THE JOURNAL OF CHEMICAL PHYSICS 142, 034303 (2015)
Theoretical investigation of stabilities and optical properties of Si12C12 clusters Xiaofeng F. Duan1,2 and Larry W. Burggraf2,a) 1
Air Force Research Laboratory DoD Supercomputer Resource Center, Wright-Patterson Air Force Base, Ohio 45433, USA 2 Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio 45433, USA
(Received 6 June 2014; accepted 18 December 2014; published online 16 January 2015) By sorting through hundreds of globally stable Si12C12 isomers using a potential surface search and using simulated annealing, we have identified low-energy structures. Unlike isomers knit together by Si–C bonds, the lowest energy isomers have segregated carbon and silicon regions that maximize stronger C–C bonding. Positing that charge separation between the carbon and silicon regions would produce interesting optical absorption in these cluster molecules, we used time-dependent density functional theory to compare the calculated optical properties of four isomers representing structural classes having different types of silicon and carbon segregation regions. Absorptions involving charge transfer between segregated carbon and silicon regions produce lower excitation energies than do structures having alternating Si–C bonding for which frontier orbital charge transfer is exclusively from separated carbon atoms to silicon atoms. The most stable Si12C12 isomer at temperatures below 1100 K is unique as regards its high symmetry and large optical oscillator strength in the visible blue. Its high-energy and low-energy visible transitions (1.15 eV and 2.56 eV) are nearly pure one-electron silicon-to-carbon transitions, while an intermediate energy transition (1.28 eV) is a nearly pure carbon-to-silicon one-electron charge transfer. [http://dx.doi.org/10.1063/1.4905542]
I. INTRODUCTION
Nanoscale silicon carbide materials that have enormous potential applications as high-power, high-frequency, hightemperature semiconductors are being synthesized.1,2 Novel high-purity, high-performance silicon carbide thin film, and nano-size electro-optical materials can be created from small molecules using processing techniques such as chemical vapor deposition, molecular beam epitaxy, or pulsed laser ablation.3,4 We anticipate that other innovative techniques of nanoassembly will be developed in the future. Just as structures of bulk solid state carbon and silicon are very different from structures of nano-scale carbon and silicon molecules, so are structures of silicon carbon molecules different from solid state silicon carbide.5–7 For small SinCm molecules, different bonding patterns confer on them a variety of stabilities and reactivity. Thus, efficient growth of desired material surfaces using sources of small molecular clusters may be either benefitted or degraded for different processing techniques and processing conditions by changes in distributions of molecular clusters. Because silicon carbide nanostructures also will likely find applications in novel optical technologies, computation of optical properties of silicon carbide nanoclusters is of interest also.8 In previous research, we described bonding influences that order isomer stabilities for small SinCm clusters at low a)Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2015/142(3)/034303/10/$30.00
temperature.5,6 The order of bond strengths C–C > C–Si > Si–Si is the dominant influence controlling isomer structures and energies. In consequence, the most stable isomers have segregated carbon regions terminated by C–Si bonds. This conclusion agrees with previous results for SinC20−n (n ≤ 8) heterofullerenes.9,10 Our modeling of the series of SinCn clusters (4 ≤ n ≤ 12) shows that globally stable isomers for the Si12C12 stoichiometry have unique structures and properties.7 This relationship between isomer structure and stability may change when the temperature increases from near zero to high temperatures, especially for larger cluster sizes. Scipioni et al.11 studied Si30C30 segregated (SE) and non-segregated (NSE) cage structures and found that SE is more stable than NSE at T = 0 K, but at T = 2000 K only the NSE structure is stable. The group of Pochet12 also studied Si30C30 cages and concluded that a fully segregated chemical ordering is more favorable than partially segregated cage. Two low-energy Si12C12 isomers have similar energies, but they have very different structures from each other and from low-energy isomers of the smaller clusters in the SinCn series. One of these isomers, structure A having no special symmetry shown in Figure 1, has a silicon region that contiguously spans the carbon segregation region. The other of these isomers, structure B in Figure 1, has a structure with high symmetry (D2h), behavior that is not observed in minimum energy isomers for other SinCn clusters with n > 4. We anticipated that these low-energy isomers for Si12C12 stoichiometry have very different optical properties involving significant charge transfer between the segregation regions. This study
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FIG. 1. Selected stable Si12C12 structures, left column is top views and right column is side views. Si atoms are dark gray and C atoms are light gray.
confirms the minimum energy isomers for Si12C12 and calculates the optical properties of these low-energy isomers. For comparison, two other Si12C12 molecule isomers having special geometries that are different from structures A and B were studied. Our approach to calculate properties of the four isomers employs density functional theory (DFT) to characterize their geometric and electronic structures, Time-Dependent DFT (TDDFT) to predict their optical absorption spectra, and
Coupled Perturbed Hartree-Fock (CPHF) to predict their dynamic polarizability spectra.
II. METHODOLOGY
In order to find the stable isomer structures for the Si12C12 cluster stoichiometry at low temperature, we employed the
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combination of Stochastic Potential Surface Search and Pseudopotential Plane-Wave Car-Parinello Simulated Annealing Simulations (PSPW-CPMD-SA) which we developed previously to search for stable SinCn clusters.7 In our previous approach, an enhanced kick program generates a random cluster structure that is then used as a seed structure for PSPWCPMD-SA simulation. This ensures that each SA simulation samples a different potential surface region in order to find the regional minimum structure. In this work, we extended the method in two respects: (a) the kick program was re-designed adding a function to kick predefined atomic groups as well as single atoms and (b) we adopted the concept of a “genetic algorithm”13,14 into the search iteration process. The details of the algorithm are as follows. At the first stage, 10 stable SiC molecule structures that were obtained from the stochastic searches were saved as initial references listed in descending energy order. Each of the structures on the list was then used as a parent structure in a new PSPW-CPMDSA simulation. In the new PSPW-CPMD-SA simulation, an extra process was added to “mutate” the parent structure. In this extra mutation process, the parent structure was heated to certain high temperature at which only part of the structure was relaxed for a time period. In this case, the more weakly bonded Si atoms were relaxed while connectivity of the more stable C fragment was restrained. After the extra mutation process is completed, the normal PSPW-CPMD-SA simulation is resumed with the new “mutated” structure as initial structure. If the energy of a new offspring structure is lower than any of the structures on the reference list, it is inserted into the list and the highest energy structure is eliminated from reference list. Then this new offspring is used as a parent for a new search. Otherwise, the offspring is discarded and a new search is started using a new seed structure randomly generated by the kick program. The first stage establishes the energy reference list after a few hundred searches. In the second stage, an energy criterion was set to focus on the lower part of reference list. Any new isomer structures which satisfy the energy criterion were taken as a parent to generate offspring structures using the same “mutate” process as above. After hundreds to thousands of screening iterations, 10 of the lowest energy Si12C12 structures were selected for further optimization using DFT with full core atomic basis sets. Specifically, B3LYP hybrid functional15 with aug-ccpVTZ basis sets16 were employed. Among the fully optimized 10 structures, we selected 4 of them to carry out optical properties calculations. For this purpose, we employed TDDFT17 with aug-ccpVTZ as well as cc-pVTZ16 basis sets. In consideration of accuracy of excited state energies, we also used PBE0 hybrid functional18 with cc-pVTZ basis sets to carry out TDDFT calculations. The absorption spectra were convoluted as a sum of pseudo-Voigt functions with a fullwidth-half-maximum value of 0.185 eV, using the software package SWizard.19 In addition to optical absorption spectra, we calculated dynamic polarizabilities for the selected stable structures using frequency-dependent CPHF20 at B3LYP/ccpVTZ and PBE0/cc-pVTZ levels of theory. We also generated the total and partial density of states (TDOS and PDOS) for the structures at B3LYP/cc-pVTZ level of theory by calculating the molecular orbital (MO) compositions in terms of the
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constituent chemical fragment using AOMix software.21,22 All the DFT and TDDFT calculations were carried out using the Gaussian 09 computational chemistry package.23
III. RESULTS AND DISCUSSION A. Stable structures
In previous work,7 we described 6 low-lying energy structures for Si12C12 isomers. The three structures at lower end of energy order feature a common carbon segregation structure having two C6 rings and one C5 ring fused together. The Si segregation network terminates the C region and forms closed concave cages. In this work, we selected the low energy structure A that is shown in Figure 1 for special study. With the improvement of the searching method, by kicking two C6 rings and 12 Si atoms to form an initial seed structure, we found low energy structures that features two separated C6 rings. Among these, we selected the lowest energy one, structure B shown in Figure 1. Unlike structure A, structure B is composed of two C segregations and two Si segregations which link together forming a capsule shape with the two Si segregation regions at both ends. B is highly symmetric having point group symmetry D2h. The total energies of molecules A and B are very similar. Using B3LYP DFT functional, with 6-311G* or 6-311+G* basis sets, the energies of these two structure are almost identical; with cc-pVTZ or aug-cc-pVTZ, structure B is slightly lower (0.10-0.11 eV) than A. The third distinct structure selected for comparison in the work has one carbon segregation region in the form of a naphthalene frame with a carbon substituted in both 2 and 6 positions, surrounded by 12 Si atoms forming a flat bowl structure having C2h point group symmetry. This structure, displayed as C in Figure 1, has somewhat higher energy than A and B, in the range of 0.33–0.43 eV. Both structures B and C were also previously optimized by Wang et al.24 calculated at BLYP/dnd level of theory. The last structure interesting for this comparison is a well-studied fullerene-like structure, the so-called fulsicene structure25,26 displayed in Figure 1 as structure D. It is a highly symmetric structure with Th point group symmetry. At the B3LYP/aug-cc-pVTZ level of theory, this structure has much higher energy (0.67-1.11 eV) than structures A, B, and C. The instability of the structure is mainly due to lack of stronger C–C bonds compared with those in molecules A, B, and C. The molecular structures described above can be categorized into four types. Structure A represents the cage type clusters formed by two segregations of C and Si atoms each spanning the other. Structure B is the type consisting of separate small groups of C and Si atoms. Structure C is the flat type cluster with nearly planar polyaromatic carbon group at the core surrounded by terminating Si atoms. With no carbon and silicon segregations in structure, the bonding pattern of structure D has analogy with SiC solid state structures. The relative energies and average bond lengths for the four clusters are listed in Table I. As one can see, the average C–C bond distance for A, B, and C is very close to conjugated C–C bonds, between 1.41 and 1.42 Å. The average Si–Si bond distances
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TABLE I. Relative total energy Erel (eV) and average bond length (Å) calculated at B3LYP/aug-cc-pVTZ level of theory. Structure A B C D
Erel
C–C
Si–Si
Si–C
0.00 −0.11 0.32 0.99
1.42 1.41 1.42
2.43 2.45 2.46
1.92 1.93 1.85 1.81
for the three structures are close to each other, in the range of 2.43-2.46 Å. For Si–C bonds, A and B have almost identical bond lengths while those of C and D are close to each other, but ∼0.08-0.11 Å shorter than those in A and B. Among the selected Si12C12 molecular isomer structures, structures A and B which have separate carbon and silicon segregations are the most stable Si12C12 isomers. The different stabilities of the 4 structures may be rationalized by consideration of bond types and overall geometry of the cluster. The greater stability of structures A and B is because they not only tend to maximize the total number of bonds but also maximize the number of strong C–C bonds in carbon segregation regions. Also the number of Si–C bonds that connect to the silicon segregation regions at the periphery is maximized. Within the limitations imposed by maximizing C–C and Si–C bonds, the number of weaker Si–Si bonds is maximized in the silicon segregation region. For example, there are 15 C–C plus 9 Si–C bonds in A, 12 C–C plus 20 Si–C bonds in B, and 13 C–C plus 10 Si–C bonds in the structure C. The much larger number of Si–C bonds in B compensates the smaller number of stronger C–C bonds in A. Structure C, less stable than A or B, maximizes the size of a single nearly planar carbon segregation region at expense of forming fewer C–C bonds than structure A and fewer Si–C bonds than structure B, and fewer Si–Si bonds than both structures A and B. Structure D predominantly forms Si–C bonds at the expense of any strong C–C bonds. In order to examine the stabilities of all four isomer structures at finite temperature, we conducted Car-Parrinello molecular dynamics (MD) simulations at T = 1100-1500 K for 250 000 time steps (30 ps) with rotational and translational motions projected out. The Root Mean Square Displacement (RMSD) of all the atoms for all four structures at 1100 K and 1500 K are displayed in Figure 2. As seen from the RMSD analysis, at 1100 K, the motions of atoms in structures B and D are rather small indicating that they are stable at that temperature. The RMSD for structure A varies around 1.0 Å but its cage structure is maintained. Shown by its rather large RMSD, structure C becomes unstable and cannot keep its flat bowl structure at this temperature. When temperature is raised to 1500 K, although the RMSD is increased, B and D maintain their original shapes, especially D which shows only small geometry distortions. At this temperature, A becomes unstable and cannot maintain its cage structure. We observed that at these elevated temperatures, the breaking and forming of Si–Si bonds occur constantly while breaking and forming either C–C or Si–C bonds happen rarely. Because of differences in bonding patterns (especially the number of Si–Si bonds and hybridization (sp3 or sp2) of Si atoms), the relative stability order for these structures, B ≥ A > C > D at T = 0 K, becomes
FIG. 2. RMSD of all atoms as a function of the Car-Parrinello MD simulation time for at T = 1100 K (bottom) and 1500 K (top) for all four structures.
B ≈ D > A > C at T = 1100 K, while at T = 1500 K, the stability order changes to D > B > A > C according to the RMSD analysis. This stability trend at high temperature is consistent with previous results for Si30C30.11 In this research, we concentrate on optical properties of several selected Si12C12 isomers that are stable at low temperature. By studying the photo-absorption spectra for the four classes of Si12C12 molecules having different bonding patterns, we gain insight into how isomer structures affect photoexcitations in different types of silicon carbide molecules. B. Optical properties
To predict the optical properties of the selected molecule types, we first performed TDDFT calculations followed by CPHF calculations. In our previous work,5 we have shown that DFT with B3LYP functional is capable to produce ground state properties for SiC clusters with good accuracy compared to photoelectron spectrometry. However, the accuracy of TDDFT calculations using normal hybrid functionals such as B3LYP for charge-transfer type of excitations are in question,27 and some suggest that recently developed range-separated hybrid functionals are a remedy. Lan and Feng28 studied density functional and basis set dependencies of excitation energy for small SiCn and SinC (n = 2-6) clusters and argued that B3LYP and aug-ccpVTZ give suitable computational accuracy. Good spectrometry data are available for the well studied SiC molecule29,30 which we use for benchmark comparisons to evaluate the predictive capabilities of TDDFT using regular hybrid functionals (B3LYP and PBE0) as well as functionals which have long range corrections (CAM-B3LYP,31 LC-BLYP, and LCPBE32). We find that accuracies of TDDFT calculations for
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TABLE II. TDDFT calculations for ground and excited states properties of SiC molecule with different functionals and cc-pV5z basis sets. State X3Π A3Σ−
a Reference b Reference
r0, Å ωe, cm−1 r0, Å ωe, cm−1 Te, cm−1
B3LYP
PBE0
CAMB3LYP
LC-PBE
LC-BLYP
Expt.
1.712 986 1.798 902 2280
1.708 998 1.779 1025 3942
1.698 1015 1.758 1073 2761
1.680 1038 1.704 1299 6114
1.679 1047 1.715 1246 3475
1.722a 965a 1.802b 861a 3733b
29. 30.
excited states of SiC vary for the different functionals. The calculated results as well as experimental values are listed in Table II. As seen in the table, B3LYP yields the best bond distance and frequency for both the ground state and first excited state. However, B3LYP predicts lower excited state energy. PBE0 yields the best excited state energies of all the functionals evaluated. This is consistent with benchmarking results of the Gordon group33 for a variety of organic molecules for which PBE0 was the best performing global hybrid generalized gradient approximation functional. Based on this result, in addition to predicting ground state and excited state properties using B3LYP functional with both cc-pVTZ and aug-cc-pVTZ basis sets, which produce accurate photoabsorption spectra, we also employed PBE0 with cc-pVTZ to calculate the excitation energies of Si12C12 molecules. PBE0 excited state energy predictions were consistently higher than B3LYP predictions. Energy results that were calculated with B3LYP and PBE0 functionals and cc-pVTZ basis sets for the four molecules: (1) highest occupied molecular orbitals-lowest unoccupied molecular orbitals (HOMO-LUMO) gaps Egap, (2) first allowed optical excitation energies Eopt, and (3) exciton binding energies Eexe are listed in Table III. The HOMO-LUMO gaps represent the quasi-particle energies of the clusters. As shown in the table, with PBE0 functional, Egap values decrease from 2.11 eV to 1.90 eV to 1.78 eV in order of A, B, and C, with increasing amount of pi-electron delocalization in the carbon segregation region. In contrast to these three structures, structure D has an Egap value as large as 3.76 eV. The Egap value of D has been reported at different levels of theory in previous works as 8.81 eV at RHF/6-31G,34 6.24 eV at RHF/6-31G(d),25 2.36 eV at generalized gradient approximations (GGA) with PBE functional and standard norm-conserving Troullier-Martins pseudopotentials.26 As for the optical absorption gap, Eopt,
TABLE III. HOMO-LUMO energy gap Egap, optical absorption gap Eopt, and exciton binding energy Eexe (eV). Egap Structure A B C D
Eopt
Eexc
B3LYP
PBE0
B3LYP
PBE0
B3LYP
PBE0
1.86 1.70 1.55 3.43
2.11 1.90 1.78 3.76
1.22 1.15 0.94 3.32
1.29 1.16 0.99 3.44
0.64 0.55 0.61 0.11
0.82 0.74 0.79 0.32
calculated with PBE0 functional, it also decreases in the order of A, B, and C, from 1.29 eV to 1.16 eV to 0.99 eV. For structure D, Eopt is much larger than the values for the other three structures, having a value as large as 3.44 eV. This clearly indicates that optical absorption gap is significantly correlated to the quasi-particle energies of the clusters. To evaluate the electron-hole pair Coulomb correlations for these four Si12C12 structures, we obtained exciton binding energies, Eexc, from the difference between the quasiparticle energy gap Egap and optical absorption gap Eopt, as listed in Table III. The Eexc values obtained with PBE0 functional for structures A, B, and C are comparable, ranging from 0.74 to 0.82 eV. On the other hand, structure D has a much smaller Eexc of 0.32 eV that is comparable from the previous reported value of 0.31 eV by Javan.26 With B3LYP functional, the values of Egap, Eopt, and Eexc for all four molecules are systematically lower than those from PBE0 functional; however, the pattern of the changes is similar to that of PBE0 functional. We obtained the optical absorption spectra of the four Si12C12 structures using TDDFT calculations using cc-pVTZ basis sets B3LYP and PBE0 functionals. Table IV lists the parameters for the first and major excitation transitions while Figures 3–6 display the absorption spectra for the A, B, C, and D structures. Comparing absorption spectra from the two functionals for each molecule in Figures 3–6, it is noticeable that the line shapes and relative positions of the absorption peaks are nearly identical, only the peaks of PBE0 are blue-shifted ∼0.1 eV or less compared to B3LYP. Therefore, the following discussions are mainly based on the results of B3LYP functional. We also did TDDFT calculations at B3LYP/aug-ccpVTZ level of theory and found the spectra are insignificantly different from those calculated using B3LYP/cc-pVTZ. Using plane wave DFT calculations, Javan26 found that the spectra of SiC nanocages can be roughly divided into a low energy region of 1.5-4.0 eV and a high energy region of 4.0-10.0 eV. Our TDDFT calculations, solving 40 singlet excited states, give absorptions for all the four structures in the energy region between 0.9 and 4.0 eV. The maximum energy of strong absorption is lowest for C and highest for D. Peak absorptivities for molecule A are weak compared to those of B and C and D. Information about 1-photon and 2-photon resonances with an applied external electric field, relevant to non-linear optical measurements, can be obtained from these optical spectra.35,36 In order to obtain the photon resonance information, we calculated dynamic polarizabilities for the four structures and displayed them in Figures 3–6. The optical excitations are closely
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TABLE IV. Parameters for the first and major transitions of B3LYP/aug-ccpVTZ calculations.
Structure
A
B
Excitation in eV
Oscillator strength f
Transition MO assignmentsa
1.22 2.76
0.0003 0.0116
2.77
0.0118
1.15 1.28 2.56 0.94
0.0012 0.0512 0.1414 0.0001
1.46
0.0676
1.74
0.0339 0.0256
3.32
0.0256
H − 0 → L + 0(97%) H − 8 → L + 0(24%) H − 6 → L + 1(19%) H − 2 → L + 4(18%) H − 6 → L + 1(63%) H − 8 → L + 0(11%) H − 0 → L + 1(100%) H − 0 → L + 0(97%) H − 2 → L + 1(93%) H − 1 → L + 0(44%) H − 0 → L + 0(43%) H − 3 → L + 2(35%) H − 0 → L + 1(20%) H − 1 → L + 1(11%) H − 4 → L + 2(11%) H − 4 → L + 2(71%) H − 1 → L + 5(55%) H − 4 → L + 3(23%) H − 2 → L + 4(36%) H − 2 → L + 5(28%) H − 5 → L + 1(26%) H − 0 → L + 4(52%) H − 3 → L + 2(23%) H − 0 → L + 5(31%) H − 1 → L + 4(28%) H − 2 → L + 4(36%) H − 1 → L + 4(18%) H − 2 → L + 5(17%) H − 0 → L + 5(44%) H − 1 → L + 4(15%)
C
D
0.0256 3.53
0.0122 0.0122
0.0122 a H−n
and L+n indicates HOMO-n and LUMO+n orbitals.
related to frontier molecular orbitals and orbital densities at the different electronic states. The PDOS and TDOS for structure A, B, and C are shown in Figures 7–10. Strong transition probability involves a large change in TDOS. The HOMO and LUMO are displayed in Figure 11. For structure A, the first excitation occurs at 1.22 eV of energy, as shown in Table IV. This is a very weak transition from HOMO to LUMO having oscillator strength of only 0.0003. The observable transitions happen in the energy range between 1.3 and 3.2 eV as shown in Figure 3. The highest absorption peak at ∼2.7 eV mostly arises from transitions from frontier orbitals (HOMO − 2, HOMO − 6, and HOMO − 8) to LUMO and LUMO + 1. The dynamic polarizabilities for this structure show no dispersions lower than 3.3 eV where a very large peak appears (Figure 3). This indicates that none of the absorption peaks up to 3.2 eV of excitation energy are one-photon resonances. The nature of the HOMO and LUMO (pictured in Figure 11) as well as the adjacent frontier MOs of structure A is reflected in PDOS and TDOS in Figure 7. The strength of the absorptions generally increases with the TDOS. Weak transitions are observed when the TDOS for the acceptor and donor states are small. For example, the lowest energy transition for structure A from HOMO to LUMO, exhibiting small TDOS for both HOMO and LUMO, as shown
FIG. 3. Absorption spectrum (bottom) and dynamic polarizability spectrum (top) for structure A.
in Figure 7, has very small oscillator strength. While the transitions for the higher energy excitations (2.76 eV and 2.77 eV embedded in a single spectral peak shown in Figure 3) have larger oscillator strengths and correspondingly larger TDOS for donor and acceptor states. These higher energy excitations are of different donor-acceptor charge transfer types. For structure A, the high energy transitions have mixed charge transfer
FIG. 4. Absorption spectrum (bottom) and dynamic polarizability spectrum (top) for structure B.
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FIG. 7. Partial and total density of states for structure A.
FIG. 5. Absorption spectrum (bottom) and dynamic polarizability spectrum (top) for structure C.
contributions dominated by one or the other type. Carbonto-silicon charge (HOMO − 8 → LUMO and HOMO − 2 → LUMO + 4) transfer dominates the 2.76 eV transition, while silicon-to-carbon charge transfer (HOMO − 6 → LUMO + 1) dominates the 2.77 eV transition, as is evident from Table IV. For structure B, all the observable absorptions occur in the energy range of 1.0-3.2 eV similar to structure A, as shown in Figure 4. Structure B has anomalous electronic transitions in
FIG. 8. Partial and total density of states for structure B.
FIG. 9. Partial and total density of states for structure C.
FIG. 6. Absorption spectrum (bottom) and dynamic polarizability spectrum (top) for structure D.
FIG. 10. Partial and total density of states for structure D.
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FIG. 11. Frontier molecular orbitals for structures A, B, C, and D.
which the HOMO to LUMO transition is at higher energy than the HOMO to LUMO+1 transition due to the exciton chargeseparation contribution. For structure B, the first excitation has weak oscillator strength of 0.0012 at 1.15 eV for absorption transition from HOMO to LUMO + 1. The HOMO-LUMO excitation having oscillator strength of 0.0512 happens to be at 1.28 eV, exactly half the energy of the strongest absorption from HOMO − 2 to LUMO + 1 at 2.56 eV with oscillator strength of 0.1414. The large absorption from HOMO − 2 to LUMO+1 compared with absorptions from HOMO to LUMO is due to much larger TDOS near the HOMO − 2 energy. The absorption transitions for structure B exhibit nearly pure charge transfer, unlike structure A; pure electronic density transfer between C regions and Si regions in structure B makes large contributions to the strong excitations. The higher and lowest energy transitions are nearly pure silicon-to-carbon transitions at 1.15 eV and 2.56 eV, while the intermediate energy transition is nearly pure carbon-to-silicon charge trans-
fer at 1.28 eV (Table IV). Unlike structure A, the dynamic polarizability spectrum of B, also shown in Figure 4, exhibits variability in the energy range of 1.0-3.2 eV. The largest polarizability change occurs at 2.55 eV which precisely corresponds to the strongest absorption in the spectrum indicating a onephoton resonance. The absorption transition at 1.28 eV is also coincident with a large change in dynamic polarizability, also indicating a one-photon resonance. The first excitation for structure C is at energy 0.94 eV. The oscillator strength for the excitation is very low with a value of 0.0001. The excitation is associated with the transitions of HOMO−1 to LUMO and HOMO to LUMO. Different from structure A and B, absorptions by structure C mainly occur in the lower energy range below 2.6 eV. The lower energy range for optical absorptions of structure C can be explained by the fact that frontier MOs of C are at lower energies compared to those of structures A and B. Two strong absorptions occur at 1.48 eV and 1.74 eV (Figure 5) which are
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mainly transitions from HOMO, HOMO − 3, and HOMO − 4 to LUMO+1 and LUMO + 2 for which TDOS is large (Figure 9). Unlike structures A and B, all the absorptions for C involve charge transfer from silicon to carbon. Influence that the large aromatic carbon region dominates transitions is illustrated by comparison of HOMO and LUMO wavefunctions in Figure 11. The dynamic polarizability from PBE0 functional for the structure C has a huge change occurred at 1.22 eV input energy, which precisely correspond to the first optical absorption peak indicating a one-photon resonance. The optical absorption of structure D was previously studied by Javan26 using local density approximation (LDA), GGA, and plane wave DFT and TDDFT methods. Using these methods, a single peak at ∼2.0 eV was found which Javan related to s-to-p orbital transitions without distinguishing carbon and silicon atom types. At variance, we found from our calculations that there are no absorptions for excitation energies below 3.0 eV (Figure 6). The highest absorption peak appears at 3.93 eV. As shown in Table IV, there are 3 degenerate transitions in both energy peaks, 3.32 eV and 3.53 eV, due to the highly symmetric electronic structure of D. The dynamic polarizability is nearly constant in the energy region below 3.0 eV. The largest polarizability accurately corresponds to the highest absorption peak indicating that it is a one-photon resonance. The optical absorptions for D are observed in the high energy region can be explained by the fact that the frontier MOs of the cluster have higher energies and the HOMO-LUMO gap is larger as compared to properties of the other three structures studied in the work. As shown in Table IV, the strong excitations mainly corresponded to the transitions between HOMO − n (n = 1-5) and LUMO + m (m = 1-5). The large difference in optical absorption between D and the other structures is the larger contribution of carbon PDOS in occupied frontier orbitals. Shown in Figure 10, the partial density of carbon atoms dominate the TDOS in the states around the occupied of frontier MOs. On the other hand, similar to other structures, the TDOS near unoccupied frontier MOs have larger contributions from PDOS of Si atoms. The DOS information shows that transition transfer charge exclusively from carbon atoms to silicon atoms in structure D. Molecules such as A and B have optical properties analogous to organic dye materials having special optical absorption and fluorescence properties which are finding utility in specialized photonics applications.
IV. CONCLUSIONS
Among the selected Si12C12 molecular isomer structures, structures A and B which have separate carbon and silicon segregations, are the most stable Si12C12 isomers. These structures are more stable because they maximize strong C–C bonds in carbon segregation regions, the Si–C bonds at the periphery and, within those constraints, maximize weaker Si–Si bonds in the silicon segregation regions. Structures like D that predominantly form Si–C bonds at the expense of C–C bonds have much higher energies at low temperatures. Structures like C that maximize the size of a single nearly planar carbon segre-
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gation region at expense of forming Si–C bonds are less stable than A or B. The zero-temperature relative stability order B ≥ A > C > D becomes B ≈ D > A > C at T = 1100 K and changes to D > B > A > C temperature T = 1500 K according to the RMSD analysis. As anticipated, the different types of Si12C12 isomer structures exhibit different optical properties dominated by differences in charge transfer, both qualitative and quantitative. Some isomers exhibit large TDOS for frontier molecular orbitals, yielding large oscillator strengths. Structures having separate carbon and silicon regions, like A, B, and C, have transitions characterized by charge transfer between the carbon and silicon regions and produce lower excitation energies than does structure D. Unlike the other more stable structures, isomer D has bonding more like solid-state SiC in which there are not separate carbon and silicon segregations and frontier orbital charge transfer is from separate carbon atoms to silicon atoms exclusively. The very large difference between isomer D and the other Si12C12 structures that we considered is the larger contribution of carbon PDOS in occupied frontier orbitals. While the lowest energy isomers, A and B, both have separate carbon and silicon segregation regions, they have very different optical properties. Structure B is unique among small stable SinCm isomers (n > 4) as regards its high symmetry, while A has no special symmetry. As regards to their optical absorptions, molecule B has absorptions with largest optical oscillator strength characterized by pure charge transfer between the silicon and carbon segregations regions, unlike mixed charge transfer transitions of isomer A. For isomer B, a strong one-electron transition in the visible blue transfers charge from the silicon region to the carbon region; while a strong one-electron transition in the near infrared, at half the energy of the blue transition, transfers charge from the carbon region to the silicon region. ACKNOWLEDGMENTS
The views expressed in this work are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the United States Government. We acknowledge financial support from Molecular Dynamics and Theory Research Program of the Air Force Office of Scientific Research managed by Dr. Michael Berman. The DoD High Performance Computing Modernization Program and the AFRL Supercomputing Resource Center (DSRC) are gratefully acknowledged for computer time and helpful support. We are grateful for helpful discussions with Professor Mark Gordon and Professor David Weeks. 1Y.
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Theoretical investigation of stabilities and optical properties of Si12C12 clusters Xiaofeng F. Duan1,2 and Larry W. Burggraf2,a) 1
Air Force Research Laboratory DoD Supercomputer Resource Center, Wright-Patterson Air Force Base, Ohio 45433, USA 2 Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio 45433, USA
(Received 6 June 2014; accepted 18 December 2014; published online 16 January 2015) By sorting through hundreds of globally stable Si12C12 isomers using a potential surface search and using simulated annealing, we have identified low-energy structures. Unlike isomers knit together by Si–C bonds, the lowest energy isomers have segregated carbon and silicon regions that maximize stronger C–C bonding. Positing that charge separation between the carbon and silicon regions would produce interesting optical absorption in these cluster molecules, we used time-dependent density functional theory to compare the calculated optical properties of four isomers representing structural classes having different types of silicon and carbon segregation regions. Absorptions involving charge transfer between segregated carbon and silicon regions produce lower excitation energies than do structures having alternating Si–C bonding for which frontier orbital charge transfer is exclusively from separated carbon atoms to silicon atoms. The most stable Si12C12 isomer at temperatures below 1100 K is unique as regards its high symmetry and large optical oscillator strength in the visible blue. Its high-energy and low-energy visible transitions (1.15 eV and 2.56 eV) are nearly pure one-electron silicon-to-carbon transitions, while an intermediate energy transition (1.28 eV) is a nearly pure carbon-to-silicon one-electron charge transfer. [http://dx.doi.org/10.1063/1.4905542]
I. INTRODUCTION
Nanoscale silicon carbide materials that have enormous potential applications as high-power, high-frequency, hightemperature semiconductors are being synthesized.1,2 Novel high-purity, high-performance silicon carbide thin film, and nano-size electro-optical materials can be created from small molecules using processing techniques such as chemical vapor deposition, molecular beam epitaxy, or pulsed laser ablation.3,4 We anticipate that other innovative techniques of nanoassembly will be developed in the future. Just as structures of bulk solid state carbon and silicon are very different from structures of nano-scale carbon and silicon molecules, so are structures of silicon carbon molecules different from solid state silicon carbide.5–7 For small SinCm molecules, different bonding patterns confer on them a variety of stabilities and reactivity. Thus, efficient growth of desired material surfaces using sources of small molecular clusters may be either benefitted or degraded for different processing techniques and processing conditions by changes in distributions of molecular clusters. Because silicon carbide nanostructures also will likely find applications in novel optical technologies, computation of optical properties of silicon carbide nanoclusters is of interest also.8 In previous research, we described bonding influences that order isomer stabilities for small SinCm clusters at low a)Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2015/142(3)/034303/10/$30.00
temperature.5,6 The order of bond strengths C–C > C–Si > Si–Si is the dominant influence controlling isomer structures and energies. In consequence, the most stable isomers have segregated carbon regions terminated by C–Si bonds. This conclusion agrees with previous results for SinC20−n (n ≤ 8) heterofullerenes.9,10 Our modeling of the series of SinCn clusters (4 ≤ n ≤ 12) shows that globally stable isomers for the Si12C12 stoichiometry have unique structures and properties.7 This relationship between isomer structure and stability may change when the temperature increases from near zero to high temperatures, especially for larger cluster sizes. Scipioni et al.11 studied Si30C30 segregated (SE) and non-segregated (NSE) cage structures and found that SE is more stable than NSE at T = 0 K, but at T = 2000 K only the NSE structure is stable. The group of Pochet12 also studied Si30C30 cages and concluded that a fully segregated chemical ordering is more favorable than partially segregated cage. Two low-energy Si12C12 isomers have similar energies, but they have very different structures from each other and from low-energy isomers of the smaller clusters in the SinCn series. One of these isomers, structure A having no special symmetry shown in Figure 1, has a silicon region that contiguously spans the carbon segregation region. The other of these isomers, structure B in Figure 1, has a structure with high symmetry (D2h), behavior that is not observed in minimum energy isomers for other SinCn clusters with n > 4. We anticipated that these low-energy isomers for Si12C12 stoichiometry have very different optical properties involving significant charge transfer between the segregation regions. This study
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FIG. 1. Selected stable Si12C12 structures, left column is top views and right column is side views. Si atoms are dark gray and C atoms are light gray.
confirms the minimum energy isomers for Si12C12 and calculates the optical properties of these low-energy isomers. For comparison, two other Si12C12 molecule isomers having special geometries that are different from structures A and B were studied. Our approach to calculate properties of the four isomers employs density functional theory (DFT) to characterize their geometric and electronic structures, Time-Dependent DFT (TDDFT) to predict their optical absorption spectra, and
Coupled Perturbed Hartree-Fock (CPHF) to predict their dynamic polarizability spectra.
II. METHODOLOGY
In order to find the stable isomer structures for the Si12C12 cluster stoichiometry at low temperature, we employed the
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combination of Stochastic Potential Surface Search and Pseudopotential Plane-Wave Car-Parinello Simulated Annealing Simulations (PSPW-CPMD-SA) which we developed previously to search for stable SinCn clusters.7 In our previous approach, an enhanced kick program generates a random cluster structure that is then used as a seed structure for PSPWCPMD-SA simulation. This ensures that each SA simulation samples a different potential surface region in order to find the regional minimum structure. In this work, we extended the method in two respects: (a) the kick program was re-designed adding a function to kick predefined atomic groups as well as single atoms and (b) we adopted the concept of a “genetic algorithm”13,14 into the search iteration process. The details of the algorithm are as follows. At the first stage, 10 stable SiC molecule structures that were obtained from the stochastic searches were saved as initial references listed in descending energy order. Each of the structures on the list was then used as a parent structure in a new PSPW-CPMDSA simulation. In the new PSPW-CPMD-SA simulation, an extra process was added to “mutate” the parent structure. In this extra mutation process, the parent structure was heated to certain high temperature at which only part of the structure was relaxed for a time period. In this case, the more weakly bonded Si atoms were relaxed while connectivity of the more stable C fragment was restrained. After the extra mutation process is completed, the normal PSPW-CPMD-SA simulation is resumed with the new “mutated” structure as initial structure. If the energy of a new offspring structure is lower than any of the structures on the reference list, it is inserted into the list and the highest energy structure is eliminated from reference list. Then this new offspring is used as a parent for a new search. Otherwise, the offspring is discarded and a new search is started using a new seed structure randomly generated by the kick program. The first stage establishes the energy reference list after a few hundred searches. In the second stage, an energy criterion was set to focus on the lower part of reference list. Any new isomer structures which satisfy the energy criterion were taken as a parent to generate offspring structures using the same “mutate” process as above. After hundreds to thousands of screening iterations, 10 of the lowest energy Si12C12 structures were selected for further optimization using DFT with full core atomic basis sets. Specifically, B3LYP hybrid functional15 with aug-ccpVTZ basis sets16 were employed. Among the fully optimized 10 structures, we selected 4 of them to carry out optical properties calculations. For this purpose, we employed TDDFT17 with aug-ccpVTZ as well as cc-pVTZ16 basis sets. In consideration of accuracy of excited state energies, we also used PBE0 hybrid functional18 with cc-pVTZ basis sets to carry out TDDFT calculations. The absorption spectra were convoluted as a sum of pseudo-Voigt functions with a fullwidth-half-maximum value of 0.185 eV, using the software package SWizard.19 In addition to optical absorption spectra, we calculated dynamic polarizabilities for the selected stable structures using frequency-dependent CPHF20 at B3LYP/ccpVTZ and PBE0/cc-pVTZ levels of theory. We also generated the total and partial density of states (TDOS and PDOS) for the structures at B3LYP/cc-pVTZ level of theory by calculating the molecular orbital (MO) compositions in terms of the
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constituent chemical fragment using AOMix software.21,22 All the DFT and TDDFT calculations were carried out using the Gaussian 09 computational chemistry package.23
III. RESULTS AND DISCUSSION A. Stable structures
In previous work,7 we described 6 low-lying energy structures for Si12C12 isomers. The three structures at lower end of energy order feature a common carbon segregation structure having two C6 rings and one C5 ring fused together. The Si segregation network terminates the C region and forms closed concave cages. In this work, we selected the low energy structure A that is shown in Figure 1 for special study. With the improvement of the searching method, by kicking two C6 rings and 12 Si atoms to form an initial seed structure, we found low energy structures that features two separated C6 rings. Among these, we selected the lowest energy one, structure B shown in Figure 1. Unlike structure A, structure B is composed of two C segregations and two Si segregations which link together forming a capsule shape with the two Si segregation regions at both ends. B is highly symmetric having point group symmetry D2h. The total energies of molecules A and B are very similar. Using B3LYP DFT functional, with 6-311G* or 6-311+G* basis sets, the energies of these two structure are almost identical; with cc-pVTZ or aug-cc-pVTZ, structure B is slightly lower (0.10-0.11 eV) than A. The third distinct structure selected for comparison in the work has one carbon segregation region in the form of a naphthalene frame with a carbon substituted in both 2 and 6 positions, surrounded by 12 Si atoms forming a flat bowl structure having C2h point group symmetry. This structure, displayed as C in Figure 1, has somewhat higher energy than A and B, in the range of 0.33–0.43 eV. Both structures B and C were also previously optimized by Wang et al.24 calculated at BLYP/dnd level of theory. The last structure interesting for this comparison is a well-studied fullerene-like structure, the so-called fulsicene structure25,26 displayed in Figure 1 as structure D. It is a highly symmetric structure with Th point group symmetry. At the B3LYP/aug-cc-pVTZ level of theory, this structure has much higher energy (0.67-1.11 eV) than structures A, B, and C. The instability of the structure is mainly due to lack of stronger C–C bonds compared with those in molecules A, B, and C. The molecular structures described above can be categorized into four types. Structure A represents the cage type clusters formed by two segregations of C and Si atoms each spanning the other. Structure B is the type consisting of separate small groups of C and Si atoms. Structure C is the flat type cluster with nearly planar polyaromatic carbon group at the core surrounded by terminating Si atoms. With no carbon and silicon segregations in structure, the bonding pattern of structure D has analogy with SiC solid state structures. The relative energies and average bond lengths for the four clusters are listed in Table I. As one can see, the average C–C bond distance for A, B, and C is very close to conjugated C–C bonds, between 1.41 and 1.42 Å. The average Si–Si bond distances
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TABLE I. Relative total energy Erel (eV) and average bond length (Å) calculated at B3LYP/aug-cc-pVTZ level of theory. Structure A B C D
Erel
C–C
Si–Si
Si–C
0.00 −0.11 0.32 0.99
1.42 1.41 1.42
2.43 2.45 2.46
1.92 1.93 1.85 1.81
for the three structures are close to each other, in the range of 2.43-2.46 Å. For Si–C bonds, A and B have almost identical bond lengths while those of C and D are close to each other, but ∼0.08-0.11 Å shorter than those in A and B. Among the selected Si12C12 molecular isomer structures, structures A and B which have separate carbon and silicon segregations are the most stable Si12C12 isomers. The different stabilities of the 4 structures may be rationalized by consideration of bond types and overall geometry of the cluster. The greater stability of structures A and B is because they not only tend to maximize the total number of bonds but also maximize the number of strong C–C bonds in carbon segregation regions. Also the number of Si–C bonds that connect to the silicon segregation regions at the periphery is maximized. Within the limitations imposed by maximizing C–C and Si–C bonds, the number of weaker Si–Si bonds is maximized in the silicon segregation region. For example, there are 15 C–C plus 9 Si–C bonds in A, 12 C–C plus 20 Si–C bonds in B, and 13 C–C plus 10 Si–C bonds in the structure C. The much larger number of Si–C bonds in B compensates the smaller number of stronger C–C bonds in A. Structure C, less stable than A or B, maximizes the size of a single nearly planar carbon segregation region at expense of forming fewer C–C bonds than structure A and fewer Si–C bonds than structure B, and fewer Si–Si bonds than both structures A and B. Structure D predominantly forms Si–C bonds at the expense of any strong C–C bonds. In order to examine the stabilities of all four isomer structures at finite temperature, we conducted Car-Parrinello molecular dynamics (MD) simulations at T = 1100-1500 K for 250 000 time steps (30 ps) with rotational and translational motions projected out. The Root Mean Square Displacement (RMSD) of all the atoms for all four structures at 1100 K and 1500 K are displayed in Figure 2. As seen from the RMSD analysis, at 1100 K, the motions of atoms in structures B and D are rather small indicating that they are stable at that temperature. The RMSD for structure A varies around 1.0 Å but its cage structure is maintained. Shown by its rather large RMSD, structure C becomes unstable and cannot keep its flat bowl structure at this temperature. When temperature is raised to 1500 K, although the RMSD is increased, B and D maintain their original shapes, especially D which shows only small geometry distortions. At this temperature, A becomes unstable and cannot maintain its cage structure. We observed that at these elevated temperatures, the breaking and forming of Si–Si bonds occur constantly while breaking and forming either C–C or Si–C bonds happen rarely. Because of differences in bonding patterns (especially the number of Si–Si bonds and hybridization (sp3 or sp2) of Si atoms), the relative stability order for these structures, B ≥ A > C > D at T = 0 K, becomes
FIG. 2. RMSD of all atoms as a function of the Car-Parrinello MD simulation time for at T = 1100 K (bottom) and 1500 K (top) for all four structures.
B ≈ D > A > C at T = 1100 K, while at T = 1500 K, the stability order changes to D > B > A > C according to the RMSD analysis. This stability trend at high temperature is consistent with previous results for Si30C30.11 In this research, we concentrate on optical properties of several selected Si12C12 isomers that are stable at low temperature. By studying the photo-absorption spectra for the four classes of Si12C12 molecules having different bonding patterns, we gain insight into how isomer structures affect photoexcitations in different types of silicon carbide molecules. B. Optical properties
To predict the optical properties of the selected molecule types, we first performed TDDFT calculations followed by CPHF calculations. In our previous work,5 we have shown that DFT with B3LYP functional is capable to produce ground state properties for SiC clusters with good accuracy compared to photoelectron spectrometry. However, the accuracy of TDDFT calculations using normal hybrid functionals such as B3LYP for charge-transfer type of excitations are in question,27 and some suggest that recently developed range-separated hybrid functionals are a remedy. Lan and Feng28 studied density functional and basis set dependencies of excitation energy for small SiCn and SinC (n = 2-6) clusters and argued that B3LYP and aug-ccpVTZ give suitable computational accuracy. Good spectrometry data are available for the well studied SiC molecule29,30 which we use for benchmark comparisons to evaluate the predictive capabilities of TDDFT using regular hybrid functionals (B3LYP and PBE0) as well as functionals which have long range corrections (CAM-B3LYP,31 LC-BLYP, and LCPBE32). We find that accuracies of TDDFT calculations for
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TABLE II. TDDFT calculations for ground and excited states properties of SiC molecule with different functionals and cc-pV5z basis sets. State X3Π A3Σ−
a Reference b Reference
r0, Å ωe, cm−1 r0, Å ωe, cm−1 Te, cm−1
B3LYP
PBE0
CAMB3LYP
LC-PBE
LC-BLYP
Expt.
1.712 986 1.798 902 2280
1.708 998 1.779 1025 3942
1.698 1015 1.758 1073 2761
1.680 1038 1.704 1299 6114
1.679 1047 1.715 1246 3475
1.722a 965a 1.802b 861a 3733b
29. 30.
excited states of SiC vary for the different functionals. The calculated results as well as experimental values are listed in Table II. As seen in the table, B3LYP yields the best bond distance and frequency for both the ground state and first excited state. However, B3LYP predicts lower excited state energy. PBE0 yields the best excited state energies of all the functionals evaluated. This is consistent with benchmarking results of the Gordon group33 for a variety of organic molecules for which PBE0 was the best performing global hybrid generalized gradient approximation functional. Based on this result, in addition to predicting ground state and excited state properties using B3LYP functional with both cc-pVTZ and aug-cc-pVTZ basis sets, which produce accurate photoabsorption spectra, we also employed PBE0 with cc-pVTZ to calculate the excitation energies of Si12C12 molecules. PBE0 excited state energy predictions were consistently higher than B3LYP predictions. Energy results that were calculated with B3LYP and PBE0 functionals and cc-pVTZ basis sets for the four molecules: (1) highest occupied molecular orbitals-lowest unoccupied molecular orbitals (HOMO-LUMO) gaps Egap, (2) first allowed optical excitation energies Eopt, and (3) exciton binding energies Eexe are listed in Table III. The HOMO-LUMO gaps represent the quasi-particle energies of the clusters. As shown in the table, with PBE0 functional, Egap values decrease from 2.11 eV to 1.90 eV to 1.78 eV in order of A, B, and C, with increasing amount of pi-electron delocalization in the carbon segregation region. In contrast to these three structures, structure D has an Egap value as large as 3.76 eV. The Egap value of D has been reported at different levels of theory in previous works as 8.81 eV at RHF/6-31G,34 6.24 eV at RHF/6-31G(d),25 2.36 eV at generalized gradient approximations (GGA) with PBE functional and standard norm-conserving Troullier-Martins pseudopotentials.26 As for the optical absorption gap, Eopt,
TABLE III. HOMO-LUMO energy gap Egap, optical absorption gap Eopt, and exciton binding energy Eexe (eV). Egap Structure A B C D
Eopt
Eexc
B3LYP
PBE0
B3LYP
PBE0
B3LYP
PBE0
1.86 1.70 1.55 3.43
2.11 1.90 1.78 3.76
1.22 1.15 0.94 3.32
1.29 1.16 0.99 3.44
0.64 0.55 0.61 0.11
0.82 0.74 0.79 0.32
calculated with PBE0 functional, it also decreases in the order of A, B, and C, from 1.29 eV to 1.16 eV to 0.99 eV. For structure D, Eopt is much larger than the values for the other three structures, having a value as large as 3.44 eV. This clearly indicates that optical absorption gap is significantly correlated to the quasi-particle energies of the clusters. To evaluate the electron-hole pair Coulomb correlations for these four Si12C12 structures, we obtained exciton binding energies, Eexc, from the difference between the quasiparticle energy gap Egap and optical absorption gap Eopt, as listed in Table III. The Eexc values obtained with PBE0 functional for structures A, B, and C are comparable, ranging from 0.74 to 0.82 eV. On the other hand, structure D has a much smaller Eexc of 0.32 eV that is comparable from the previous reported value of 0.31 eV by Javan.26 With B3LYP functional, the values of Egap, Eopt, and Eexc for all four molecules are systematically lower than those from PBE0 functional; however, the pattern of the changes is similar to that of PBE0 functional. We obtained the optical absorption spectra of the four Si12C12 structures using TDDFT calculations using cc-pVTZ basis sets B3LYP and PBE0 functionals. Table IV lists the parameters for the first and major excitation transitions while Figures 3–6 display the absorption spectra for the A, B, C, and D structures. Comparing absorption spectra from the two functionals for each molecule in Figures 3–6, it is noticeable that the line shapes and relative positions of the absorption peaks are nearly identical, only the peaks of PBE0 are blue-shifted ∼0.1 eV or less compared to B3LYP. Therefore, the following discussions are mainly based on the results of B3LYP functional. We also did TDDFT calculations at B3LYP/aug-ccpVTZ level of theory and found the spectra are insignificantly different from those calculated using B3LYP/cc-pVTZ. Using plane wave DFT calculations, Javan26 found that the spectra of SiC nanocages can be roughly divided into a low energy region of 1.5-4.0 eV and a high energy region of 4.0-10.0 eV. Our TDDFT calculations, solving 40 singlet excited states, give absorptions for all the four structures in the energy region between 0.9 and 4.0 eV. The maximum energy of strong absorption is lowest for C and highest for D. Peak absorptivities for molecule A are weak compared to those of B and C and D. Information about 1-photon and 2-photon resonances with an applied external electric field, relevant to non-linear optical measurements, can be obtained from these optical spectra.35,36 In order to obtain the photon resonance information, we calculated dynamic polarizabilities for the four structures and displayed them in Figures 3–6. The optical excitations are closely
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TABLE IV. Parameters for the first and major transitions of B3LYP/aug-ccpVTZ calculations.
Structure
A
B
Excitation in eV
Oscillator strength f
Transition MO assignmentsa
1.22 2.76
0.0003 0.0116
2.77
0.0118
1.15 1.28 2.56 0.94
0.0012 0.0512 0.1414 0.0001
1.46
0.0676
1.74
0.0339 0.0256
3.32
0.0256
H − 0 → L + 0(97%) H − 8 → L + 0(24%) H − 6 → L + 1(19%) H − 2 → L + 4(18%) H − 6 → L + 1(63%) H − 8 → L + 0(11%) H − 0 → L + 1(100%) H − 0 → L + 0(97%) H − 2 → L + 1(93%) H − 1 → L + 0(44%) H − 0 → L + 0(43%) H − 3 → L + 2(35%) H − 0 → L + 1(20%) H − 1 → L + 1(11%) H − 4 → L + 2(11%) H − 4 → L + 2(71%) H − 1 → L + 5(55%) H − 4 → L + 3(23%) H − 2 → L + 4(36%) H − 2 → L + 5(28%) H − 5 → L + 1(26%) H − 0 → L + 4(52%) H − 3 → L + 2(23%) H − 0 → L + 5(31%) H − 1 → L + 4(28%) H − 2 → L + 4(36%) H − 1 → L + 4(18%) H − 2 → L + 5(17%) H − 0 → L + 5(44%) H − 1 → L + 4(15%)
C
D
0.0256 3.53
0.0122 0.0122
0.0122 a H−n
and L+n indicates HOMO-n and LUMO+n orbitals.
related to frontier molecular orbitals and orbital densities at the different electronic states. The PDOS and TDOS for structure A, B, and C are shown in Figures 7–10. Strong transition probability involves a large change in TDOS. The HOMO and LUMO are displayed in Figure 11. For structure A, the first excitation occurs at 1.22 eV of energy, as shown in Table IV. This is a very weak transition from HOMO to LUMO having oscillator strength of only 0.0003. The observable transitions happen in the energy range between 1.3 and 3.2 eV as shown in Figure 3. The highest absorption peak at ∼2.7 eV mostly arises from transitions from frontier orbitals (HOMO − 2, HOMO − 6, and HOMO − 8) to LUMO and LUMO + 1. The dynamic polarizabilities for this structure show no dispersions lower than 3.3 eV where a very large peak appears (Figure 3). This indicates that none of the absorption peaks up to 3.2 eV of excitation energy are one-photon resonances. The nature of the HOMO and LUMO (pictured in Figure 11) as well as the adjacent frontier MOs of structure A is reflected in PDOS and TDOS in Figure 7. The strength of the absorptions generally increases with the TDOS. Weak transitions are observed when the TDOS for the acceptor and donor states are small. For example, the lowest energy transition for structure A from HOMO to LUMO, exhibiting small TDOS for both HOMO and LUMO, as shown
FIG. 3. Absorption spectrum (bottom) and dynamic polarizability spectrum (top) for structure A.
in Figure 7, has very small oscillator strength. While the transitions for the higher energy excitations (2.76 eV and 2.77 eV embedded in a single spectral peak shown in Figure 3) have larger oscillator strengths and correspondingly larger TDOS for donor and acceptor states. These higher energy excitations are of different donor-acceptor charge transfer types. For structure A, the high energy transitions have mixed charge transfer
FIG. 4. Absorption spectrum (bottom) and dynamic polarizability spectrum (top) for structure B.
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FIG. 7. Partial and total density of states for structure A.
FIG. 5. Absorption spectrum (bottom) and dynamic polarizability spectrum (top) for structure C.
contributions dominated by one or the other type. Carbonto-silicon charge (HOMO − 8 → LUMO and HOMO − 2 → LUMO + 4) transfer dominates the 2.76 eV transition, while silicon-to-carbon charge transfer (HOMO − 6 → LUMO + 1) dominates the 2.77 eV transition, as is evident from Table IV. For structure B, all the observable absorptions occur in the energy range of 1.0-3.2 eV similar to structure A, as shown in Figure 4. Structure B has anomalous electronic transitions in
FIG. 8. Partial and total density of states for structure B.
FIG. 9. Partial and total density of states for structure C.
FIG. 6. Absorption spectrum (bottom) and dynamic polarizability spectrum (top) for structure D.
FIG. 10. Partial and total density of states for structure D.
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FIG. 11. Frontier molecular orbitals for structures A, B, C, and D.
which the HOMO to LUMO transition is at higher energy than the HOMO to LUMO+1 transition due to the exciton chargeseparation contribution. For structure B, the first excitation has weak oscillator strength of 0.0012 at 1.15 eV for absorption transition from HOMO to LUMO + 1. The HOMO-LUMO excitation having oscillator strength of 0.0512 happens to be at 1.28 eV, exactly half the energy of the strongest absorption from HOMO − 2 to LUMO + 1 at 2.56 eV with oscillator strength of 0.1414. The large absorption from HOMO − 2 to LUMO+1 compared with absorptions from HOMO to LUMO is due to much larger TDOS near the HOMO − 2 energy. The absorption transitions for structure B exhibit nearly pure charge transfer, unlike structure A; pure electronic density transfer between C regions and Si regions in structure B makes large contributions to the strong excitations. The higher and lowest energy transitions are nearly pure silicon-to-carbon transitions at 1.15 eV and 2.56 eV, while the intermediate energy transition is nearly pure carbon-to-silicon charge trans-
fer at 1.28 eV (Table IV). Unlike structure A, the dynamic polarizability spectrum of B, also shown in Figure 4, exhibits variability in the energy range of 1.0-3.2 eV. The largest polarizability change occurs at 2.55 eV which precisely corresponds to the strongest absorption in the spectrum indicating a onephoton resonance. The absorption transition at 1.28 eV is also coincident with a large change in dynamic polarizability, also indicating a one-photon resonance. The first excitation for structure C is at energy 0.94 eV. The oscillator strength for the excitation is very low with a value of 0.0001. The excitation is associated with the transitions of HOMO−1 to LUMO and HOMO to LUMO. Different from structure A and B, absorptions by structure C mainly occur in the lower energy range below 2.6 eV. The lower energy range for optical absorptions of structure C can be explained by the fact that frontier MOs of C are at lower energies compared to those of structures A and B. Two strong absorptions occur at 1.48 eV and 1.74 eV (Figure 5) which are
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mainly transitions from HOMO, HOMO − 3, and HOMO − 4 to LUMO+1 and LUMO + 2 for which TDOS is large (Figure 9). Unlike structures A and B, all the absorptions for C involve charge transfer from silicon to carbon. Influence that the large aromatic carbon region dominates transitions is illustrated by comparison of HOMO and LUMO wavefunctions in Figure 11. The dynamic polarizability from PBE0 functional for the structure C has a huge change occurred at 1.22 eV input energy, which precisely correspond to the first optical absorption peak indicating a one-photon resonance. The optical absorption of structure D was previously studied by Javan26 using local density approximation (LDA), GGA, and plane wave DFT and TDDFT methods. Using these methods, a single peak at ∼2.0 eV was found which Javan related to s-to-p orbital transitions without distinguishing carbon and silicon atom types. At variance, we found from our calculations that there are no absorptions for excitation energies below 3.0 eV (Figure 6). The highest absorption peak appears at 3.93 eV. As shown in Table IV, there are 3 degenerate transitions in both energy peaks, 3.32 eV and 3.53 eV, due to the highly symmetric electronic structure of D. The dynamic polarizability is nearly constant in the energy region below 3.0 eV. The largest polarizability accurately corresponds to the highest absorption peak indicating that it is a one-photon resonance. The optical absorptions for D are observed in the high energy region can be explained by the fact that the frontier MOs of the cluster have higher energies and the HOMO-LUMO gap is larger as compared to properties of the other three structures studied in the work. As shown in Table IV, the strong excitations mainly corresponded to the transitions between HOMO − n (n = 1-5) and LUMO + m (m = 1-5). The large difference in optical absorption between D and the other structures is the larger contribution of carbon PDOS in occupied frontier orbitals. Shown in Figure 10, the partial density of carbon atoms dominate the TDOS in the states around the occupied of frontier MOs. On the other hand, similar to other structures, the TDOS near unoccupied frontier MOs have larger contributions from PDOS of Si atoms. The DOS information shows that transition transfer charge exclusively from carbon atoms to silicon atoms in structure D. Molecules such as A and B have optical properties analogous to organic dye materials having special optical absorption and fluorescence properties which are finding utility in specialized photonics applications.
IV. CONCLUSIONS
Among the selected Si12C12 molecular isomer structures, structures A and B which have separate carbon and silicon segregations, are the most stable Si12C12 isomers. These structures are more stable because they maximize strong C–C bonds in carbon segregation regions, the Si–C bonds at the periphery and, within those constraints, maximize weaker Si–Si bonds in the silicon segregation regions. Structures like D that predominantly form Si–C bonds at the expense of C–C bonds have much higher energies at low temperatures. Structures like C that maximize the size of a single nearly planar carbon segre-
J. Chem. Phys. 142, 034303 (2015)
gation region at expense of forming Si–C bonds are less stable than A or B. The zero-temperature relative stability order B ≥ A > C > D becomes B ≈ D > A > C at T = 1100 K and changes to D > B > A > C temperature T = 1500 K according to the RMSD analysis. As anticipated, the different types of Si12C12 isomer structures exhibit different optical properties dominated by differences in charge transfer, both qualitative and quantitative. Some isomers exhibit large TDOS for frontier molecular orbitals, yielding large oscillator strengths. Structures having separate carbon and silicon regions, like A, B, and C, have transitions characterized by charge transfer between the carbon and silicon regions and produce lower excitation energies than does structure D. Unlike the other more stable structures, isomer D has bonding more like solid-state SiC in which there are not separate carbon and silicon segregations and frontier orbital charge transfer is from separate carbon atoms to silicon atoms exclusively. The very large difference between isomer D and the other Si12C12 structures that we considered is the larger contribution of carbon PDOS in occupied frontier orbitals. While the lowest energy isomers, A and B, both have separate carbon and silicon segregation regions, they have very different optical properties. Structure B is unique among small stable SinCm isomers (n > 4) as regards its high symmetry, while A has no special symmetry. As regards to their optical absorptions, molecule B has absorptions with largest optical oscillator strength characterized by pure charge transfer between the silicon and carbon segregations regions, unlike mixed charge transfer transitions of isomer A. For isomer B, a strong one-electron transition in the visible blue transfers charge from the silicon region to the carbon region; while a strong one-electron transition in the near infrared, at half the energy of the blue transition, transfers charge from the carbon region to the silicon region. ACKNOWLEDGMENTS
The views expressed in this work are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the United States Government. We acknowledge financial support from Molecular Dynamics and Theory Research Program of the Air Force Office of Scientific Research managed by Dr. Michael Berman. The DoD High Performance Computing Modernization Program and the AFRL Supercomputing Resource Center (DSRC) are gratefully acknowledged for computer time and helpful support. We are grateful for helpful discussions with Professor Mark Gordon and Professor David Weeks. 1Y.
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