Study of the Fundamental Units of Novel Semiconductor Materials

Article pubs.acs.org/JPCA

Study of the Fundamental Units of Novel Semiconductor Materials: Structures, Energetics, and Thermodynamics of the Ge−Sn and Si− Ge−Sn Molecular Systems A. Ciccioli and G. Gigli* Dipartimento di Chimica, Sapienza Università di Roma, Roma, P.le A. Moro 5 00185, Italy S Supporting Information *

ABSTRACT: The binary GeySnz and ternary SixGeySnz molecular systems containing up to five atoms were investigated by means of density functional theory and coupled cluster calculations. The minimum energy structures were calculated and higher energy isomers are also proposed. The atomization energies of the ground state isomers were calculated by the CCSD(T) method with correlation consistent basis sets up to quadruple-ζ quality. The resulting values were extrapolated to the complete basis set limit and corrected by an approximate evaluation of the spin−orbit effect. Energetic properties such as binding, fragmentation and mixing energies, and HOMO−LUMO gap were analyzed as a function of the cluster size and composition. By using empirically adjusted atomization energies and DFT harmonic frequencies, the thermal functions were evaluated, and a thermodynamic database for the Si−Ge−Sn system was built, containing data for 55 gaseous species. On this basis, equilibrium calculations were performed in the temperature interval 1600−2200 K aimed at predicting the composition of the gas phase under various conditions. The results presented here can be of interest to improve the microscopic knowledge of Ge−Sn and Si−Ge−Sn materials, which are among the most promising candidates for advanced applications in the field of electronic and optoelectronic components, both as epitaxially grown layers and as nanocrystal quantum dots. and Ge2Sn)17 where upper limits are also given for the atomization energy of the Ge3Sn and Ge4Sn species. Binary Ge−Sn anion clusters have also been observed in a laser vaporization experiment.18 The diatomic species GeSn has been studied theoretically by Andzelm et al.19 using local-spindensity calculations. Finally, GenSn (n = 1−4) clusters, under the constraint of well-defined point-group symmetry, have been investigated computationally using the density functional approach.20 Just before submitting this article, a DFT study by Samanta et al. on the GenSnm species with n + m ≤ 5 appeared in the literature21 where, as far as the DFT calculations are concerned, a treatment similar to that here presented by us was used, with a slightly larger basis set (augcc-pVTZ-PP rather than cc-pVTZ-PP). Finally, to the best of our knowledge, no experimental or computational work has been devoted to ternary Si−Ge−Sn molecules and microclusters. In this work, we present a computational study of the aforementioned systems up to the pentatomic species. Our attention is especially focused on providing reliable estimates of their stability, both energetic and thermodynamic (i.e., taking entropies into account). In order to better put into perspective

1. INTRODUCTION While silicon and silicon oxide have been the cornerstones of microelectronics technology, in the past decade, new binary Ge−Si1−4 and ternary C−Si−Ge5 intragroup 14 materials have also been considered as functional components. Interest has been directed to the Si−Ge−Sn alloys6,7 as well, as these materials incorporate less extreme components in terms of electronegativity and bonding type compared to C−Si−Ge. In recent years, Ga−As has been used as the prototypical system for optoelectronic devices, exploiting the direct band gap property of group 13−15 materials. However, toxicity and cost issues suggested a search for alternative materials based on the group 14 elements; hence interest has developed into the difficult epitaxial growth of Si−Ge−Sn alloys, the deposition of Ge−Sn alloys on silicon, and the investigation of molecularbased synthetic approaches.6−12 In addition, binary Ge−Sn and ternary Si−Ge−Sn alloys are considered to be prospective materials for infrared detectors.13 As a consequence, in recent years, there have been theoretical investigations of the electronic structure of Si−Ge−Sn alloys and of the influence of composition fluctuations.14−16 The elemental building blocks of these binary (Ge−Sn) and ternary (Si−Ge−Sn) systems, i.e., the clusters and molecules, have received less attention. As far as the binary system is concerned, the only experimental work is that of Gingerich and co-workers on the determination of the atomization energy of small Ge−Sn molecules (GeSn, GeSn2, © 2012 American Chemical Society

Received: January 18, 2012 Revised: May 18, 2012 Published: May 29, 2012 7107

dx.doi.org/10.1021/jp300624z | J. Phys. Chem. A 2012, 116, 7107−7122

The Journal of Physical Chemistry A

Article

discussion was given on the rationale of the computational strategy adopted. The main conclusions reached were that the geometrical and vibrational parameters, especially for the evaluation of the thermodynamic properties, could be reasonably well predicted at the B3LYP/cc-pVTZ level, whereas the electronic energies could be better evaluated with the CCSD(T) approach. Moreover, it has been shown that CCSD(T) energy values calculated at the optimized B3LYP geometries did not significantly differ from the those evaluated at the CCSD(T) optimized structures. In view of the large number of species treated here, a simpler approach has been adopted, deriving the separation in energy between the ground state and higher energy isomers at the B3LYP/cc-pVTZ level. In order to substantiate this procedure, in selected cases, we performed single point CCSD(T) calculations at the B3LYP optimized geometries. The pertinent data are reported in Table 1, where it can be seen that the differences with the B3LYP// B3LYP results are not large.

the results obtained here, the properties of the binary clusters of the Si−Ge and Si−Sn systems are also included in the discussion. Si−Ge species containing up to five22−24 and seven atoms25 have been investigated theoretically in recent years, while SinGen clusters (with n up to 42) have been studied with a parametrized density functional method.26 An experimental study has been published on clusters up to tetratomic.27 For the sake of consistency, these species are here recalculated with the same methods used for the Ge−Sn and Si−Ge−Sn systems. With regard to the Si−Sn molecules, clusters up to four atoms were studied both experimentally and computationally in a previous article by our group28 while, in the present article, calculations are extended to the pentatomic species.

2. COMPUTATIONAL METHODS In this work, the B3LYP density functional theory (DFT) method has been employed to optimize the structures of the species under study. Minima in the potential energy surfaces were checked by evaluation of harmonic frequencies. The energies of the ground state structures are further evaluated at higher level with single point CCSD(T) calculations (Coupled−Cluster including Single, Double, and perturbative Triple excitations). This strategy, aimed at reducing the computational cost, has been proved to be satisfactory in the cases of pure Ge,29 mixed Ge−Si,22 and mixed Si−Sn28 molecular species. The computations were carried out using the Gaussian03 program package.30 Correlation consistent basis sets of triple-ζ quality were employed for all the elements involved: silicon (cc-pVTZ), germanium (cc-pVTZ-PP), and tin (cc-pVTZ-PP).31 For Si, all the 14 electrons are included in the valence space. For both Ge and Sn, 22 electrons are treated explicitly, while a small core relativistic pseudo potential31 accounts for the 10 and 28 inner electrons, respectively. Finally, in order to extrapolate the CCSD(T) atomization energies to the complete basis set (CBS) limit, the corresponding doubleand quadruple-ζ basis sets were also employed.31 Tight convergence criteria were used for both self-consistent field convergence and geometry optimizations. As better detailed in section 4.1, an evaluation of the spin−orbit corrections to be applied to our results has been made, at the DFT level, with the NWChem 5.1 software package.32 In dealing with microclusters, the number of possible isomers and structures becomes rapidly large on increasing the number of atoms as well as the diversity of elements considered. In this work, a reasonably large and comprehensive search for the true minimum energy structures has been made for the singlet states up to approximately 1 eV from the ground state. Local minimum structures for spin triplet states have been searched starting from the singlet state structures restricting the search, as a general rule, to singlets within 0.5 eV from the fundamental one. Triplet states below 0.5 eV from the singlet ground states were found only for the triatomic species. As previously mentioned, the general computational method used here, i.e., DFT complemented with higher level calculations such as CCSD(T),22,25 has been used recently to study the binary Si−Ge species. Here, our goal is to give an overview of the structures, energetics, and thermodynamics of the species under scrutiny with special emphasis on the new class of ternary Si−Ge−Sn compounds. To this end, a compromise between accuracy and computational cost has been adopted by analyzing how well the experimentally known parameters of similar molecules can be reproduced. In our previous work on the Si−Sn species,28 a somewhat detailed

Table 1. Energies ΔE (eV) of Selected Higher Energy Isomers with Respect to the Ground State: Single Point CCSD(T) Calculations at the B3LYP Optimized Geometries Compared with the B3LYP Results molecule Ge2Sn SiGeSn Si2Ge2Sn

state 3

B2 A′ 3 A′ 1 A 1 A′ 1

symmetry

ΔE CCSD(T)// B3LYP

ΔE B3LYP// B3LYP

C2v Cs Cs C1 Cs

0.20 0.14 0.30 0.08 0.41

0.14 0.26 0.32 0.15 0.44

It might be argued that a multireference treatment would have given a better description of the electronic structures under study. However, as a consequence of the validation tests referred to above, such a treatment was deemed not to be strictly necessary. In this connection, it is of interest to note that, in most of our CCSD(T) calculations, the T1 diagnostic was computed to be smaller than 0.02, a value that has been suggested as warranting some caution in the interpretation of single reference results. Exceptions were the Si2Ge2Sn, Si2GeSn2, Si4Sn, and Si3Sn2 species with T1 values 0.021, 0.022, 0.029, and 0.022, respectively. As a further argument in support of the strategy adopted here to obtain reliable estimates of the low lying structures at a reasonable computational cost, it is useful to comment on the theoretical results recently published for the SiSn molecule,33 already studied experimentally by our group.28 In that article, MRDCI calculations including relativistic effective core potentials for many low-lying Λ−S states are reported as well as a number of Ω states. In Table 2, we compare these results for the ground and the first excited state with (a) those obtained in our previous work28 at the CCSD(T)/cc-pVTZ level; (b) the energy computed here at the B3LYP/ccpVTZ level; and (c) the only reported experimental value.34 The 3Σ ground state predicted by our B3LYP and CCSD(T) calculations is confirmed by MRDCI results, with a small (115 cm−1) splitting of the Ω components. Moreover, on looking at the data reported in Table 2 for the first excited state, it is apparent that, as far as the evaluation of thermodynamic functions is concerned, the simple B3LYP approach seems to reproduce the correct ordering of the states and to provide a fairly good value of the energy separation. 7108



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Table 2. Energies (cm−1) of the Ground and First Excited State of the SiSn Molecule Computed at Various Levels of Theory (See Text) MRDCIa state

Λ−S states

Ω states

Ω averaged states

experimentalb

CCSD(T)/cc-pVTZc

B3LYP/cc-pVTZd

1228

1628

568.5

1568

77

0

0

0

+

Π

3

790

Σ

3

a

1890 (0 ) 1628 (0−) 1512 (1) 414 (2) 115 (1) 0 (0+)

0 b

c

d

Reference 33. Reference 34. Reference 28. This work.

3. MOLECULAR STRUCTURES In Tables 3 and 4 we present the ground state computed geometries, vibrational parameters and dipole moments of all the binary Ge−Sn and Si−Sn, as well as the ternary Si−Ge−Sn species. The corresponding molecular structures are displayed in Figures 1−3, together with those of the isomers within 0.5 eV from the ground states. Data and structures for the isomers of energy up to 1 eV can be found in the Supporting Information in Tables 1−5 and Figures 1−5. 3.1. Triatomic Species. For the triatomic species here under study, namely, the two Ge−Sn species Ge2Sn and GeSn2 and the ternary molecule SiGeSn, the stable structures found up to 0.5 eV, i.e., those subsequently used in evaluating their thermodynamic properties (vide infra), are displayed in Figure 1. In drawing these structures, we used an arbitrary set of cutoff

energies of the Ge−Ge, Sn−Sn, and Ge−Sn bonds. The Ge− Ge bond energy can be placed in the approximate range 261− 197 kJ/mol from the Ge235a and Ge335b atomization energies. The Ge−Sn bond energy in the GeSn molecule has been determined to be 230 kJ/mol.17 The Sn−Sn bond energy can be estimated from the range 183−147 kJ/mol found in the Sn236 and Sn328 molecules. Therefore, in this case, the asymmetrical Ge2Sn structure preserves the presence of a strong Ge−Ge bond, as the terminal germanium atom is not involved in another bond. The rather unusual L-shaped geometry in Ge2Sn can also be explained with the orbital arrangement first analyzed by Wielgus et al.23 for the Si2Ge molecule. In brief, the argument is that the molecular orbital concept for two bonds almost perpendicular to each other implies two mutually orthogonal π orbitals with the lone pairs on the terminal atoms. This, with an assumed sp hybridization on the central atom, leads in our case to a π bond (Ge−Ge) orthogonal to the molecular frame and to the in-plane Ge−Sn π bond. The existence of the L-shaped structure instead of an allene type linear bonding is related to this arrangement of orbitals. We note that the C2v structure of Ge2Sn proposed as the lowest energy isomer in the very recent work by Samanta et al.21 corresponds to our isomer 0.58 eV above the Cs L-shaped ground state (see Table S1 in the Supporting Information), whereas the Cs ground state and the C2v isomer at 0.14 eV found by us (see Figure 1 and again the same Table in the Supporting Information) were apparently not identified in ref 21. On the contrary, the ground state for the GeSn2 molecule here determined is the same as reported in ref 21, with practically identical geometry and vibrational frequencies, as expected in view of the very similar basis sets used in the two works. The type and sequence of higher energy isomers are the same as those found for the Si−Sn triatomic molecules,28 but in this case, energies are always lower by a roughly constant factor of 1.7. Cyclic structures with bonds between all the atoms are found only in the triplet states. Simple energetic arguments can be also invoked in trying to rationalize the stable configurations found for the singlet states of the ternary molecule SiGeSn. Here, the ground state is predicted to be in a Ge−Si−Sn configuration, with the configuration Si−Ge−Sn at 0.26 eV. The heteroatomic bond energies here at work can be estimated from the corresponding diatomic molecules as 295.4 kJ/mol (SiGe),23 233 kJ/mol (Si− Sn),28 and 230 kJ/mol (GeSn).17 Therefore, a delicate energetic balance only very slightly favors the ground state configuration with Si−Ge and Si−Sn bonds, compared to the higher energy configuration with Si−Ge and Ge−Sn bonds. The Si−Sn−Ge configuration with the two weakest bonds, Si−

Figure 1. Structure and relative energy of the triatomic Ge−Sn and Si−Ge−Sn species, obtained at the B3LYP/cc-pVTZ level of theory; Si in black, Ge in gray, and Sn in white.

distances to define whether a bond is to be displayed: 2.48, 2.60, 3.11, 2.57, 2.77, and 2.86 Å for the Si−Si, Ge−Ge, Sn−Sn, Si−Ge, Si−Sn, and Ge−Sn distances, respectively. The above values of the Si−Si, Sn−Sn, and Si−Sn distances were already used by us in ref 28. In evaluating the Ge−Ge, Si−Ge, and Ge− Sn cutoff distances, the experimental bond lengths of the diatomic molecules Ge2 (2.368 Å) and SiGe (2.34 Å) were taken into account as well as the GeSn interatomic distance here calculated at the CCSD(T)/cc-pVTZ (2.56 Å) and CCSD(T)/cc-pVQZ (2.54 Å) levels. These values have been conventionally and arbitrarily increased by 10%. These cutoff distances are here defined and used for clarity of representation, with no chemical meaning implied. As shown in Figure 1, the predicted ground states for the Ge2Sn and GeSn2 molecule are similar to those found in the triatomic Si−Sn28 and Si−Ge22,23,25 species: a symmetric structure for GeSn2 and an asymmetric L-shaped Ge−Ge−Sn structure for Ge2Sn. Similarly to the Si−Sn and Si−Ge cases, this difference can be rationalized by examining the different 7109



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Figure 2. Structure and relative energy of the tetratomic Ge−Sn and Si−Ge−Sn species (upper section) and pentatomic Si−Ge−Sn species (lower section), obtained at the B3LYP/cc-pVTZ level of theory; Si in black, Ge in gray, and Sn in white. All the species are singlet states.

Table 3. Molecular Parameters of the Ground States of the Triatomic and Tetratomic Ge−Sn and Si−Ge−Sn Species Calculated at the B3LYP/cc-pVTZ Level (For Higher Energy States, See the Supporting Information)a,b molecule

state

symmetry

Ge2Sn SiGeSn GeSn2 Ge3Sn

1

A1 1 A′ 1 A1 1 A1

Cs Cs C2v C2v

Ge2Sn2

1

Ag

D2h

GeSn3

1

A1

C2v

Si2GeSn

1

A1

C2v

SiGe2Sn

1

A′

Cs

SiGeSn2

1

A′

Cs

rSi−Sic

rGe−Gec

rGe−Sic

2.315 2.253 2.463 (2) [2.644] [2.672]

μ

HOMO− LUMO gap

1.19 1.19 1.10 1.02

2.24 2.40 2.22 2.29

0.00

2.32

0.31

2.14

2.600 (2)

1.14

2.32

2.704

2.571

1.09

2.33

2.625 [2.805]

2.661

1.67

2.07

rGe−Snc 2.503, 3.250 3.226 2.503 2.667 (2)

rSi−Snc 2.446

3.437

2.657 (4) 2.618 (2) [2.847]

[2.466]

2.391 (2) 2.498

rSn−Snc

2.363 [2.556] 2.343

2.904 (2)

2.928

ωe 86(A′), 244(A′), 297(A′) 95(A′), 363(A′), 424(A′) 75(A1), 240(A1), 250(B2) 50(B1), 109(B2), 159(A1) 223(B2), 228(A1), 268(A1) 44(B3u), 98(B2u), 138(Ag) 20(B3 g), 220(Ag), 237(B1u) 41(B1), 92(A1), 131(A1) 156(B2), 197(A1), 240(B2) 67(B1), 103(B2), 118(A1) 151(B2), 156(A1), 205(A1) 120(A′), 127(A′), 175(A′) 275(A′), 299(A′), 458(A″) 57(A″), 111(A′), 165(A′) 200(A′), 236(A′), 372(A′)

a Interatomic distances (r) in Å, dipole moments (μ) in Debye, HOMO−LUMO gap in eV, harmonic frequencies (ωe) in cm−1. bThe corresponding structures are displayed in Figure 1 and in the upper part of Figure 2. cWhenever necessary, the number of identical bond distances is given in parentheses. Diagonal bond lengths are given within square brackets.

similar to that used for the corresponding Si−Sn28 species. Indeed, on energetic grounds, it can be argued that the difference in the atomization energies of Ge335b (589.8 kJ/mol) and Sn328 (440 kJ/mol) would favor the preservation of the Ge3 frame in the Ge3Sn molecule (although the Ge−Ge diagonal distance of 2.644 Å is slightly larger than the chosen conventional cutoff value of 2.60 Å), whereas the number of Ge−Sn interactions would be maximized in the GeSn3 structure. In comparing our results with those recently made available21 a situation similar to the above-discussed triatomic clusters emerges: the C2v fan-like ground state of Ge3Sn reported in ref 21, where the Ge3 frame is lost, corresponds to our isomer 0.18 eV above the ground state depicted in Figure 2 (see also Table S2 in the Supporting Information); the structure determined in ref 21 for the ground state of GeSn3

Sn and Ge−Sn, is rather largely destabilized and found at 0.92 eV. 3.2. Tetratomic Species. In the upper part of Figure 2, we present the structures of the Ge−Sn and Si−Ge−Sn tetratomic molecules up to 0.5 eV, subsequently used in the evaluation of their thermodynamic properties (vide infra). The molecular parameters of the ground states are reported in Table 3. Compared to the triatomic species, the number of possible isomers increases considerably. All the molecules are found to be planar in their ground states as was already found for the Si− Sn28 and Si−Ge22,23,25 analogues. This is an interesting result because the onset of a tridimensional evolution could in principle appear in the tetratomic molecules. With regard to the Ge−Sn species, the ground state structures (Figure 2) can be discussed along a line of reasoning 7110



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Table 4. Molecular Parameters of the Ground States of the Pentatomic Species of the Ge−Sn, Si−Ge−Sn, and Si−Sn Systems Calculated at the B3LYP/cc-pVTZ Level (For Higher Energy States, See the Supporting Information)a,b rGe−Gec

rGe−Sic

rGe−Snc

rSi−Snc

state

Ge4Sn

1

C2v

3.288 [2.475] (4)

3.490 (2) [2.659] (2)

Ge3Sn2

1

C2v

[2.479] (2)

3.472 (2)

A1

A1

symmetry

rSi−Sic

molecule

rSn−Snc

3.671

μ

gap

ωe

0.72

2.94

82(B1), 100(A1), 121(A1), 160(B2), 178(A2), 203(B2),

0.68

2.92

[2.664] (4) Ge2Sn3

1

D3h

[2.669] (6)

3.650 (3)

0.00

3.01

GeSn4

1

C3v

[2.655] (3)

3.722 (3) [2.863] (3)

0.21

2.70

Si3GeSn

1

Cs

1.12

3.05

A′

A1

A′

[2.333] (2)

3.198

3.445

3.347

[2.406] (2) SiGe3Sn

1

A′

Cs

3.263

[2.396] (2)

[2.488] (2) SiGeSn3

1

C3v

Si2Ge2Sn

1

C2v

Si2GeSn2

A1

A1

1

A1

3.239

C2v

[2.601] (2)

3.465 (2)

[2.411] (2)

[2.589]

2.683 (3)

2.599 (3)

3.440 (2)

[2.601] (2)

3.427 (2)

[2.607] (4)

1

Cs

Si4Sn

1

C2v

3.098 [2.332] (4)

3.379 (2) [2.599] (2)

Si3Sn2

1

C2v

[2.335] (2)

3.365 (2)

A1

A1

[2.493]

0.75

3.06

[2.398]

3.449 (2) [2.677] (2)

[2.595] (2)

3.629 (3)

3.624

3.647

3.626

0.04

3.07

0.78

3.19

292(A′), 312(A″), 342(A′) 79(E), 79(E), 107(A1), 165(E), 165(E), 185(A1), 276(E), 276(E), 300(A1) 97(B1), 112(A1), 139(A1),

3.06

243(B2), 258(A2), 298(B2), 310(A1), 327(B1), 366(A1) 85(A1), 103(B1), 125(A1),

0.74

0.71

3.02

1.34

2.99

1.25

2.91

[2.606] (4) A1′

Si2Sn3

1

D3h

[2.611] (6)

3.609 (2)

0.00

3.12

SiSn4

1

C3v

[2.587] (3)

3.697 (3) [2.876] (3)

0.23

2.79

A1

350(A′), 357(A″), 438(A′) 89(A″), 106(A′), 130(A′), 175(A′), 193(A″), 218(A′),

SiGe2Sn2

A′

157(A2), 158(B2), 193(B2), 207(B1), 213(A1), 243(A1) 74(E′), 74(E′), 102(A1′), 152(E″), 152(E″), 170(A2″), 202(E′), 202(E′), 216(A1′) 66 (E), 66(E), 90(A1), 133(E), 133(E), 151(A1), 193(E), 193(E), 201(A1) 103(A′), 149(A′), 165(A′), 245 (A″), 274(A″), 314(A′),

[2.672]

[2.408] (4)

213(A1), 239(B1), 252(A1) 73(A1), 92(B1), 110(A1),

236(B2), 239(A2), 283(B2), 299(B1), 308(A1), 349(A1) 79(A′), 97(A″), 118(A′), 170(A″), 173(A′), 207(A′), 283(A″), 288(A′), 327(A′) 128(B1), 158(A1), 195(A1), 246(B2), 323(A1), 330(A2), 376(B2), 427(B1), 449(A1) 88(A1), 145(B1), 150(A1), 241(A2), 241(B2), 301(B1), 323(A1), 353(B2), 431(A1) 84(E′), 84(E′), 113(A1′), 230(E″), 230(E″), 257(A2″), 292(E′), 292(E′), 325(A1′) 71(E), 71(E), 96(A1), 135(E), 135(E), 156(A1), 282(E), 282(E), 292(A1)

Interatomic distances (r) in Å, dipole moments (μ) in Debye, HOMO−LUMO gap in eV, harmonic frequencies (ωe) in cm−1. bThe corresponding structures are displayed in the lower part of Figure 2 and in Figure 3. cDistances, with reference to the trigonal bipyramid structures of Figure 3, are reported as follows: the bond lengths between the apex and base atoms are given in square brackets; the number of identical bond distances is given in parentheses. a

the presence, either in the HOMO − 2 or in the HOMO − 3 orbitals, of a π-type orbital in which two electrons are delocalized over the four atoms (see the orbital pictures in the Supporting Information). With regard to the higher energy isomers, their type and sequence are also the same as observed

is in agreement with ours. Our lowest energy isomer with the Ge3 frame preserved was apparently not identified in ref 21. For the Ge2Sn2 molecule, the ring is stabilized by the Ge−Sn interactions. The planarity of these molecules can be traced back, as was observed for the Ge−Si23 and Si−Sn28 species, to 7111



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for the corresponding Si−Sn species.28 The range of the singlet state energies is here compressed compared to the Si−Sn case, similarly to what happens for the triatomic molecules. As in the Si−Sn system, two tridimensional structures of higher energy have been also found within 1 eV (see Figure S2 in the Supporting Information). The D2h ground state structure of Ge2Sn2 is not reported in ref 21. At this point, we note that, for three Ge−Sn species (Ge2Sn, Ge3Sn, and Ge2Sn2), we find ground states that have been apparently overlooked in ref 21. This can be possibly related to the procedure they used. Indeed, in order to reduce the gamut of possible structures to be optimized by DFT, in ref 21, a two-step strategy was adopted, based on a preliminary screening performed by the Metropolis Monte Carlo and Genetic Algorithm approaches. In the ternary tetratomic Si−Ge−Sn molecules (see again Figure 2), the previously invoked energetic considerations can be summarized by observing that, in the ground state structures, the silicon and germanium atoms preferentially occupy the highest possible coordination position, as a result of the greater strength of their chemical bonds. Nevertheless, as already noted, this does not produce a tridimensional structure: the shorter bonds of planar structures are always preferred. As a consequence, a symmetrical C2v structure occurs for the only Si2GeSn species. In other words, it can be said that, in the ternary species, the only diagonal homonuclear bond that is preserved in building the tetratomic structures is the strong silicon−silicon bond. Finally, we note that, as better detailed in section 3.4, in all the tetratomic species, but SiGeSn2, the atoms on the shorter diagonal are negatively charged as already was found in the corresponding clusters of the Si−Sn28 and Si− Ge23 systems. 3.3. Pentatomic Species. The pentatomic species here studied are those of the Ge−Sn and Si−Ge−Sn systems, as well as those of the Si−Sn system, the latter to complement our previous experimental and computational study of the smaller Si−Sn molecules.28 The molecular parameters of the most stable isomers are reported in Table 4, and the structures within 0.5 eV from the ground state are displayed in the lower part of Figure 2 for the ternary Si−Ge−Sn species and in Figure 3 for the binary Ge−Sn and Si−Sn species. The complete computational results of all the isomers with energy within 1 eV from the ground states and spin multiplicity 1 and 3 can be found in the Supporting Information. On increasing the size of the microclusters, the number of possible geometries increases substantially. Within the limit of 1 eV, the structures found for all the pentatomic species are of the trigonal bipyramid type. While, in the case of the Ge−Sn system, all the possible singlet isomers based on the bipyramidal frame lie in this energy range, the same does not occur for the Si−Sn species, due to the aforementioned larger energy spacings. As a general rule, in the ground state structures, the germanium or silicon atoms are placed in the apical positions, thus maximizing the number of the Ge−Sn and Si−Sn bonds. The ground state structures we found for the Ge−Sn pentatomic species agree well with the recent results of ref 21, whereas, within the energy region of 0.5 eV here of primary interest, we find some additional stable structures. In the Si−Ge−Sn pentatomic molecules, the ground state structures follow the same rule with silicon, first, and germanium, second, preferentially occupying apical positions. The ternary nature of these species reduces the symmetry and the resulting trigonal bipyramids are, even if slightly, distorted. Nevertheless, the energetic balance between the various bonds

Figure 3. Structure and relative energy of the pentatomic Ge−Sn and Si−Sn species obtained at the B3LYP/cc-pVTZ level of theory; Si in black, Ge in gray, and Sn in white. All the isomers shown are singlets with the exception of the triplet state of Si4Sn at 0.47 eV.

generates a number of symmetrical structures, with the C3v SiGeSn3 as the most symmetrical one. The overall pattern of the ground state symmetries can be better appreciated in Figure 4 where they are reported in a diagram reminiscent of the

Figure 4. Symmetries of the ground state structures of the pentatomic clusters.

representation used for ternary phase diagrams in alloy systems. In constructing this figure, the structures previously published for the Si−Ge22,23,25 pentatomics have been used in addition to the results obtained here, after checking them by reoptimizing the geometries with the cc-pVTZ basis set here employed. Two islets of symmetrical geometries are shown in the diagram, which can be traced back to the presence of silicon or germanium atoms in apical positions. It is of interest to note here that similar trigonal bipyramidal structures are also observed in the propellanes known for 7112



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Table 5. NPA Charges (e) Computed at the B3LYP/cc-pVTZ(-PP) Level molecule

Si-1

SiGeSn Si2GeSn SiGe2Sn SiGeSn2 Si3GeSn SiGe3Sn SiGeSn3 Si2Ge2Sn Si2GeSn2 SiGe2Sn2

−0.558 −0.355 −0.415 0.042 −0.324 −0.379 −0.613 −0.340 −0.453 −0.493

Si-2

Si-3

−0.355

−0.324

−0.340 −0.453

0.069

Ge-1 0.145 0.229 0.209 −0.432 0.164 −0.276 −0.471 0.138 0.123 −0.370

Ge-2

Ge-3

−0.243

0.134 0.138 0.116

0.134

Sn-1 0.413 0.482 0.448 0.346 0.415 0.387 0.361 0.404 0.391 0.373

Sn-2

Sn-3

0.043

0.361

0.361

0.391 0.373

considered to be related to the strength required to perturb the electronic structure and therefore to the chemical reactivity. We note that, in all cases, the gap calculated for the ground state structure is smaller than that of the corresponding first higher energy isomer, the only exception being the Ge2Sn2 species. In Figures 5 and 6, we inquiry into the trends of the computed gap on varying the composition. In particular, Figure

silicon, germanium, and tin; moreover, comparable structures have been found in larger clusters as the Ge14 species in metalloid compounds.37 3.4. Charges and HOMO−LUMO Gaps. Some further insight into the molecular and electronic structure of the species under study can be gained with the outcomes of the simple DFT treatment at the triple-ζ level. NPA (Natural Population Analysis) charges have been evaluated with the B3LYP functional and the cc-pVTZ(-PP) basis sets. For simplicity, details are here reported for the only ternary species (Table 5). Silicon is found to be negative in almost all the cases and tin positive in all cases. The charge of germanium is dependent on its position in the molecular skeleton. A simple general rule can be drawn from the charge values of Table 5. With the exception of the triatomic SiGeSn molecule, all the species here considered have planar quadrilateral or bipyramidal structures, for tetratomic and pentatomic molecules, respectively. In both the structure types, it is possible to identify two atoms at the opposite sides along the shortest distance; in all the pentatomic molecules, this corresponds to the apical positions. The general rule is that these very two atoms defined above are found to be negatively charged irrespectively of their nature. It is to be mentioned that this finding has been reported by Wielgus et al.23 for the similar tetratomic GexSiy species. In our case, the only exception is the tin atom in the SiGeSn2 molecule, which, however, is only slightly positive. The simple electronegativity concept does not allow to predict the observed charge transfer. Indeed, in all the various scales available, the electronegativity increases from silicon to germanium and decreases from germanium to tin (1.90, 2.01, and 1.96, Pauling; 1.74, 2.02, and 1.72, AllredRochow; 2.25, 2.50, and 2,44, Mulliken; 1.916, 1.994, and 1.824, Allen). The same behavior is also found in all the GexSny and SixSny molecules here studied, where the apical or shortest diagonal positions of the molecular skeleton are preferentially occupied by Si, Ge, and Sn atoms in this order. As a consequence, when in such positions, atoms are always negatively charged. The HOMO−LUMO gaps derived from the DFT calculations by the simple Kohn−Sham orbital eigenvalues are reported in Tables 3 and 4 for the ground states, as well as in Tables S1−S5 of the Supporting Information for the higher energy isomers. These values represent a rather crude estimate. Indeed, besides the limited significance that can be attributed to the eigenvalues of the individual orbitals in the framework of the DFT approach, some sort of scaling should probably be applied38,39 to get quantitative information. Nevertheless, the behavior of the as such gap values on varying the composition of the clusters can be of interest. Indeed, the width of the gap is

Figure 5. Trends of the HOMO−LUMO gaps computed at the B3LYP/cc-pVTZ level on varying the composition of binary tetra- and pentatomic clusters.

5 illustrates the behavior of the gap for the pentatomic and tetratomic molecules on progressively substituting one of the atomic species with the other while keeping constant the number of atoms. Maxima are observed for the species with the highest symmetry: namely, the D3h structures of Si2Ge3, Ge2Sn3, and Si2Sn3 for the pentatomic molecules (Figure 4); the D2h structure for the Me2Me2′ molecules for the tetratomic ones. These molecular species, being, accordingly to the HOMO− LUMO gaps, the most stable ones with respect to the neighbors, are on this basis predicted to play a role as building blocks of the corresponding novel materials. In Figure 6, the effect of the progressive addition of a single atomic species to both binary and ternary species is studied, starting from a triatomic molecule. Within the limits of the very short series available, the initial part of a probably alternating behavior can be seen. This behavior, already observed for a number of small clusters,40 is here, to the best of our knowledge, for the first time described for ternary species. In both the sequences 7113



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molecules. To this end, the energy difference between the lowest J level of the3P state and the J-averaged 3P state,41 as calculated from the experimental atomic data of ref 45, has been employed. The resulting values in kJ/mol are 1.79, 11.59, and 29.53 for the Si, Ge, and Sn atoms, respectively. Mixing between the 3P0 and 1S0 states as well as the 3P2 and 1D2 states was not taken into account because it would have needed a correction based on first principles computation and would have moderately affected only the tin atom. The spin−orbit coupling in the molecular species, which were all found to have closed shell singlet ground states, should be partially quenched by the molecular field and, in view of its likely smaller importance, has been neglected. In the case of the homonuclear trimers Si3,28,46−50 Ge3,51−53 and Sn3,28,54,55 singlet and triplet states were reported to compete for the ground state, with singlet states generally favored except for Sn3 (see our previous work28,56 for a brief discussion). Regarding the completeness of the basis sets, an extrapolation of the calculated atomization energies to the complete basis set (CBS) limit can be done provided that a consistent series of basis sets is employed. As detailed in the computational methods section, to address the CBS extrapolation, CCSD(T) computations with double, triple, and quadruple-ζ quality basis sets were done at the B3LYP/cc-pVTZ optimized geometries reported in Tables 3 and 4. This simplified procedure has been proved to work well in the case of the SixSny molecules (x + y ≤ 4).28 The extrapolation was done using two methods: the mixed exponential/Gaussian CBS formula proposed in ref 57 and the n−3 dependence58 where n is the cardinal number of the cc-pVXZ basis sets. Total energies were used in both cases. For simplicity, only the mixed exponential/Gaussian data are reported in Table 6. Results of the second method were found to be only very slightly larger within a factor of 1.01. The atomization energies (ΔatH) of the tri-, tetra-, and pentatomic species of the Si−Ge−Sn system calculated with both CBS CCSD(T) and B3LYP DFT methods are reported in Table 6. From a comparison with the few experimental values available (see Table 6), an overestimation of the atomization energies provided by the CBS extrapolated CCSD(T) values is apparent. The deviation from the experimental values increases on going from Si to Ge and Sn clusters in a rather linear way. This behavior parallels that observed for the SixSny species, whose results, for ease of comparison, are also reported in Table 6. This increase of the deviations going down in the group 14 and, therefore, with increasingly heavy atoms, is interesting and calls for an explanation. Most probably it is correlated with the increase of the importance of relativistic effects in the chemical bonds. Indeed, as an example, the bond is known to be progressively destabilized by relativistic effects along the group 14, with a decrease in the dissociation energy of homonuclear diatomics up to half its nonrelativistic value in Pb2.59 In spite of the scalar contribution and spin−orbit corrections adopted, which largely reduce in a systematic way the computed atomization energies, agreement with the few experimental data available is not achieved. It is to be concluded that these species seem to require a more rigorous relativistic treatment. It can also be noted that, of the two methods here adopted in order to evaluate the spin−orbit corrections, SO2 is found to provide final results in better agreement with the available experimental values. In such a situation, taking into account that our final goal is to evaluate the overall thermodynamic properties of the molecular species as well as

Figure 6. Trends of the HOMO−LUMO gaps computed at the B3LYP/cc-pVTZ level on increasing the size of the binary and ternary species.

reported in Figure 6, a minimum is observed for the tetratomic species, which are therefore predicted as the most reactive ones. The overall behavior of the energy gaps on varying the composition of the species can also be appreciated, here once more, in the ternary-like diagrams reported in the Supporting Information. The issue of the chemical reactivity of the Si−Ge− Sn species will also be addressed in the thermodynamic section (vide infra) taking into account also the entropic contributions, so evaluating this property under equilibrium conditions. Furthermore, in the same section, possible growth paths will be examined on the basis of the exothermicity concept.

4. ENERGETICS 4.1. Atomization Energies. In evaluating the atomization energies derived from our computations, after deriving the ΔatH0 values with the zero-point energies evaluated with the harmonic frequencies reported in Tables 3 and 4, it is necessary to deal with two issues: the spin−orbit coupling for atoms and molecules41 and the completeness of the basis sets. In our calculations, relativistic effects are taken into account by the use of core (spin-free) scalar pseudopotentials. Two approaches have been used in order to evaluate the spin dependent relativistic contribution. In the first approximation,42 hereafter called SO1, it has been taken advantage of effective core potentials43 developed to include both the scalar (spinfree) and the spin−orbit (spin-dependent) relativistic effects. To this end, it has been used the capability of the NWChem software package32 to deal with this type of ECPs through the SODFT module. The corrections in energy, ΔESO, were evaluated at the B3LYP/cc-pVTZ-PP level as the differences in the atomization energy computed with the SO-ECPs and with the ECPs including only the scalar contribution. Following Zaitsevskii et al.,44 this correction could be further approximately incorporated into the results of the scalar calculations by adding the DFT ΔESO to the CCSD(T) potential curves. In our case, an additional simplification was used because we work, as previously detailed, with single point CCSD(T)// B3LYP energy values, which makes it unnecessary to evaluate the ΔESO corrections as a function of the geometry. In the second approach, in the following called SO2, the spin−orbit correction was included by adjusting the calculated atomization energies with estimated spin−orbit couplings of atoms and 7114



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Table 6. Atomization Energies in kJ/mol: As Such (ΔatHe), Adjusted for the Zero Point Energy (ΔatH0), Corrected for the Spin−Orbit Coupling (See Text) with Two Approximations (ΔatHe-SO1) and (ΔatHe-SO2), Empirically Adjusted (ΔatH0,adj, See Text), and Experimental (ΔatH0,exptl)a ΔatHe Si3 Ge3 Ge2Sn GeSn2 Sn3 Si2Sn SiSn2 SiGeSn

Si4 Ge4 Ge3Sn Ge2Sn2 GeSn3 Si3Sn Si2Sn2 SiSn3 Sn4 Si2GeSn SiGe2Sn SiGeSn2

Si5 Ge5 Ge4Sn Ge3Sn2 Ge2Sn3 GeSn4 Sn5 Si4Sn

ΔatH0

677.3 720.4 586.1 672.6 563.0 655.1 538.2 636.4 489.4 589.6 623.5 686.4 572.6 650.0 603.7 674.3

669.9 713.0 582.0 668.5 559.2 651.4 534.8 633.1 487.0 587.1 617.3 680.3 567.8 645.2 598.4 669.1

1084.2 1165.2 951.2 1115.4 917.6 1085.2 882.2 1053.5 842.2 1018.0 1016.1 1118.3 943.5 1069.4 873.7 1025.5 798.8 982.0 987.1 1106.6 952.6 1095.4 890.6 1045.9

1072 1153 944.5 1108.7 911.4 1079.0 876.5 1047.8 837.0 1012.9 1005.8 1108.0 935.1 1061.0 867.2 1019.0 794.2 977.4 978.0 1097.5 944.9 1087.7 883.8 1039.0

1426.4 1554.9 1256.1 1510.5 1223.7 1482.8 1191.0 1455.5 1158.1 1428.9 1109.3 1389.5 1359.2 1514.5 1058.1 1349.0

1410.7 1539.2 1246.2 1500.6 1214.4 1473.5 1182.3 1446.9 1150.1 1420.8 1102.0 1382.2 1343.5 1498.7 1051.5 1342.4

ΔatH0-SO1 Triatomic Species 669.9 713.0 565.4 651.9 530.1 622.3 493.5 591.8 434.3 534.4 599.3 662.3 531.2 608.6 574.4 645.1 Tetratomic Species 1072 1153 922.3 1086.5 876.3 1043.9 828.5 999.8 775.9 951.8 986.9 1089.1 897.4 1023.3 810.9 962.7 721.6 904.8 953.4 1072.9 915.0 1057.8 840.7 995.9 Pentatomic Species 1410.7 1539.2 1217.4 1471.8 1171.6 1430.7 1125.3 1389.9 1078.6 1349.3 1016.6 1296.8 1324.0 1479.2 952.1 1243.0 7115

ΔatH0-SO2 664.5 707.6 547.2 633.7 506.5 598.7 464.2 562.4 398.4 498.5 584.2 647.1 506.9 584.4 555.5 626.2 1064.8 1145.8 898.1 1062.3 847.1 1014.7 794.3 965.5 736.9 912.7 970.9 1073.1 872.4 998.3 776.8 928.6 676.1 859.3 933.3 1052.8 890.4 1033.2 811.4 966.6 1401.7 1530.2 1188.3 1442.6 1138.5 1397.6 1088.5 1353.0 1038.3 1309.1 972.2 1252.4 1306.8 1462.0 903.9 1194.7

ΔatH0,adjb

ΔatH0,exptl 716.5

716.5 589.8 589.8 579 549.9 521 508.8 440 440.0 625.6 633.5 550.2 548.4 595.0 1160 1160.0 969.8 969.8 954 (upper limit) 918.0 864.7 807.8 1039.2c 1056.5 955.2 950.9 860.2 850.3 750.2 750.2 1009.5 963.2 892.5 1559 1559.0 1313 1313.0 1300 (upper limit) 1253.0 1193.4 1134.4 1062.8 990 990.0 1444.1 dx.doi.org/10.1021/jp300624z | J. Phys. Chem. A 2012, 116, 7107−7122


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Table 6. continued ΔatHe Si3Sn2 Si2Sn3 SiSn4 Si3GeSn SiGe3Sn SiGeSn3 Si2Ge2Sn SiGe2Sn2 Si2GeSn2

1290.8 1474.0 1220.8 1433.4 1144.4 1393.7 1331.7 1509.0 1265.4 1493.6 1190.6 1431.4 1304.4 1503.5 1228.1 1462.2 1262.7 1468.2

ΔatH0 1277.2 1460.4 1209.4 1422.0 1135.3 1384.6 1317.4 1494.7 1254.3 1482.5 1180.9 1421.6 1291.5 1490.7 1217.7 1451.8 1250.6 1456.0

ΔatH0-SO1

ΔatH0-SO2

Pentatomic Species 1237.5 1420.7 1148.7 1361.3 1055.5 1304.8 1291.9 1469.2 1217.1 1445.3 1114.9 1355.6 1259.8 1459.0 1166.2 1400.3 1204.6 1410.0

1212.8 1395.9 1117.3 1329.8 1015.4 1264.7 1270.9 1448.2 1188.2 1416.4 1078.9 1319.6 1235.2 1434.4 1133.7 1367.8 1176.4 1381.8

ΔatH0,adjb

ΔatH0,exptl

1331.3 1218.5 1106.7 1398.6 1303.5 1176.6 1353.1 1239.8 1285.5

For each molecule are reported in the first line the B3LYP values calculated with the cc-pVTZ basis sets; in the second line the CBS extrapolated values calculated with the CCSD(T) method at the B3LYP optimized geometry. The CBS extrapolation has been made with a three-point formula using the cc-pVDZ, cc-pVTZ, and cc-pVQZ basis sets (see text). bThe ΔatH0,adj values are the recommended atomization energies. cThis value is a correction of that (1046.1 kJ/mol) reported in ref 28 where an error occurred in the calculation of the thermodynamic function for this species. The revised value is still well within the reported error limits of ±19.9 kJ/mol. a

In our case, the average binding energy can be written, for example, Eb(n) = ΔatH0°(SixGeySnz)/(n) (n = x + y + z) for the ternary mixed clusters, and Eb(n) = ΔatH0°(GeySnz)/(n) (n = y + z) for the binary species. For the homonuclear species, the binding energy is simply the ratio between the atomization energy and the number of atoms; again, for example: Eb(n) = ΔatH0°(Gen)/n. For the binary species, the trend of Eb with the number of atoms can be followed along a series representing the doping of homonuclear clusters of increasing size with an extra atom, as reported in the lower part of Figure 7. The

their abundances under equilibrium conditions, and considering that the sought abundances are largely dependent on the atomization energies, we deemed it useful to devise a simple method, which could provide an empirical refinement of the calculated atomization energies. To this end, the deviations of the calculated atomizations (ΔatH0-SO2) of the homonuclear trimers, tetramers, and pentamers, for which experimental values are available, were evaluated ((Δ[ΔatH0](Sin), Δ[ΔatH0](Gen), Δ[ΔatH0](Snn); n = 3, 4, and 5), and a simple linear interpolation was subsequently used to estimate the corrections to be applied to the mixed species (x + y + z = n): Δ[Δat H 0(Si xGe ySnz)] = (x/n)Δ[Δat H0 (Si n)] + (y/n)Δ[ΔatH0(Gen)] + (z/n)Δ[ΔatH0(Snn)]. The resulting values (ΔatH0,adj) are reported in column 6 of Table 6 and are here proposed as the most reliable evaluation of the atomization energies. As already mentioned, in Table 6, we also report the atomization energies calculated at the B3LYP/cc-pVTZ level. After applying the SO correction, the DFT values, lacking a full account of the correlation, are always lower than the experimental data. A more detailed analysis shows that the average of the absolute deviations of these values from the experimental data is somewhat larger than in the case of the CCSD(T) results. Moreover, the DFT deviations are greater in a larger number of cases. On the contrary, the uncorrected DFT results approximate better the known experimental values.28 A partial compensation of the neglect of spin−orbit coupling and the incomplete account of correlation could be at work. 4.2. Stability. The overall picture of the relative stability of the molecular species presented in Table 6 can be analyzed by the concepts commonly used in cluster science. The relative stability of the clusters of different size can be analyzed by use of the average binding energy, Eb(n), and the fragmentation energy, D(n,n − 1).

Figure 7. Trend of the calculated binding energies (see text) with the cluster size. 7116



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ΔEmix = −⎡⎣Δat H0°(SixGeySnz)

expected behavior is observed in this series, with the growth process occurring with increasing energy. By comparing the Eb(n) values of the binary species with those of the homonuclear ones, it is apparent how the doping with tin on silicon or germanium clusters, especially the smaller ones, lowers their stability. On the contrary, once again as expected, doping with silicon or germanium increases the stability of tin clusters to a progressively lower extent for larger clusters. By extending this approach, we report in Figure 7 also the analogous trend for the ternary clusters, where the role of the dopant species is played in this case by a pair of different atoms. We note that, in this series, the energetic effect accompanying the doping of germanium clusters is negligible. The fragmentation energy is given by the simple differences of the atomization energies along a series; for example: D(n,n − 1) = ΔatH0°(GenSn) − ΔatH0°(Gen−1Sn) for a binary species or D(n,n − 1) = ΔatH0°(SinGeySnz) − ΔatH0°(Sin−1GeySnz) for a ternary one. The resulting values are reported in Figure 8.

− Δat H0°(SixSi ySiz)x /(x + y + z) − Δat H0°(GexGeyGez)y/(x + y + z) − Δat H0°(SnxSn ySnz)z /(x + y + z)⎤⎦ /(x + y + z)

A negative value of this parameter indicates a cluster that is energetically stable with respect to the homonuclear elemental clusters of similar size. The resulting values are reported in Figures 9 and 10 in the same diagrammatic form as used in

Figure 8. Trend of the calculated fragmentation energies (see text) with the cluster size.

Besides the maximum previously observed28 in the SiSnn sequence, a distinctive maximum for n = 4 is apparent in all except the SiGeSnz series. This is a clear indication of a greater relative stability. Interestingly, these tetratomic clusters, where the onset of a three-dimensional evolution might be found are, on the contrary, calculated to be always planar in their ground states. The special stability of these clusters seems therefore related to their capability to retain the planar structure. However, the short series available from the present calculations do not permit further investigation. Another concept sometimes used in cluster science is the mixing energy, which gives an indication of the mixing vs segregation tendency of the components. The mixing energy for a binary system, defined in ref 60, attempts to measure the change in energy occurring when two Men and Mem′ fragments, removed form the homonuclear MenMem and Men′Mem′ clusters, are brought together to form the MenMem′ mixed species. By extending this idea to our ternary system, the mixing energies per atom22 can be written as

Figure 9. Mixing energies in kJ/mol for the triatomic (upper diagram) and tetratomic (lower diagram) species.

Figure 4. It can be seen that, while a general stabilization is reached in the mixed triatomic and pentatomic clusters, two notable exceptions occur for the Si3Sn and SiGeSn2 tetratomic species.

5. THERMODYNAMICS 5.1. Analysis of Equilibrium Abundances. The molecular parameters here calculated at the B3LYP/cc-pVTZ level open the way for a thermodynamic analysis of the species under study. First, the thermodynamic functions can be computed with the methods of statistical thermodynamics. In particular, the heat content functions (HCF0 = HT° − H0°) and the Gibbs energy functions (GEF0 = −(GT° − H0°)/T) have been evaluated by taking into account the isomers located up to 0.5 eV from the ground state. All the singlet and triplet isomers have been treated, for simplicity, as truly different electronic 7117



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+ Sn2(g) = GeSn2(g) + Ge(g)). With the dissociation energies D0,at°(Ge2) = 260.7 kJ/mol35a and D0,at°(Sn2) = 183.4 kJ/ mol,36 we finally obtain the updated values ΔatH0°(Ge2Sn) = 579 kJ/mol and ΔatH0°(GeSn2) = 521 kJ/mol, still almost within the originally proposed error limits. The upper limits for the atomization energies of the Ge3Sn and Ge4Sn molecules are similarly revised to 957 and 1309 kJ/mol, respectively The above derived thermodynamic properties provide us with quantitative information on the thermodynamic stability of each species and thus permit an evaluation of their relative abundances under equilibrium conditions. This information is important to predict the role that the binary and ternary species can play both in the thermal stability and in the vapor deposition processes of the corresponding materials. To this end, the thermodynamic properties of the homonuclear Sin, Gen, and Snn as well as of the binary Si−Ge species up to five atoms were also taken into account. For the Sin, Gen, and Snn, the pertinent data were taken from refs 28 (Si2, Si3, Sn2, and Sn3) and 56 (Ge2 and Ge3), evaluated from molecular parameters reported previously28 (Si4 and Sn4), or obtained here with B3LYP calculations (Si5, Sn5,Ge4, and Ge5). The experimental atomization energies of a number of Si−Ge species were revised by Wielgus et al.,23 and their proposed values have been employed here together with the Gibbs energy functions evaluated by the same authors. For the other Si−Ge species, where no experimental values are available, an average of the atomization energies proposed in ref 23 has been adopted and empirically adjusted to smooth the trend with the number of atoms, also including the homonuclear species: 1144, 1063, 1496, 1459, 1423, and 1371 kJ/mol for GeSi3, Ge3Si, GeSi4, Ge2Si3, Ge3Si2, and Ge4Si, respectively. In these cases, for the sake of consistency, the Gibbs energy functions have been evaluated by recalculating the molecular parameters at the B3LYP/ccpVTZ level. The equilibrium calculations were carried out with the Thermo-Calc software package,63 which is based on the minimization of the Gibbs free energy for a given set of constraints. A total of 55 species were included in the database, i.e., the three atomic species and all the binary and ternary species containing silicon, germanium, and tin up to the pentatomic clusters. In setting the initial conditions for the equilibrium calculations, an obvious choice is to use the equilibrium vapor pressures of the three pure elements at a given temperature. Experimentally, this would correspond to withdrawing the saturated vapor from three sources, each containing an element held at the temperature of interest and permitting the mixed vapor to attain chemical equilibrium at the same temperature. Computations were performed in the range of temperature 1600−2200 K, with the corresponding initial pressures in the ranges 5.1 × 10−5−2.2 × 10−2, 5.0 × 10−6−4.9 × 10−3, and 6.3 × 10−8−2.7 × 10−4 bar for tin, germanium, and silicon, respectively. The resulting behavior of the equilibrium molar fractions is represented in Figure 11 where, in order to keep the plot understandable, the atomic species are not reported, and the aggregated species were grouped on the basis of the total number of atoms (upper curves) and of the constituent elements (lower curves) (a complete list of the calculated molar fractions and partial pressures at 2000 K is provided in the Supporting Information). As expected, the aggregated species are found to be, in the temperature range explored, between about 2 and 9% of the total pressure, with the atoms largely prevailing. Using as representative the data at 2000 K, the di-, tri-, tetra-, and

Figure 10. Mixing energies in kJ/mol for the pentatomic species.

levels. Indeed, a study of the possible interconversion and related barriers between the different isomers would largely complicate the problem without significantly improving the overall accuracy of both the thermal function evaluation and the equilibrium predictions, the latter being mainly limited by uncertainties in the atomization energies. For the very same reason, the simple rigid rotor−harmonic oscillator approximation has been employed. On the contrary, whenever relevant, the HCF0 and GEF0 have been evaluated by taking into account the interdependence of electronic and internal motions and, therefore, making use of the vibrational and rotational parameters of the pertinent electronic state. In this approach,61 the partition function Q is expressed by the formula Q = Qtr(T,p)∑iqrot,i qvib,i gi e−Δεi/RT, where Qtr(T,p) is the translational partition function, and qrot,i, qvib,i, gi, and Δεi are the rotational partition function, the vibrational partition function, the degeneracy, and the energy with respect to the ground state of the ith electronic state. This refinement is needed, in particular, when the symmetry number of the structures considered is different.28 The species of interest here, where this more accurate approach has been adopted, were Ge2Sn, Ge2Sn2, Ge2Sn3, Si2Sn3, Si2GeSn, SiGe2Sn, SiGeSn3, Si2Ge2Sn, Si2GeSn2, and SiGe2Sn2. The complete list of the thermal functions at various temperatures for the species GexSny (x + y = 2,3,4,5), SixSny (x + y = 5) and SixGeySnz (x + y + z = 3,4,5) is reported in the Supporting Information. As already mentioned, an experimental study17 is available on some of the GexSny species here studied. In that article, the original KEMS (Knudsen Effusion Mass Spectrometry) data were analyzed with thermodynamic functions evaluated with intuitive structures and estimated bond distances and force constants. The proposed values of the atomization energies were 490 ± 22 kJ/mol (GeSn2) and 559 ± 27 kJ/mol (Ge2Sn). Upper limits were also given for the Ge3Sn and Ge4Sn species as 996 ± 32 kJ/mol and 1389 ± 42 kJ/mol, respectively. The more reliable thermodynamic functions here computed allow a reanalysis of the original data. With our Gibbs energy functions for the GexSny species and those taken from the IVTANTHERMO database62 for Ge(g), Sn(g), Ge2(g), and Sn2(g), we obtain the third-law enthalpies of reactions −57.4 kJ/mol (2Ge2(g) + Sn(g) = Ge2Sn(g) + 2Ge(g)) and −77.2 kJ/mol (Ge2(g) 7118



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equilibration temperature of the overall vapor at 2000 K and the initial pressure of Si(g) at the corresponding saturation value, while varying the Ge(g) and Sn(g) in a rather wide range below the respective vapor pressures at 2000 K. With no claim of deriving rigorously optimized conditions, the most favorable input pressures were found to be log P(Sn)/bar = −3.0 and log P(Ge)/bar = −4.1, corresponding to vapors equilibrated with the pure elements at T = 1850 and T = 1760 K for Sn and Ge, respectively. In such conditions, the monatomic species represent around 98.9% of the vapor, and the di-, tri-, tetra-, and pentatomic molecules are in the ratio 100:21:7:0.1. The relative abundances of the tri- and tetratomic species, resulting from these initial conditions are reported in Figure 12. This

Figure 11. Molar fractions of the Si−Ge−Sn species estimated from thermodynamic data. Atoms are not reported, and the aggregated species were grouped on the basis of the total number of atoms (upper curves) and of the constituent elements. Total pressures are 5.4 × 10−5, 6.5 × 10−4, 4.8 × 10−3, and 2.5 × 10−2 bar at T = 1600, 1800, 2000, and 2200 K, respectively.

pentatomic species are computed to be in the ratio 100:64:29:1, while the subgroups of species with the same constituent elements resulted to be in the relative amounts Six/Gex/Snx/ SixGey/GeySnz/SixSnz/SixGeySnz ≈ 0:9:28:3:100:3:2. These results illustrate that the role of larger aggregates and ternary species is not entirely negligible. With regard to the latter, at 2000 K, their overall partial pressure is 4.6 × 10−6 bar, to be compared with the total pressure of 4.8 × 10−3 bar, and the triatomic and tetratomic molecules account for 99.5% of the total ternary species. Among them, SiGeSn predominates, with relative ratios equal to 100:24:9:4 for the SiGeSn/Si2GeSn/ SiGe2Sn/SiGeSn2 sequence. As mentioned, the above-reported results were obtained by taking the vapor pressures of pure elements as initial conditions. However, it is of interest to look for different initial conditions which could maximize, at equilibrium, the relative abundances of the ternary clusters with respect to the other aggregated species. Indeed, while the absolute maximum partial pressure of these species is quite obviously attained with initial conditions corresponding to the saturated pure vapors, the question arises if the molar fraction of ternary species could be increased by setting initial conditions of undersaturation for one or more elements. This can be realized, in practice, with the pure components acting as sources of vapor each at its optimized temperature and allowing the collected vapor mixture to attain equilibrium at a higher temperature. In particular, we performed a series of calculations by setting the

Figure 12. Molar fractions at T = 2000 K of the triatomic and tetratomic species with respect to the corresponding total triatomic and tetratomic molar fractions. The values were calculated with input pressures of the elements that maximize the relative abundance of ternary species (see text).

figure illustrates in a compact view which ternary species, and to what extent, may be relevant in the overall vapor, and which of the binary molecular species would dominate in those conditions (a complete list of the calculated molar fractions and partial pressures at 2000 K for these initial conditions is provided in the Supporting Information). The values reported, besides the role of the ternary species, do show the relatively large importance of some binary Si−Ge species (especially of the Si3Ge molecule), resulting from the interplay between binding energies, entropies. and also initial partial pressures, the latter being unfavorable to the silicon-containing species. Needless to say, equilibrium experiments by the Knudsen Effusion Mass Spectrometric method28 or fragmentation/ reaction processes experiments with the Fourier Transform 7119



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Ion Cyclotron Resonance mass spectrometric technique64 would be of great value to follow up the energetic and thermodynamic results here presented. A suitable starting point for the design of the demanding experiments aimed at detecting and studying the yet unobserved tetratomic and pentatomic species could be the work done on the GenSim and SnnGem metalloid clusters.65,66 5.2. Evolution Paths. Having treated the equilibrium properties of the molecular species under scrutiny, quite naturally the question arises on the possible ways by which they may form. While, on one hand, nonequilibrium conditions may be likely involved, on the other, a kinetically based analysis would require the rather complex study of the multidimensional potential surface, which is outside the scope of this contribution. Nevertheless, at least for the most interesting case of the ternary species, it is of some interest to consider the relative energies and to hypothesize the paths of formation on the basis of the exothermicity67 of each step of the route. Indeed, with the reasonable simplifying hypothesis of a series of steps where one single atom is added to the molecular frame, an exothermically favored route can be identified in the gamut of the possible ones. Of course, the initial and final points of each possible route are fixed at the energy of the isolated atoms and of the polyatomic species under consideration, respectively. The exothermically favored route was selected as the one that at each subsequent step involves the most exothermic addition of an atom, ultimately leading to the formation of the sought molecule. These routes, which illustrate, under the given hypothesis, the evolution of the moieties, were derived taking advantage of the atomization enthalpies reported in Table 6 and in ref 23 and are shown in the scheme of Figure 13, where the parenthood relationship between the ternary species can be appreciated. In general, the initial part of the evolution path is ruled by the larger energy of the silicon−silicon and silicon− germanium bonds. All the final steps from the tetratomic to the pentatomic species involve the addition of a tin atom. It is also

of some interest to note how the so defined lowest routes include all the four ternary species with 3 and 4 atoms.

6. CONCLUSIONS We have presented a theoretical investigation of molecules belonging to the Ge−Sn binary and the Si−Ge−Sn ternary systems, up to the pentatomic species, with the goal to provide fundamental information on the simplest units of the corresponding materials, which are currently considered promising candidates for a number of applications as functional components in electronic and optoelectronic devices. By using the DFT approach, the geometrical structures and the harmonic frequencies of the tri-, tetra-, and pentatomic clusters were calculated for the ground state and higher energy isomers. The atomization energies of the ground states were computed by the CCSD(T) method with basis sets up to QZ quality, extrapolated to the CBS limit, and corrected for the spin−orbit contribution by two different approximate approaches. A set of empirically adjusted atomization energies are proposed as the most reliable values. Other properties of the species under study were evaluated, such as NPA charges, HOMO−LUMO gaps, binding energies, fragmentation energies, and mixing energies. On the basis of the atomization energies and thermal functions here derived, a thermodynamic database was created containing all the 55 species of the Si−Ge−Sn system up to pentatomic clusters, and equilibrium calculations were carried out in the temperature range 1600−2200 K to predict the composition of the gas phase under various conditions. Finally, possible formation routes of the ternary species were identified on the basis of exothermiticity and of a simple hypothesis of subsequent steps in which one atom is progressively added to the molecular frame.

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ASSOCIATED CONTENT

S Supporting Information *

Tables of the computed molecular parameters of the triatomic, tetratomic, and pentatomic species calculated at the B3LYP/ccpVTZ level up to 1 eV from the ground states; Gibbs energy functions and heat content functions of the molecules GexSny (x + y = 3,4,5), SixSny (x + y = 5), SixGeySnz (x + y + z = 3,4), and SixGeySnz (x + y + z = 5); equilibrium molar fractions and partial pressures of the gaseous atomic and molecular species at 2000 K with initial conditions corresponding to the saturated vapor pressures of the elements; computed structures and relative energy of the triatomic, tetratomic, and pentatomic species; figures of the molecular orbitals of GexSny (x + y = 4) at the B3LYP/cc-pVTZ level; HOMO−LUMO gaps. This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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Figure 13. Evolution paths for the formation of the Si−Ge−Sn species. For molecules with an equal number of atoms, the difference between the atomization enthalpy of each species and that of the species with the lowest atomization energy is reported, so indicating a normalized increase in the exothermicity on going from one species to that with an additional atom. The numbers in the top part of the figure are the exothermicities of each step along the evolution of the least exothermically favored path (SiGe−SiGeSn−SiGeSn2−SiGeSn3).

ACKNOWLEDGMENTS This work has been funded by Sapienza Università di Roma, under the project “Studio di nuove molecole con rilevanti effetti relativistici sul legame.” All the calculations were performed on the computer facilities at CASPUR (Consorzio interuniversitario per le applicazioni di supercalcolo per università e ricerca) in 7120



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the framework of the HPC Grant std11-505 (“Studio relativistico di molecole intermetalliche pesanti”).

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