Temperature effect on the structural stabilities and electronic properties of X22H28 (X=C, Si and Ge) nanocrystals: A first-principles study

Temperature effect on the structural stabilities and electronic properties of X22H28 (X=C, Si and Ge) nanocrystals: A first-principles study Xiao-Lin Deng,1 Yu-Jun Zhao,1,2 Ya-Ting Wang,1 Ji-Hai Liao,1 and Xiao-Bao Yang1,2,a 1Department of Physics, South China University of Technology, Guangzhou 510640, People’s Republic of China 2Key Laboratory of Advanced Energy Storage Materials of Guangdong Province, South China University of Technology, Guangzhou 510640, People’s Republic of China


I. INTRODUCTION
Recently, there has been a great interest in hydrogenated diamond nanocrystals, 1,2 where hydrogenated diamond nanocrystals were isolated and synthesized. 3 Because of the biocompatibility and ultra-high hardness, hydrogenated diamond nanocrystals showed potential applications in the pharmaceutical industry. [4][5][6] Hydrogenated diamond nanocrystals can also be used as fluorescent label and photoelectric devices 7,8 owing to their high luminous efficiency. Characteristic optical properties 9 evolution for the hydrogenated diamond nanocrystals as a function of size, shape, and symmetry in the subnanometer regime have been measured in the gas phase. Theoretically, the simulated optical adsorption by combining first-principles calculations and Important Sampling Monte Carlo methods in the basic diamond nanocrystals is in quantitative agreement with the experiment, demonstrating compelling evidence for the role of quantum nuclear dynamics in the photophysics. 10 The indirect band gap of silicon (Si) limits its applications on optoelectronics, while Si nanostructures (such as porous silicon, 11 Si nanoparticles, 12 Si nanocrystals, 13 and Si nanocrystals embedded in Si oxide 14,15 ) have exhibited visible photoluminescence at room temperature 14 due to the quantum confinement effect. Compared to bulk Si, [16][17][18][19] there are few studies for the temperature effect on the Si nanocrystals. Franceschetti 20 theoretically calculated temperature dependence of the gap of Si nanocrystals using constant temperature molecular dynamics (MD) methods. Hartel et al. 21 investigated the temperature-dependent gap of the Si nanocrystals, which were embedded in Si substrates. Similarly, germanium (Ge) nanocrystals have also stimulated extensive researches about the preparative technique 22,23 and the fundamental principles since the photoluminescence of Ge quantum dot. 24 Due to the fact of smaller gap, higher carrier mobility, and lighter effective mass, Ge nanocrystals can be used in charge storage, 25 infrared optics 26 and optoelectronics. 27 Especially, Ge is a candidate of green environment material, 28,29 which is non toxic compared with nanocrystals containing Pb, Cd, and Hg.
In our previous works, 30,31 we have studied that the ground states of hydrogenated group IV nanocrystals X m H n (X=C, Si, and Ge), as a function of the chemical potential of hydrogen. In this work, we use X 22 H 28 as an example to investigate the structural stabilities and electronic properties as a function of temperature with consideration of vibrational entropy effect. X 22 H 28 contains four face-fused cages, with three isomers 9 that are one, two, and three dimension structures (1D, 2D, 3D) respectively. The vibration free energies according to the calculated phonon spectrum and total free energies obtained from the constant-temperature molecular dynamics 32,33 methods were used to compare the relative stabilities of X 22 H 28 isomers, where the typical modes are shown to be dominant to the structural stability. Furthermore, we obtained the gap variance of X 22 H 28 from the constanttemperature molecular dynamics, where there is significant gap reduction as the temperature increases from 0 K to 300 K with the decrements are 0.2 /0.5 /0.6eV for C/Si/Ge nanocrystals respectively. In addition, we not only consider the distribution of Highest Occupied Molecular Orbital (HOMO) and Lowest Unoccupied Molecular Orbital (LUMO) levels at zero temperature, but also the temperature effect on atomic attributions to HOMO and LUMO levels.

II. COMPUTATIONAL METHODS
The first-principle calculations of X 22 H 28 nanocrystals were based on density functional theory (DFT) method implemented in the Vienna ab initio simulation package 34,35 (VASP). The generalized gradient approximation (GGA) functional 36,37 was employed for the exchange-correlation energy. With a mesh of 1×1×1, all the structures are fully relaxed by the conjugate gradient minimization and the convergence of the forces on each atom is less than 0.01eV/Å. The cutoff energy is 520eV (360eV) for carbon (silicon and germanium) nanocrystals and the vacuum distance is set to be 15Å. Using Nosé-thermostat 32,33 approach, we have performed the constant-temperature molecular dynamics simulations with the duration of 7 ps and the time step of 1fs. We recorded the energy gaps and the total free energies of hydrogenated C/Si/Ge nanocrystals after 3ps. At different temperature, the gaps and total free energies were obtained by averaging the corresponding values of every MD step. For the vibrational frequency calculations, 38 the higher accuracy is needed, so the corresponding cutoff energy was set to 550eV (400eV) for C (Si, Ge) nanocrystals and the convergence of the forces on each atom is less than 10 -7 eV/Å.

III. RESULTS AND DISCUSSIONS
In Sec. IIIA, we compare the relative stabilities for the three isomers of X 22 H 28 according to the vibration free energies and the total energies, where the low frequency vibrational modes are also shown to be crucial to the structural stabilities. In Sec. IIIB, the gap reduction of X 22 H 28 is discussed, and the relation between the gap and the variance of bond length is also analyzed. In Sec. IIIC, we show the distribution of HOMO and LUMO for X 22 H 28 , with analyzing the atomic attributions to HOMO and LUMO levels at various temperature.

A. Temperature effect on the stability
There are four isomers for X 22 H 28 , two of which are chirality. Thus we only consider three configurations (X 22 H 28 (S1-2D), X 22 H 28 (S2-1D), X 22 H 28 (S3-3D)), 9 as shown in the top panels of Fig.1. We find that the total energy of X 22 H 28 (S3) is the lowest compared to those of X 22 H 28 (S1) and X 22 H 28 (S2) at 0K through the first-principles calculation. To study the thermodynamics properties of nanocrystals, we consider the vibration free energies under the quasi-harmonic approximation, which can be written as 39 Here E 0 is the total energy at 0 K and ω i is the frequency of different vibrational mode, as both can be easily obtained from DFT calculations. The second term on the right side of Eq. (1) is zero point energy, which makes a positive contribution to the vibration free energies. is the reduced Planck constant, and k is the Boltzmann constant. We define the relative vibration free energies (∆F vib = F vib − F 0 vib (X 22 H 28 (S3))), where F 0 vib (X 22 H 28 (S3)) is the vibration free energy of X 22 H 28 (S3) at T=0K.
The ∆F vib of X 22 H 28 isomers as a function of temperature is shown in the panels of the middle row of Fig. 1. The vibrational free energy of X 22 H 28 (S3) is the lowest among three configurations, indicating that X 22 H 28 (S3) is the most stable one at T= 0 ∼ 300K. In order to further confirm this, we obtain the total free energies (F tot ) of these configurations at different temperature by averaging the energies of the last four thousand MD steps, where the relative total free energies (∆F tot = F tot − F 0 tot (X 22 H 28 (S3))) are also shown in the bottom panels of Fig.1. From the MD simulations and the vibration free energies, X 22 H 28 (S3) is the most stable structures among three configurations for X=C and Si at 0-300K. However, the differences in the free energies among these isomers are larger in the MD simulation as the temperature increases, compared to that from the vibration free energy of Eq.(1), which is under the quasi-harmonic approximation. For Ge 22 H 28 , the MD simulations show that there might be a transition from Ge 22 H 28 (S3) to Ge 22 H 28 (S1) when the temperature exceeds 60K, while Ge 22 H 28 (S3) is the most stable one among three configurations at 0-300K according to the vibration free energy.
In our calculations, we have found that the low frequency vibrational modes make a main contribution to the vibration free energies as the temperature increases according to Eq. (1). We have displayed the lowest frequency vibrational modes and corresponding vibrational frequency of X 22 H 28 in Fig.2, which indicates that the lowest frequency vibrational modes are similar in the same configuration of C 22 H 28 , Si 22 H 28 , and Ge 22 H 28 . Besides, the vibrational frequency of C 22 H 28 is the largest, and the one of Ge 22 H 28 is the smallest in the same configuration. Besides, the atoms near the surface are more important to the low frequency vibrational modes compared to the atoms inside.

B. Temperature dependence of the energy gap
The energy gap is one of the most important electronic properties of nanocrystals, while the materials are always measured experimentally at specific temperature (e.g. room temperature). We have obtained the gap of X 22 H 28 nanocrystals at different temperature (T=100K, 200K and 300K) by averaging the values of the last four thousand MD steps (shown in Fig. 3), where the gap reduction depends on both the shape and the group-IV elements. The gap decrement of C 22 H 28 is the smallest at the same temperature, while the one of Ge 22 H 28 is the largest among these nanocrystals. For example, the gap reduction of X 22 H 28 (S1) at T= 300K is 0.190eV, 0.388eV, 0.592eV for C, Si, and Ge respectively, where there are similar phenomena for the X 22 H 28 (S2) and X 22 H 28 (S3). Meanwhile, the shape is also important to the gap reduction, where the decrement of X 22 H 28 (S2) is smallest among these nanocrystals. For example, the gap reduction at T= 300K is 0.388eV, 0.304eV, 0.509eV for Si 22 H 28 (S1), Si 22 H 28 (S2), and Si 22 H 28 (S3), respectively. However, the difference between the gap reduction of X 22 H 28 (S1) and X 22 H 28 (S3) are not obvious for C and Ge.
We have also calculated the average variance of all the bond lengths of every MD step for the last 4000 MD steps compared with their corresponding bond lengths at zero temperature. The correlation between the variance of the bond length and temperature is also shown in Fig.3. We find that the variance of the bond length enlarges as the temperature increases, while the gap decreases. There are similar results for the three nanocrystals. Besides, C 22 H 28 has the smallest gap reduction and variance of bond length, while Ge 22 H 28 has the largest. Thus, the gap reduction might be mainly attributed to the variance of bond length.

C. Temperature effect on the charge distributions
In order to study the temperature effect on the electronic properties, we firstly analyzed the distribution of HOMO and LUMO levels at T= 0 K, as shown in the Fig. 4 three types of Si atoms are the smallest, while they are larger for Si 22 H 28 (S1) and Si 22 H 28 (S3). Note that the gap reduction is the smallest in Si 22 H 28 (S2) as the temperature increases, while it is larger for Si 22 H 28 (S1) and Si 22 H 28 (S3). There are similar phenomenon for C 22 H 28 and Ge 22 H 28 , which would provide an understanding that the gap reduction is smaller in X 22 H 28 (S2) compared to that in X 22 H 28 (S1) and X 22 H 28 (S3).

IV. CONCLUSIONS
In summary, we have investigated the temperature effect on the structural stabilities and electronic properties of X 22 H 28 by the first-principles calculations by considering vibrational entropy effect. The differences in the free energies among the isomers are larger in the MD simulation as the temperature increases, compared to that under the quasi-harmonic approximation. There is a significant gap reduction for the X 22 H 28 as the temperature increases, where the decrement of C 22 H 28 's gap is the smallest and that of Ge 22 H 28 is the largest. The shape is also important to the gap reduction, since the decrement of one dimension structure (X 22 H 28 -1D) is smallest among these three kinds of isomers. In the one dimension structure, the contribution differences from the inner and surface atoms to the HOMO and LUMO levels among the three types of X atoms are the smallest, while they are larger for the two (X 22 H 28 -2D) and three dimension (X 22 H 28 -3D) structures. Our finding would provide a better understanding of the temperature effect on the properties of small nanocrystals.