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https://journals.aps.org/prb/abstract/10.1103/PhysRevB.74.153203
DOI: 10.1103/PhysRevB.74.153203
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Thermodynamics of type-I and type-II Si clathrates at zero pressure: Monte Carlo simulations
Caetano R. Miranda*and Alex Antonelli†
Instituto de Física “Gleb Wataghin,” Universidade Estadual de Campinas, Caixa Postal 6165, CEP 13083-970, Campinas, São Paulo, Brazil
共Received 20 April 2006; revised manuscript received 5 September 2006; published 27 October 2006兲
We have investigated the thermodynamic and structural properties of types I and II silicon clathrates through Monte Carlo simulations. Using efficient methods to determine free energies, we studied the stable and metastable relations between the various phases of Si, namely, crystalline, liquid, amorphous, and the two types of clathrates, at zero pressure. We determined the melting point of Si46共type I兲 and Si34共type II兲 clathrate
structures to be at 1482± 25 and 1522± 25 K, respectively. Our result for the melting point of Si34is in good
agreement with the experimental value of 1473 K. Our results also indicate that both clathrate forms are more stable than amorphous silicon for any temperature up to their melting point.
DOI:10.1103/PhysRevB.74.153203 PACS number共s兲: 65.40.⫺b, 65.20.⫹w, 64.70.⫺p, 64.60.My
I. INTRODUCTION
Since they were first synthesized 40 years ago, the open-framework silicon compounds called clathrates1 have been
the focus of considerable research. The first silicon clathrates consisted of sodium atoms trapped inside silicon cages. Starting from these early semiconductor clathrates, a whole new class of Si-based materials has been developed by re-placing Si by Ge, Sn and Ga, and having other guest atoms encapsulated in their cages such as Sr and Ba. These mate-rials display striking properties, such as superconductivity,2
very low thermal conductivity,3,4 large band gap,5 and low
compressibility or large hardness,6 which make them attrac-tive for applications.
Although silicon clathrates can in principle exist in a va-riety of different structures,7 so far two of them have been
synthesized and studied in detail, namely, types I and II. The structure of the clathrate type I is a combination of six tetra-kaidecahedra 共24-atom cages兲 and two dodecahedra 共20-atom cages兲 per unit cell. The resulting lattice is simple cubic with 46 atoms in the primitive cell. The cubic unit cell of the clathrate type II results from the arrangement of two hexa-kaidecahedra共28-atom cage兲 and four dodecahedra, compris-ing 136 atoms altogether. The clathrate type II can also be seen as 34 atoms arranged in a fcc unit cell.
Recently, a guest free type II clathrate of silicon was ob-tained by Griko and collaborators8 by successive vacuum
treatment and density separation of NaxSi136-based materials. This remarkable achievement brought to light a new form of elemental silicon, which is surprisingly stable, even upon heating to anneal extended defects, whose presence enable the removal of sodium atoms. This new material has a large band gap共1.9 eV兲 and remarkably low phonon thermal con-ductivity that could be used in electronics applications. The melting point of this silicon form has been reported to be 1473 K,9 which, although significantly lower than that of
crystalline silicon in the diamond structure共cd-Si兲, demon-strates the extraordinarily high thermal stability of this ma-terial.
Thus, it became clear that in order to devise new synthesis routes for these materials a better understanding of their ther-modynamic properties is highly desirable. Computer
simula-tions have been extensively used to that end. Munetoh et
al.10 observed through molecular dynamics simulations that
both guest free clathrates Si34 and Si46 could be grown via solid phase epitaxy 共SPE兲 from amorphous/clathrate inter-faces. The same group also found that the melting tempera-tures for these clathrate forms would be lower than that of
cd-Si.11 Based on these results, they proposed that silicon
guest free clathrates could be obtained by liquid-phase epi-taxy共LPE兲. Both studies10,11 were performed using the
Ter-soff potential13 to model the interatomic forces in silicon,
which is known to overestimate the melting temperature of
cd-silicon共2550 K兲. It was found that the melting point for
Si34 and Si46 to be at 1590 and 1570 K 共after correction to account for the overestimation of the melting temperature兲, respectively. Moriguchi et al.11 have also studied
thermody-namic properties, such as Gibbs free energy and entropy of both clathrate forms, within the harmonic approximation 共HA兲.
Since silicon clathrate structures have atomic volumes larger than that of cd-Si, it has been suggested that these structures could be thermodynamically stabilized by negative pressures. Wilson and McMillan12 have explored, through
molecular dynamics simulations, the negative pressure re-gion of the phase diagram of silicon. According to their cal-culations, at negative pressures the Si34 structure becomes stabilized relative to cd-Si. They propose that, if the silicon sample can be stretched before and during melting, equilib-rium growth of Si34 will occur from l-Si, as the clathrate phase becomes more stable than and comparable in volume with the cd-Si structure. They have also determined the melt-ing point of Si34 at zero pressure to be 1507 K. Wilson and McMillan used a Stillinger-Weber three-body potential14for
silicon in their simulations.
More recently, Kaczmarski, Bedoya-Martínez, and Hernández15 presented a very thorough study of the phase diagram of silicon using a tight-binding approach. The melt-ing point at zero pressure of Si34 was found to be 1424 K, with the melting point of cd-Si for this tight-binding model of silicon being at 1551 K.
In this work, we present a computational study of the thermodynamic properties at zero pressure of different phases of silicon, namely, both clathrate forms Si34and Si46,
cd-Si, l-Si, and amorphous silicon 共a-Si兲. The calculated
Gibbs free energies allow us to determine the stable and metastable relations between these various phases. From these results we have determined the melting point for the two clathrate forms. We also discuss the implications of our findings for possible routes for the growth of these new forms of elemental silicon.
II. METHODOLOGY AND COMPUTATIONAL DETAILS We have chosen to use the environment dependent inter-atomic potential16 共EDIP兲 in our calculations because it
pro-vides the most comprehensive description of the three well known forms of silicon, namely, cd-Si, l-Si, a-Si.17,18In
or-der to compare some of our results with those in Ref.11, we have also carried out calculations using the Tersoff potential for silicon.13
We used in our free energy calculations the reversible scaling 共RS兲 method, which allows the computation of the free energy for a broad temperature range using only one computer simulation.19,20. The implementation of the RS
method requires a reference value for the free energy at a given temperature. This free energy reference was obtained using the adiabatic switching共AS兲 method,21,22 i.e., by
cal-culating the work done to transform the system of interest into a reference system. We have used an Einstein crystal 共collection of harmonic oscillators兲 as a reference system for the solid forms and an inverse 12th-power fluid for the liquid phase. Aside from computational efficiency, these method-ologies also take into account all anharmonic effects.
The computer simulations were carried out using the iso-baric Monte Carlo 共MC兲 method.23 The MC method was
implemented using the Metropolis algorithm. The RS-MC runs used typically 5⫻105MC sweeps. Each RS-MC sweep consists in attempting to move each one of the atoms in the computational cell plus one attempt to change the volume of the computational cell.
We have employed 2⫻2⫻2 supercells for Si34 共272 at-oms兲 and Si46 共368 atoms兲 and a supercell with 216 atoms for diamond structures subject to periodic boundary condi-tions. The finite size effects were estimated by performing simulations using 3⫻3⫻3 supercells for both Si34and Si46 and computational cells with 512 atoms for cd-Si. No sig-nificant differences in the thermodynamic and structural properties were observed. In the case of a-Si, we used a perfectly fourfold coordinated 216-atom simulation cell ob-tained by the procedure of bond-switching proposed by the WWW procedure.24The systems were kept at zero external
pressure during the entire simulations.
III. RESULTS AND DISCUSSION
The importance of anharmonic effects on the thermal properties of the clathrate forms can be seen in Fig. 1. It displays the Gibbs free energy of Si34 and Si46 obtained through RS-MC simulations in comparison with the results in Ref.11using the HA. The inset in Fig.1shows the Gibbs free energy of cd-Si obtained from RS-MC simulations and the harmonic approximation in comparison with
experimen-tal results.25 These results were obtained using the Tersoff
potential.13 It is clear from Fig.1that at high temperatures,
the anharmonic effects become very important and the deter-mination of the melting temperature, for example, can change significantly if these effects are not taken into ac-count.
We now turn the attention to the Gibbs free energy calcu-lations in order to determine the stable and metastable rela-tions between the various forms of silicon. Figure2 depicts the Gibbs free energy for Si34, Si46, l-Si, a-Si, and cd-Si. Our RS-MC simulations include only the vibrational entropy con-tribution to the Gibbs free energy, in the case of a-Si, there is an additional contribution to the entropy, namely, the con-figurational entropy. Vink and Barkema26 have shown that
perfectly fourfold coordinated amorphous Si structure has a configurational entropy of 0.93kB/ atom. Since no atomic
re-FIG. 1. Gibbs free energy of Si34and Si46at zero pressure using RS-MC simulations and HA calculations from Ref.11. The inset shows the Gibbs free energy of cd-Si obtained using RS-MC and HA in comparison with experimental results from Ref.25. Details in the text.
FIG. 2. 共Color online兲 Gibbs free energy of the various phases of silicon. The inset shows the intersection of the l-Si curve with those of the other forms of silicon.
BRIEF REPORTS PHYSICAL REVIEW B 74, 153203共2006兲
arrangements were observed in our simulations of the amor-phous structure, we assume the configurational entropy to be independent of the temperature. The configurational entropy contribution was included in our calculations of the Gibbs free energy of a-Si depicted in Fig.2.
From Fig.2it is clear that the Gibbs free energies of the clathrates Si34 and Si46are greater than that of cd-Si for the entire temperature range, therefore, at zero pressure, both clathrate structures are metastable. From Fig.2 we can ex-tract the melting temperatures from the crossing of the Gibbs free energy curves for these three forms of silicon with that corresponding to l-Si. For the EDIP model, cd-Si melts at 1582± 25 K, which is about 100 K below the experimental temperature. It should be pointed out the agreement with the experimental melting temperature of 1687 K is quite good, considering that data for l-Si was not used in the parametri-zation of the EDIP model.18For the EDIP model, the melting
of the clathrate structures Si34 and Si46 occurs at 1522± 25 and 1482± 25 K, respectively. There is a good agreement with the experimental melting temperature of Si34, estimated to be at 1473 K.9 These results suggest that the clathrate
forms of silicon could, in principle, be grown by LPE from clathrate seeds containing guest atoms, as proposed by Moriguchi et al.,11although it may be very difficult to
pre-vent supercooled l-Si from crystallizing as cd-Si. In TableI
we summarize the relevant results for the thermodynamic properties of types I and II silicon clathrates in comparison with those of cd-Si, at their respective melting temperatures. Regarding the computer simulation studies of the melting point of Si34, it is important to note that, except for the case of the Tersoff potential, which largely overestimates the melting temperature of cd-Si, the results obtained using the Stillinger-Weber potential,12the tight-binding approach15and
the EDIP model共this work兲 are in the range of ±50 K around the estimated experimental melting point of Si34. This spread corresponds to only ±3.3% of the estimated experimental melting temperature of Si34at 1473 K.
Also from Fig.2 one can see that, at zero pressure, the clathrate form Si34 is more stable than the Si46 form for the interval of temperatures we studied, and that both structures are more stable than a-Si. These results are consistent with previous findings in the literature10 that suggest that the
clathrate forms could be obtained from a-Si by SPE. These findings also explain the switch from the growth of Si46 to Si34 observed during SPE growth of the former in MD simulations.10Our results also suggest that this switch could
be avoided by lowering the temperature of growth, that would possibly prevent the system to overcome the kinetic barrier between the two clathrate forms.
The melting temperature depends on the interplay be-tween enthalpy and entropy contributions to the Gibbs free energy. Figure 3 displays the vibrational entropy for Si34, Si46, and cd-Si. It shows that the vibrational entropy of both clathrate forms are essentially identical, of the order of the magnitude of the numerical fluctuations. In the high tempera-ture regime, the entropy of the clathrates is greater than that of cd-Si by 0.2kB/ atom. This result is to be expected, since
the open structure of the clathrates would lead to a larger vibrational entropy. We have also performed the same calcu-lations using the Tersoff potential. These calcucalcu-lations showed that the entropy of the clathrates is basically identical to that of cd-Si. This agrees with those obtained by Moriguchi et
al.11 using the HA. These results show that although these
potentials give similar results for some properties of silicon, the descriptions of anharmonic effects can be quite different. A very recent calculation of the entropy difference between
cd-Si and Si34, employing the local density approximation,27
found it to be 0.16kB/ atom at 600 K, which agrees with our
EDIP calculation.
IV. CONCLUSIONS
We have studied the stability and metastability relations between Si34, Si46, a-Si, l-Si, and cd-Si at zero pressure through MC computer simulations. The calculations indicate that, in the entire temperature range Si34 is more stable than Si46. The melting point of Si34 was determined to be at 1522± 25 K, which is in good agreement with the experi-mental melting temperature of Si34, estimated to be at 1473 K,9 whereas the melting point of Si
46 occurs at 1482± 25 K. These results suggest that Si guest free clathrate forms could be grown by LPE, although it would be very difficult to avoid crystallization of supercooled l-Si as
cd-Si. We also found that both Si clathrate structures are
more stable than a-Si for any temperature. Our results
sug-TABLE I. Thermodynamic properties of Si34, Si46, and cd-Si at
their respective melting temperatures and zero external pressure us-ing RS-MC and the EDIP approach. Table shows meltus-ing tempera-tures Tm, absolute entropies of the solid phases Ss, latent heat of fusion⌬Hsl; and volume of the solid phases Vs.
Si34 Si46 cd-Si
Tm共K兲 1522 1482 1582
Ss共kB/ atom兲 7.57 7.50 7.65 ⌬Hsl共eV/atom兲 0.39 0.36 0.42 Vs共Å3/ atom兲 23.07 22.70 20.15
FIG. 3. Calculated entropy of the two clathrate forms Si34and Si46, cd-Si, using the EDIP approach, and experimental data for cd-Si from Ref.25.
gest that Si clathrate forms could be obtained from a-Si by SPE. Furthermore, since our simulations indicate that Si34 and Si46 are more stable than a-Si for any temperature, the switch in the growth of Si46 to Si34 during simulations of SPE共Ref.10兲 could be avoided if the SPE growth were done
at a lower temperature, because it would become more diffi-cult for the system to overcome the kinetic barrier between the two clathrate structures.
ACKNOWLEDGMENTS
We acknowledge the financial support by the Brazilian funding agencies FAPESP, CNPq, and FAEP-UNICAMP. We thank M. F. Thorpe for kindly providing the WWW amor-phous structure. The computational work was partially done at CENAPAD/São Paulo.
*Email address: cmiranda@mit.edu; Current address: MIT-DMSE, BLDG 13-5001, 77 Mass. Ave., Cambridge-MA, 02138.
†Email address: aantone@ifi.unicamp.br
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