transition. But what determines thismelting temperature? The cohesive energies (Ecoh) for galliumclusters within the range of 7–36 atoms are illustrated in Fig. 4;the smaller clusters have been previously published.10,18 Ecoh

2 2.25 2.5 2.75 3 3.25 3.5

n1/3

1.9

2

2.1

2.2

2.3

2.4

2.5

Eco

h(e

V/a

tom

)

NeutralCation

3.2 3.25 3.3

2.43

2.46

2.49

Ecoh

450

750

Tm

(K)

Sim Tm

Exp Tm

1

1.2

Hfu

s(e

V/a

tom

)Hfus

8

33

79

1011

12

20

20

3634

Δ

Δ

FIG. 4. (Color online) The cohesive energies of both neutral andcationic clusters within the range of 7–36 atoms are shown. Inset: Thecohesive energies are compared to the latent heats, and the simulatedand experimental Tm, where the higher-temperature peaks are usedfor experimental Ga34

+ and Ga35+ data points.

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is generally proportional to the cube root of the number ofatoms, although the 33- and 34-atom cationic clusters havehigher cohesive stability relative to neighboring sizes.

The inset of Fig. 4 illustrates the strong correlationbetween the simulated cohesive energy, latent heat, andmelting temperature for the 32–36 atom cations. We havealso calculated the relative entropy of melting (not shown)and find that the trends parallel the latent heat and aresmall in magnitude, thus serving only to “damp” energeticeffects. The correlation between cohesive energy and latentheat has been previously established for small aluminum andsodium clusters,36 where local maxima in Ecoh arise from acombination of electronic and geometric shell closings. Here,the local maximum for Ga34

+ represents neither a geometricnor electronic closed-shell structure. The experimental data2

and previous theoretical work9 show Ga34+ to be a relative

stability minimum. The difference between the two calculatedvalues of Ecoh is attributable to small differences in the averagenearest-neighbor distance of only ∼0.03 A. It is thereforeevident that the small cohesive energy differences for clustersin this size range should be interpreted with caution, especiallywhen considering finite temperature behavior.

While the correlation between latent heat and meltingtemperature is as expected,37 it is interesting to note thatsimulated latent heats deviate from the experimental Tm

trends for only Ga33+ and Ga36

+—the same clusters whosesimulated specific heat curves required different shifts to matchexperiment. This observation leads to an explanation for thedifferent temperature shifts that is consistent with the excellentagreement of the nature of these simulated melting curves:Both Ga33

+ and Ga36+ correspond to geometric shell closings9

and are contained by quasiperfect surfaces reminiscent of theextended (010) surfaces of bulk gallium. It is a well-knowndeficiency of gradient-corrected density functionals that theyunderestimate the surface energies of metals consistently by∼30%.38 This underestimated surface energy correspondsto an overestimation of the surface stability, and thus anoverestimation of the Tm of these two clusters. It is alsopossible that the overstabilization of the surface leads to agreater problem in achieving ergodicity, due to the lowerprobability of defect creation at the surface necessary for theonset of melting. The distinction between these two effects isa matter for further investigation.

Without invoking nonergodicity, the relative overestimationof Tm is explained by the greater extent of the perfect surface

in the Ga36+ cluster. Assuming 90 K is the standard shift for

the GGA-PW91 functional, Ga36+ is shifted by 200 K while

Ga33+ is shifted by 115 K. This observation also supports the

origin of the consistent shift for clusters lacking a geometricclosed-shell GM structure: The 90 K difference arises froma shift in the energy scale of the functional. This conclusionis consistent with the underestimated cohesive energy of bulkgallium as calculated by GGA-PW91: 2.74 eV compared tothe experimental 2.81 eV.39

In conclusion, we have shown that density-functionaltheory (DFT)-based MD simulations accurately reproducethe overall nature of the experimental specific heat curves.For clusters exhibiting a double specific heat maximum,analysis of the temperature dependent dynamics assign themelting transition to the higher-temperature peak. A structuralreorganization associated with the lower-temperature peak canbe understood based on the ground state structures and theproximity of these cluster sizes to a geometric shell closing.Comparison of the simulated specific heats of clusters ofgallium and aluminum35 demonstrates a stronger dependenceon the geometric structure for gallium. This difference hints atthe origin of their very distinct bulk structures.

Highlighting the important consequences of an oftenignored limitation of DFT, we observe that clusters exhibiting ageometric shell closing exhibit different shifts in comparison toexperimental data2 than those without a closed-shell structure.These results suggest that the imbalanced treatment of differentinteractions within DFT, even in the comparison of surfaceand bulklike bonding, is a serious issue for the extension offirst-principles calculations to increasingly complex systems.Given that DFT is not systematically improvable, a firmunderstanding of the limits of accuracy of each functionalis crucial for an accurate analysis of first-principles results.Nonetheless, we demonstrate that careful analysis of the causesand extent of agreement with experiment can provide a firmbasis for the interpretation of still challenging experimentalresults.

We gratefully acknowledge the Marsden Fund of theRoyal Society of New Zealand under Contract No. IRL0801and the New Zealand eScience Infrastructure, particularlythe BlueFern (University of Canterbury, NeSI67) and Pan(University of Auckland, NeSI28) supercomputer teams, fortheir excellent support.

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