PHYSICAL REVIEW B 82, 075209 共2010兲
First-principles calculations of carbon clathrates: Comparison to silicon and germanium clathrates Damien Connétable* CIRIMAT UMR 5085, CNRS/INP/UPS, Ecole Nationale d’Ingénieurs en Arts Chimiques et Technologiques (ENSIACET) 4, allée Émile Monso, BP 44362, F-31030 Toulouse Cedex 4, France 共Received 19 December 2009; revised manuscript received 18 June 2010; published 26 August 2010兲 We employ state-of-the-art first-principles calculations based on density-functional theory and densityfunctional perturbation theory to investigate relevant physical properties and phase diagram of the free guest type-I 共X-46兲 and type-II 共X-34兲 carbon clathrates. Their properties and those of silicon and germanium diamonds, and clathrates have been computed and compared within the same approach. We briefly present and discuss their structural, cohesive, and electronic properties. In particular, we present different results about electronic properties of carbon clathrates. From the symmetry analysis of electronic states around the band gap, we deduce their optical properties, and we forecast the effects of hypothetical-doped elements on their electronic band gap. We then report first-principles calculations of vibrational, thermodynamical, and elastic properties. Whereas vibrational properties of Si and Ge systems can be linked through their atomic weight ratio, we show that the vibrational properties of carbon structures differ strongly. Raman and infrared spectra of all clathrates are also calculated and compared. The effects of pressure and temperature on thermodynamical properties 共heat capacity, entropy, thermal expansion, etc.兲 within static and quasiharmonic approximations are investigated. It is shown that thermodynamical properties of carbon clathrates and diamond present a similar evolution up to high pressures 共100 GPa兲 and over a large range of temperatures 共关0, 1500兴 K兲. Then we deduce the equilibrium phase diagram 共P , T兲 of C-2/C-34/C-46. We conclude the paper with a presentation of elastic properties computed from acoustic slopes. DOI: 10.1103/PhysRevB.82.075209
PACS number共s兲: 82.75.⫺z, 78.30.Na, 63.20.⫺e, 65.40.De
I. INTRODUCTION
Clathrates have emerged these last decades and have appeared to belong to be a family of compounds with high potentials. The main feature of clathrates is the possibility of tuning their structures: either strongly doped systems 共until 20%兲 or substituted systems. Their properties are then correlated with this tuning: a high thermoelectrical power,1–4 adapted electronic features 共either n, p, or both type semiconductors in function to the intercalated and substituted elements兲,5–8 a large electron-phonon coupling,9,10 high stability under pressure.11 Some clathrates present several of these properties: one can cite, for example, the barium or the hypothetical iodine-doped silicon clathrates.5 Doping elements are changing the carrier concentration in free guest clathrates intrinsic bands and modifying the position of the Fermi level in the system. Similarly, carbon element exists under a large variety of allotropic structures: from threedimensional 共diamond, Si-C, B-C-N alloys, etc.兲, twodimensional 共graphene, graphite, intercalated graphene兲, one-dimensional 共nanotubes兲, to zero-dimensional networks 共fullerenes, clusters兲. Each of these structures exhibits actual or potential applications in diverse applied fields. Carbon clathrates should combine the originality of clathrates 共tuning, high doping兲 with a light element. Due to their speculative existence, the literature on carbon clathrates is not exhaustive. The published studies deal with their lattice parameters,12–15 bulk modulus, stability, phase diagram,13,16 electronic band structures,17,18 plastic features,19 and about their potential superconductivity.10,18,20,21 Studies of their fundamental vibrational and associated properties remain nevertheless limited within a full density-functional-theory 共DFT兲 approach. As it has been shown, carbon clathrates 1098-0121/2010/82共7兲/075209共12兲
should yield large critical temperature and have a hardness similar to diamond. It is also significant to explore any special features in the thermal expansion, heat capacity, Raman, infrared spectra, and phase diagram of these unusual expanded-framework semiconducting crystals. The purpose of this paper is to present and discuss, by means of a first-principles approach, the properties of two empty carbon clathrates -C-34 and C-46-, and to compare them to other empty clathrates of group IV 共Si and Ge兲. The first part of the manuscript is briefly devoted to the structural, energetic, and electronic properties, which are presented and compared to those of the literature. An analysis of symmetries of electronic states around the band gap of carbon clathrates is in particular displayed. In the second part, using the density-functional perturbative theory, we present the properties calculated from dynamical properties: an analysis of phonon spectra, of Raman and infrared intensities, a study of the effect of temperature and pressure on thermodynamical properties and the 共T , P兲 phase diagram of C-2/C-34/C-46. We conclude by a discussion on their elastic properties. II. PRESENTATION
Clathrates are composed of polyhedral cages which are sharing their faces. The atoms are linked together through sp3-type covalent bonds. The nearest-neighbor distances and the angles between bonds are approximately the same as those found in the diamond systems. Their main difference with diamonds is the presence of 87% of pentagonal cycles, where there are only hexagonal cycles. These pentagonal cycles are expected to be at the origin of the specific electronic and dynamical properties of clathrates.22 Besides, the originality of cagelike materials is the possibility of a huge
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©2010 The American Physical Society
PHYSICAL REVIEW B 82, 075209 共2010兲
DAMIEN CONNÉTABLE TABLE I. Wyckoff positions of type-I and type-II clathrates. ¯ m: No 227兲 X-34 共Fd3 8a共2兲 32e共8兲 96g共24兲
x , y , z = 0 and =1 / 4 x , y , z = −0.092 x , y = −0.058; z = 0.246
¯ n: No 223兲 X-46 共Pm3 6c 16i 24k
x = 1 / 4; y = 0; z = 1 / 2 x , y , z = 0.184 x = 0; y = 0.308; z = 0.117
endohedral doping. Clathrates can thus be doped either by inserting 共intercalation兲 until one atom in each cage, which corresponds to 20% of doping, or by substitution. Doping of clathrates is thus much easier than doping of the diamond phase 共only some percent by substitution兲. The type-I clathrate, namely, X-46, has a simple cubic ¯ n 共No. 223兲 lattice with 46 atoms per unit cell and the Pm3 group symmetry 共see Table I兲. The type-II clathrate, called X-34 or X-136, is a face center cubic lattice with 34 atoms in ¯ m 共No. 227兲 symmetry. the unit cell and the Fd3 III. THEORETICAL FRAMEWORK
Calculations have been performed using the DFT-based package 共Ref. 23兲. The local density approximation, through the Perdew-Zunger parameterization24 of the Ceperley-Alder exchange correlation functional,25 and norm-conserving Troullier-Martins26 pseudopotentials factorized in Kleinman-Bylander form27 have been employed. To optimize the structures, the Brillouin zone has been sampled by a 2 ⫻ 2 ⫻ 2 Monkhorst-Pack grid28 which corresponds to one and two irreducible k points for X-46 and X-34, respectively. The wave functions are expanded on plane-wave basis with kinetic energies up to a real-space grid with 50 Ryd 共for carbon兲 and 20 Ryd 共for germanium and silicon兲 energies cutoff. For the electronic density of states 共eDOS兲 and the formation energies, we have adopted finest grids 共10⫻ 10 ⫻ 10兲 with higher energies cutoff 共60 Ryd for carbon and 30 Ryd for Si/Ge systems兲. The phonon frequencies have been determined by the dynamical matrix, Born effective charges, and dielectric constants, which have been computed using the linear response theory of the DFT.29,30 In order to generate the force constants matrices and carry out the inverse Fourier transformation, 2 ⫻ 2 ⫻ 2 q grids have been employed for static properties 共with 6 ⫻ 6 ⫻ 6 k grids mesh, and 50 and 20 Ryd of energies cutoff for carbon and silicon/germanium systems, respectively兲. For diamond phases, we have increased the k and q grids 共15⫻ 15⫻ 15 and 8 ⫻ 8 ⫻ 8, respectively兲. The vibrational density of states has been calculated on 20⫻ 20 ⫻ 20 grids with the tetrahedron method. QUANTUM-ESPRESSO
TABLE II. Carbon, silicon, and germanium diamond and clathrates phases lattice parameters 共in Å兲, bulk modulus 共in GPa兲, and formation energy 共in meV/atom兲. The evolution of the atomic volume taken as reference diamond phase ␦V is given 关␦V = Vat共X-2兲 / Vat共system兲兴. System
ao
C-2 C-34 C-46 Si-2 Si-34 Si-46 Ge-2 Ge-34 Ge-46
3.58 共3.57a兲 9.60 共9.56b兲 6.65 共6.80d兲 5.41 共5.43a兲 14.53 共14.62f兲 10.08 共10.15g兲 5.58 共5.66a兲 15.02 共15.13h, 15.21i兲 10.40 共10.47h兲
␦V
Bo
1.00 460 共545a兲 0 b 0.86 393 共398 兲 150 共134c兲 0.87 395 共398e兲 110 共144c兲 1.00 97 共99a兲 0 0.87 86 共84g兲 60 共80g兲 0.88 87 共84g兲 70 共90g兲 a 1.00 75 共77 兲 0 0.88 68 共67j, 61h兲 40 共50h兲 0.89 68 共68j, 62h兲 50 共50h兲
aReference
fReference
bReference
gReference
31. 17. cReference 12. d Reference 20. eReference 19.
Ef
32. 33. hReference 34. i Reference 35. jReference 36.
predictions for the germanium 共see Refs. 34 and 36兲, silicon 共Refs. 12, 22, 32, and 33兲, and carbon elements 共Refs. 12–14, 17, 19, and 20兲. The electronic band structures 共eBS兲 and eDOS of carbon and silicon systems are plotted in Fig. 1 共electronic properties of germanium clathrates are not presented here兲. We found that all empty clathrates are semiconductors with a large gap. Band gaps 共⌬gap, Table III兲 are found greater in Ge/Si clathrates than in diamond phases, contrary to carbon clathrate, in agreement with previous DFT values.17,32,36 We provide a qualitative interpretation of the variation in this electronic band gap for silicon and carbon elements 共the case of germanium is different due to the presence of d core states兲: from chemical and crystallographic arguments. Gaps in ⌫ are found smaller in clathrates than in diamonds. Since
IV. STRUCTURAL AND ELECTRONIC PROPERTIES
In the Table II we report the optimized lattice parameters, the bulk modulus 共fitted by a Murnaghan’s equation of state37兲 as well as the cohesive energies of the corresponding species 共X = C, Si, and Ge兲 and crystallographic structures 共X-2 = diamond, X-34, and X-46兲. These results are found in good agreement with the experimental data and theoretical
FIG. 1. Electronic band structures 共in eV兲 and density of states. The Fermi level has been put at the top of the valance band.
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FIRST-PRINCIPLES CALCULATIONS OF CARBON… TABLE III. Band gaps 共⌬gap兲 in eV. C-2 3.9 共5.48a兲 Si-2 0.62 共1.17a兲 Ge-2 0.34 共0.74a兲
C-34 3.7 共5.15b兲 Si-34 1.25 共1.85b, 1.9d兲 Ge-34 0.95 共0.80c兲
C-46 3.7 共5.25b兲 Si-46 1.3 共1.9b兲 Ge-46 1.4 共1.46e, 1.31c兲
共Ref. 31兲. b GW simulations 共Refs. 17 and 18兲. c DFT simulation 共Ref. 36兲. d Experiment 共Ref. 38兲. eDFT simulation 共Ref. 39兲. aExperiment
clathrates are highly isotropic systems comparatively to the diamond, the dispersion of the electronic bands is weak outside ⌫ 共flat bands兲. Moreover, in the case of carbon, shells are less sensitive than silicon to the crystallographic environment 共no internal p shells, and the system is more compact兲, the band gap is smaller in carbon clathrates than in diamond. Contrarily, silicon is more affected by its crystallographic environment and the band gap is then reduced in diamond. For sake of completeness, the irreducible representation of electronic states around band gaps have been analyzed, as already done for silicon clathrates.8 For a given structure, the symmetries are found to be roughly the same for carbon and silicon clathrates: along different crystallographic directions, the states of the bottom of conduction bands and the top of valence bands have the same symmetries 共see Table IV at ⌫兲. Only some states, near the top of the valence bands, change. From these results and from our previous analysis of silicon clathrates, we deduce that, for carbon clathrates, the optical transitions should be forbidden by means of symmetries in a dipole approximation. Furthermore, the effect of doped elements on electronic properties of carbon clathrates can be forecast by symmetry. Only the bottom of the conduction TABLE IV. Symmetry classification of the valence and conduction bands around the electronic band gap for X-46 and X-34 systems where X = Si and C. The energies 共in eV兲 are calculated at the ⌫ point 共symmetry Oh兲. The reference energies have been set at the top of the valence bands in the ⌫ point.
E v3 E v2 E v1 Ec1 Ec2
E v3 E v2 E v1 Ec1 Ec2
C-46
Si-46
−0.19共Eu兲 −0.05共T1g兲 0.00 共A1u兲 +4.06共T1u兲 +5.34共A2g兲
−0.11共Eu兲 −0.10共T2g兲 0.00 共A1u兲 +1.45共T1u兲 +1.82共A2g兲
C-34
Si-34
−0.43共T2u兲 −0.17共T1g兲 0.00 共Eu兲 +3.63共T1u兲 +6.40共A2u兲
−0.10共T2u兲 −0.01共Eu兲 0.00 共T1g兲 +1.46共T1u兲 +2.22共A2u兲
states should be hybridized by states of the intercalated element. So, the electronic gap should be increased like in silicon clathrates. V. DYNAMICAL PROPERTIES A. Vibrational properties 1. Phonon dispersion curves
Our results for the phonon-dispersion curves 共vband兲 of diamond and clathrates 共C, Si, and Ge elements兲 along several symmetry lines together with the corresponding vibrational density of states 共vDOS兲 are displayed Fig. 2. Numerical values at the ⌫ point are listed in Tables V and VI for X-34 and X-46 clathrates, respectively, and compared with literature data. The vibrational spectra of diamond phases— not reported here—are in agreement with experimental data 共discrepancy in frequencies less 1%兲. According to the Oh group symmetry and the Wyckoff positions of atoms in the network 共see Table I兲, the symmetry of the vibrations 共noted vSM兲 are the following:
冦 冦
vSM6c = A2g 丣 Eg 丣 T1g 丣 T2g 丣 T1u 丣 T2u vSM16i = A2g 丣 A1g 丣 2Eg 丣 3T1g 丣 3T2g 丣
A1u 丣 A2u 丣 2Eu 丣 3T1u 丣 3T2u
冧 冧
共1兲
vSM24k = 2A2g 丣 2A1g 丣 4Eg 丣 4T1g 丣 4T2g 丣 A1u 丣
A2u 丣 2Eu 丣 5T1u 丣 5T2u
for X-46 and for X-34
vSM8a = T1u 丣 T2g SM = A v 32e 1g 丣 A2u 丣 Eg 丣 Eu 丣 T1g 丣 2T1u 丣 2T2g 丣
T2u
共2兲
vSM96g = 2A1g 丣 A1u 丣 A2g 丣 2A2u 丣 3Eg 丣
3Eu 丣 4T1g 丣 5T1u 丣 5T2g 丣 4T2u
From selections rules43 and the knowledge of symmetries, we have forecast the Raman and infrared frequencies 共noted in the tables R and IR, respectively兲. The group theory assumes that, for cubic systems belonging to the Oh group symmetry, the A1g, Eg, and T2g irreducible representations could be Raman active while the infrared modes should have the T1u symmetry. For carbon clathrates, no theoretical values have been reported, except for the frequencies of active Raman modes in the hypothetical lithium-doped clathrate.16 In the case of other elements, theoretical frequencies have been already calculated 共see values reported in Tables V and VI兲. Raman measurements have been only done in the case of Si-34.40,41,44 Our results are in excellent agreement with those of the literature. We note that the vibrational dispersion curves for Si and Ge clathrate are very close, while vband and vDOS of carbon clathrates differ significantly from these systems. The difference between Si and Ge systems can be rationalized using their respective atomic weights,45 as already mentioned for diamond.46 The interatomic forces of C-34 and C-46 clathrates are different from the other clathrates in spite of a similar sp3 hybridization.
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FIG. 2. Vibrational band structures 共in cm−1兲 and density of states. We also report the contribution of atoms in 6c / 16i and 8a / 32e Wyckoff positions for C-46 and C-34, respectively, in thin lines and dashed lines.
In carbon clathrates, the transverse acoustic modes are no longer flat unlike in other Si/Ge systems, which explain why the sharp features of the vDOS 共subband兲 are lacking. As in Si/Ge clathrates, most of tetrahedral semiconductors display a peculiar flatness in the transverse acoustic frequencies region 共these bands are correlated with long range of inter atomic forces兲.46 In diamond, this difference is less tagged due to its higher compacity. A comparative analysis of the frequencies in diamond and clathrates reveals that there is in C-34/C-46 two “redshifts” 共for optical and acoustic modes兲 and one “blueshift” in acoustic modes and one “redshift” in optical modes in Si/Ge clathrates. The redshift is associated with the weakening of the X-X bonds between clathrates and diamonds. The frustration of the optical modes, due to the presence of pentagonal cycles, induces this shift of 40 cm−1 toward low frequencies in C-34/C-46 and Si/Ge clathrates. The number of optical modes, associated with the asymmetric vibrations of two neighbors atoms, decreases with the frustration. We can also note a gap in Si/Ge optical frequencies. This gap does not exist in carbon clathrates. The maximum of the optical branches is not located at the zone center for both C-34 and C-46 as in carbon diamond, contrary to other systems, which results in sharp peaks in vDOS for C-34, C-2 and—but less pronounced–for C-46. It is due to the overbending of the optical branches and has been associated, in diamond,47 to a sufficiently large interatomic force constants between second-nearest neighbor.
2. Raman shifts
Following Ref. 48, the Raman tensor and Raman spectra have been calculated from first-principles approaches. Raman intensities are plotted on Figs. 3 and 4 for type-I and type-II clathrates, respectively. The Raman cross section 共at the bottom兲 is convoluted with a Gaussian broadening and the Raman total intensities of powders are determined following Ref. 49 共Lorentzian convolution兲. A 10 cm−1 is used as damping constants. The parallel and perpendicular Raman intensities to the incident polarization have been calculated from the depolarization ratio r 共r = 3 / 4 for Eg and T1g, and r = 0 for A1g兲 and added on Figs. 3 and 4. As already mentioned, only the Raman spectrum of the free guest Si-34 has been investigated experimentally.40,41,44 Raman spectra has been calculated for Si-34 by Dong50 and Ge-34/Ge-46 共Ref. 34兲 using a bond-polarizability model. In Ge systems, the peaks are located at the same positions 共shifted, around 10 cm−1 different兲. In Si-34, three peaks are observed around 120, 250–300, and 470 cm−1, as claimed by Guyot.44 The calculated spectra is similar to the experimental spectra obtained by Tang et al.41 and Nolas et al.40 However, some peaks are missing: in Tang measurements, one A1g mode at around 320 cm−1, one T2g mode at 467 cm−1, or in Nolas measurements, the Eg and A1g modes at 454 cm−1 and 480 cm−1, respectively, what makes the comparison sometimes difficult. For example, we arrive to reproduce the vanishing of the A1g mode in parallel polarized Stokes Ra-
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FIRST-PRINCIPLES CALCULATIONS OF CARBON…
TABLE V. Frequencies 共in cm−1兲 and symmetries of the vibrational modes in ⌫ for X-34. C-34
Si-34
q,
Sym.
0 513 538 562 641 657 690 721 776 827 847 863 869 874 894 906 913 927 947 958 1003 1020 1032 1046 1056 1076 1094 1107 1114 1137 1145 1171 1175 1181 1189 1198 1200 1204 1218 1227 1245 1250
TIR 1u TR2g ERg T1g T1g T2u Eu TIR 1u T2u TR2g TIR 1u TIR 1u T1g A2u AR1g A2g TR2g Eu TR2g ERg TR2g A2u AR1g TIR 1u Eu T2u T1g A1u TIR 1u ERg T2u A2u TR2g T2u AR1g TIR 1u TR2g Eu TIR 1u T1g ERg TR2g
q,
Sym.
0 119 131 133 149 154 163 171 174 177 180 181 194 268 270 283 294 318 326 362 368 381 397 397 405 416 421 427 447 456 456 460 462 463 468 473 475 475 478 482 482 485
TIR 1u TR2g T1g ERg T2u T1g Eu TIR 1u T2u T1g TR2g A2g TIR 1u TIR 1u TR2g A2u Eu AR1g TR2g ERg T2u TIR 1u A2u AR1g TR2g A2u T1g TIR 1u Eu AR1g T2u TIR 1u Eu ERg TR2g TR2g TIR 1u T1g A1u ERg T2u TR2g
Ge-34
Expt. 共Refs. 40 and 41兲
117/120
TR2g
130/133
ERg
184/165
TR2g
271/272
TR2g
共316兲/320 324/333 360/362
AR1g TR2g ERg
387/383 401/404
AR1g TR2g
454/
AR1g
466/454 /467 472/471
ERg TR2g TR2g
480/
ERg
488/490
TR2g
man scattering spectra, at 200 cm−1 the relative intensity of the T2g mode is correctly reproduced in comparison to other peaks but at low frequency we have two peaks with the same intensity.
q,
Sym.
0 69 76 77 86 87 93 95 96 96 97 98 104 156 159 166 173 187 191 211 216 223 233 234 237 244 246 250 265 271 272 273 275 276 279 282 283 284 285 287 289 290
TIR 1u TR2g T1g ERg T2u T1g Eu T2u TIR 1u TR2g T1g A2g TIR 1u TIR 1u TR2g A2u Eu AR1g TR2g ERg T2u TIR 1u AR1g A2u TR2g A2u T1g TIR 1u Eu T2u AR1g TIR 1u Eu ERg TR2g TR2g T1g TIR 1u A1u ERg T2u TR2g
Theo. 共Ref. 34兲 0 54 72 74 79 81 81 84 88 90 91 93 97 154 157 162 170 180 187 205 210 218 225 227 234 236 243 248 261 268 269 270 271 273 274 275 276 279 281 282 283 284
TIR 1u TR2g T1g ERg TIR 1u T2u T1g TR2g Eu T1g T2u A2g TIR 1u TIR 1u TR2g A2u Eu AR1g TR2g ERg T2u TIR 1u A2u AR1g TR2g A2u T1g TIR 1u Eu T2u TIR 1u AR1g Eu ERg TR2g TIR 1u TR2g T1g TR2g A1u ERg T2u
The carbon spectra are significantly different from those of silicon or germanium networks: for C-34 only one mode A1g should present a significant active frequency, in parallel measurement. In contrast a perpendicular mea-
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TABLE VI. Frequencies 共in cm−1兲 and symmetries of the vibrational modes in ⌫ for X-46 systems. C-46
Si-46
q,
Sym.
0 457 505 576 588 591 623 623 637 682 695 716 732 752 781 818 824 851 879 878 882 909 920 946 948 955 992 1007 1015 1025 1033 1055 1058 1062 1075 1082 1088 1088 1092 1129 1140 1143 1152 1158 1167 1170 1192
TIR 1u TR2g T1g T2u ERg TIR 1u A2u T1g A2g T2u Eu ERg T1g TR2g TIR 1u T2u ERg Eu TR2g A2g TIR 1u T1g TR2g TIR 1u AR1g A2g T2u TIR 1u A1u T2u T1g TIR 1u AR1g ERg T1g T2u TR2g TIR 1u ERg A2u T2u Eu A2g TR2g TIR 1u ERg A1u
q,
Sym.
0 114 120 129 140 140 143 147 153 166 169 170 173 175 178 178 180 194 223 239 247 301 303 305 325 326 334 355 370 375 383 384 408 414 418 418 432 442 450 459 461 462 465 466 467 467 468
TIR 1u TR2g T1g A2u T1g T2u TIR 1u ERg Eu T1g T2u TR2g TIR 1u A2g TR2g Eu T1g TIR 1u ERg A2g T2u TIR 1u ERg TR2g T2u A1u AR1g T1g T2u TIR 1u A2g A2u TR2g ERg TIR 1u T1g AR1g T2u TIR 1u ERg T2u Eu TIR 1u TR2g T2u Eu T1g
Ge-46 Theo. 共Ref. 42兲 0 114 126 143 144 157 164 172 182 194 197 198 208 220 224 225 236 237 240 246 250 307 309 314 332 345 349 372 390 393 403 409 423 429 431 441 445 467 483 484 489 492 493 495 495 498 498
TIR 1u TR2g T1g T2u TIR 1u T1g TIR 1u ERg Eu T2u T1g TR2g TIR 1u Eu ERg A2g TR2g T1g A2g T2u TIR 1u Eg TIR 1u TR2g T2u AR1g A1u T1g T2u TIR 1u A2g A2u ERg T2u TR2g T1g AR1g T2u TIR 1u ERg T2u TIR 1u A2g T2u T2u Eu Eg
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q,
Sym.
0 65 70 76 79 81 82 82 86 93 93 94 94 96 96 96 97 103 131 138 142 176 177 179 190 193 195 209 217 220 223 226 240 242 242 245 251 260 268 272 274 275 277 277 277 278 278
TIR 1u TR2g T1g A2u T1g T2u TIR 1u ERg Eu T1g A2g TR2g T2u TR2g T1g TIR 1u Eu TIR 1u ERg A2g T2u ERg TIR 1u TR2g T2u A1u AR1g T1g T2u TIR 1u A2g A2u TR2g TIR 1u ERg T1g AR1g T2u TIR 1u ERg T2u Eu TIR 1u Eu TR2g TIR 1u T2u
Theo. 共Ref. 34兲 0 62 63 67 69 71 74 77 78 79 82 83 89 89 89 89 91 93 131 138 139 170 172 176 186 189 190 206 212 215 218 223 235 235 238 240 247 258 263 268 271 272 272 272 273 273 275
TIR 1u TR2g T1g ERg TIR 1u T2u T1g TR2g A2u A2g Eu T1g TIR 1u TR2g T1g T2u Eu TIR 1u ERg T2u A2g ERg TIR 1u TR2g T2u AR1g A1u T1g T2u TIR 1u A2g A2u TIR 1u TR2g ERg T1g AR1g T2u TIR 1u ERg T2u T1g TR2g TIR 1u T2u Eu TIR 1u
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FIRST-PRINCIPLES CALCULATIONS OF CARBON… TABLE VI. 共Continued.兲 C-46
q, 1192 1198 1198 1201 1202 1220 1223 1229 1260 1260
Si-46 Sym. T1g Eu T2u TR2g TIR 1u T1g ERg TR2g T2u AR1g
q,
Sym.
468 469 470 470 471 474 479 480 481 481
TR2g TIR 1u ERg A1u T1g TR2g T2u ERg A2g AR1g
Ge-46 Theo. 共Ref. 42兲 TR2g
498 498 500 504 504 505 511 512 515 516
T1g TIR 1u Eu T1g TR2g AR1g A1u TR2g T2u
surement must show more signals 共four groups of peaks兲. For C-46, we obtain a spectrum with six equivalent peaks. 3. Infrared spectra
We have calculated the infrared spectra from dynamical matrix, we have plotted it on the Fig. 5. Our results on sili-
q,
Sym.
278 279 279 279 281 281 284 284 285 286
T1g TR2g ERg A1u T1g TR2g ERg A2g T2u AR1g
Theo. 共Ref. 34兲 275 275 276 277 278 278 279 279 282 282
Eu ERg TR2g A1u T1g TR2g ERg A2g T2u AR1g
con clathrates are in agreement with those published by Li et al.,51 in spite of that the frequencies found by these authors are slightly shifted from ours. The infrared spectra are less sensitive to the element than Raman spectra. The spectra depend on the dynamical properties, which results from a strong similarity between systems. 4. Polarizability and dielectric constants
Finally, we report the polarizability 共␣兲 and the permittivity 共⑀⬁兲 in Table VII. Quantitatively, dielectric constants are
FIG. 3. Raman spectra of X-46 共from left to right: C, Si, and Ge兲 clathrates: from top to bottom figures: perpendicular, parallel, total Raman intensities, and total cross section convoluted with a uniform Gaussian broadening having 10– 5 cm−1 width. We have used a damping constant ⌫i = 10 cm−1 for all modes. Peaks are labeled in function to their symmetries.
FIG. 4. Raman spectra of X-34 共from left to right: C, Si, and Ge兲 clathrates, see Fig. 3 for labels.
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J TABLE VII. Dielectric constants 共⑀⬁兲 and polarizabilities ␣ = ␣JI 共in Å3 / atom兲.
FIG. 5. Infrared spectra of X-46 and X-34 共from left to right: C, Si, and Ge兲 clathrates.
in agreement with the values found by Blase17 within a GW approximation. The polarizability is lighter greater in clathrates than in diamond, and the Born effective charge is small 共less ⬍0.1 for Born charges兲 as in diamond phases, which accounts for the lack of LO-TO splitting.
C-2 C-34 C-46 Si-2 Si-34 Si-46 Ge-2 Ge-34 Ge-46
1. Static approximation
This approximation consists of a calculation of the vibrational free energy 共Fvib兲 at the optimized lattice parameter,
␣
5.7 共5.5a兲 4.8 共4.7b兲 4.9 13.6 共11.7a兲 10.0 共9.8b兲 10.4 18.3 共15.8a兲 12.4 12.8
0.84 0.86 0.87 3.73 4.06 4.03 4.40 4.67 4.65
Experiment 共Ref. 31兲. GW calculations, Ref. 17.
a
bTheory
where the effect of the temperature on the lattice parameter is neglected. The Helmholtz free energy, for nonmetallic systems, is then given by
B. Thermodynamical properties
The thermodynamical properties of clathrates and diamonds under pressure and temperature have been investigated from the interatomic constants. To compute these properties, two approaches for comparison were performed: the static approximation and the quasiharmonic approximation 共QHA兲. Only carbon systems have been studied within the framework of the QHA.
⑀⬁
F共T兲 = Uo + Fvib共T兲,
共3兲
where Uo is the internal energy of the cell at the optimized volume and Fvib共T兲 = kBT 兺 ln关2 sinh共q,兲兴 q,
共4兲
the vibrational free energy 共q, = បq, / 2kBT兲. The summation on the vibration frequencies has been performed on 20 ⫻ 20⫻ 20 tetrahedron grids. The heat capacity and the entropy have been evaluated following the relations:
FIG. 6. Variation in the heat capacity 共Cv兲, free energy 共F兲, and entropy 共S兲 in clathrates and diamond in the static approximation, from bottom to top: carbon, silicon, and germanium systems. Dots represent the experimental points 共C p, and S兲 of carbon diamond 共Ref. 52兲.
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FIG. 7. Variation in the lattice parameter with respect to the temperature at 0 GPa.
Cv共T兲 = − T
冋
d2Fvib q, 2 = − kB 兺 dT q, sinh共q,兲
S共T兲 = Svib共T兲 = −
册
2
,
dFvib . dT
共5a兲
共5b兲
S, Cv, and F are plotted Fig. 6 for C, Si, and Ge systems, and compared with experimental data 共C p and S兲 for carbon diamond.52 Until around 800 K, the static approximation reproduces accurately the heat capacity and entropy of carbon diamond. The results, reported for Si/Ge clathrates are in good agreement with those published by Biswas53 and Tang.41 In conclusion, according to the static approximation, diamond and clathrates behavior are, for a given element, slightly different. 2. QHA
If the anharmonic effects are neglected at the level of the QHA, the thermal expansion is restored by introducing the 共anharmonic兲 effect of the temperature and the pressure on lattice parameters and thermodynamic properties in the Eq. 共3兲. Isothermal fourth-order Birch-Murnaghan binding energy relation has been used to fit the free energy 9 F共a,T兲 = Fo + BoVou2 + Au3 + Cu4 + O关u5兴, 8
共6兲
where u = 共Vo / V兲2/3 − 1. For a given temperature, the free energy has been fitted by a spline-type approximation. The pressure is then deduced from the following relation:
冏
P共T,V兲 = −
F共T,V兲 V
冏
.
V共T , P兲 is thus interpolated from P共T , V兲, the Gibbs energy G共T , P兲 is computed as the function of the pressure and temperature. The thermal dilatation coefficient and the Grüneisen parameter are deduced from
共7兲
T
FIG. 8. Variation in the linear thermal ␣共T兲 for C-2 共dots兲, C-46 共line兲, and C-34 共dashed lines兲 with the temperature for different pressures: 0 ⬍ P ⬍ 60 GPa 共left兲, and 100, 200 GPa 共right兲.
␣v共T, P兲 =
冏 冏
1 V V T
,
共8兲
␥共T, P兲 = VBo␣/Cv
共9兲
P
while the heat capacity C p and Cv = U / T are linked by C p共T, P兲 = Cv + ␣2vBoVT.
共10兲
To calculate Fvib we have restricted the summation on q points at the only ⌫ point for carbon systems. The temperature dependence of the lattice parameter and thermal expansion, respectively, are plotted in Figs. 7 and 8 up to 1500 K at 0 GPa. The P-V-T relationship of diamond and clathrates is reported in Fig. 9 for two temperatures. The pressure and temperature behavior of C p, Cv, and ␣ of the three systems are very similar 共Fig. 10兲. Differences occur at high pressure 共up to 150 GPa兲, where the lattice parameter of C-46 is reduced contrary to C-34 and C-2, for which the related reduction occurs at higher pressure 共around 700 GPa兲.54 The Grüneisen parameters of Raman modes ln共 兲 关␥i = − ln共V兲i 兴 have been calculated and reported in Table VIII. For carbon diamond one found ␥ = 1.01 for the Raman active mode at 1307 cm−1, in agreement with experimental measurement.55 Contrary to Si-34,41 all Grüneisen parameters of carbon clathrates are positive. The Debye temperatures 共D兲, of clathrates, calculated and reported in Table IX, are 10% smaller than the diamond temperature.
FIG. 9. Calculated P-V-T relationship of diamond and clathrates for two temperatures 共0 K and 1500 K兲 taken as reference the volume at 0 K and 0 GPa.
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FIG. 10. Variation in the heat capacity 共Cv and C p兲 and Grüneisen parameter 关␥ = ln共⌰兲 / ln共V兲兴 at 0 GPa. We have enlarged around 1000 K experimental and theoretical C p of carbon diamond 共dots represent experimental data 共Ref. 52兲, dashed lines simulations兲.
C. (P , T) Phase diagram
We have investigated the phase diagram of carbon phases, by assuming the same Gibbs energies for carbon diamond with carbon clathrates, the temperature of the transition for a pressure. Figure 11 shows the phase diagram from 0 to 1500 K. At 0 K,the transition pressures Pt are −21 and −28 GPa for C-46 and C-34, respectively. These negative pressures agree with those calculated by Perottoni et al.13 TABLE VIII. Calculated Raman frequencies 共i兲 and Grüneisen parameters 关␥i = − ln共i兲 / ln共V兲兴 for C-34 and C-46.
共using a Hartree-Fock approximation兲, and Rey et al.16 Gibbs’ phase rule for one component system shows that two phases coexist along a curve in the T , P plane, where the Clausius-Clapeyron equation 共dPt / dT = ⌬S / ⌬V兲 is the coexistence curve. We found dPt / dT ⬎ 0 for C-46/C2 and C-34/ C2. Negative pressure values are expected in those kinds of transitions implying expanded phases compared to diamond. In absolute values, the transition pressures are found much larger than those calculated for Si-2/Si-46 共−6 GPa兲 and Ge2/Ge-46 共−2.4 GPa兲. D. Elasticity
C-34
C-46
i 共cm−1兲
␥i
538 958 1137 1245
0.52 1.14 1.24 1.17
i 共cm−1兲
␥i
588 716 824 1062 1092 1170 1223
0.76 1.01 1.49 1.44 1.62 1.22 0.88
948 1058 1260
1.18 1.75 1.02
457 752 879 920 1088 1158 1201 1229
0.48 0.60 0.49 0.93 1.05 1.20 0.97 0.99
Eg
The elastic constants 共Cij兲 have been derived from vibrational acoustic dispersion slopes presented above. In cubic systems, the elastic tensor can be described by three independent elastic constants 共see Ref. 31兲: namely, C11, C12, and C44. The slope of the acoustic branches along the 关100兴 direction are related to C11 and C44 = v关100兴 l
冑
C11 关100兴 ,v = t
冑
C44 ,
共11a兲
where v关100兴 and v关100兴 are, respectively, the longitudinal and l t transverse velocities of the sound and the density. For C12, C +2C +4C we use the 关111兴 direction: v关111兴 = 11 312 44 . l
冑
A1g 894 1032 1189
1.59 1.11 0.80
513 827 913 947 1003 1175 1200 1250
0.26 1.19 0.35 1.22 1.32 1.36 0.96 0.97
T2g
FIG. 11. Equilibrium phase diagram 共P , T兲 of carbon diamond and clathrates. 075209-10
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TABLE IX. The densities 共in kg/ m3兲, the elastic constants 共in GPa兲, the bulk modulus 共in GPa兲, the factors of anisotropy Ca, the shears moduli G 共in GPa兲, the Young modulus Y along the 关100兴 direction 共in GPa兲, the Poisson ratio , the Young modulus along the 关111兴 direction 共in GPa兲, and the Debye temperatures 共in K兲 for each system.
C-2 C-34 C-46 Si-2 Si-34 Si-46 Ge-2 Ge-34 Ge-46
C11
C12
C44
Bo
Ca
G
Y 100
Y 111
D
3593 3098 3129 2381 2074 2098 5566 4902 4953
1092 共1076a兲 966 935 共960b兲 167 共166a兲 135 共137c兲 140 共135c兲 134 共130a兲 114 115
135 共125a兲 87 127 共117b兲 63 共64a兲 55 共57c兲 47 共58c兲 54 共48a兲 46 37
585 共576a兲 437 462 共472b兲 78 共79a兲 44 共39c兲 47 共40c兲 66 共68a兲 36 40
458 381 396 103 82 78 82 69 63
0.81 0.99 0.87 0.62 0.92 0.97 0.59 0.94 0.96
475 439 403 48 40 46 39 34 39
1060 952 903 125 103 116 102 88 97
0.11 0.08 0.11 0.30 0.29 0.25 0.29 0.29 0.24
1231 947 996 185 111 118 156 92 101
2230 2083 1996 555 516 551 315 301 320
aExperiment,
Ref. 31. Theory, Ref. 19. cTheory, Ref. 33. b
For diamond, simulations reproduce with accuracy experimental data 共see Table IX兲. The differences for clathrates between previous calculations of elastic constants and ours simulations are small. The elastic constants of clathrates are always smaller 共10– 20 %兲 than diamond analogs. From elastic constants, experimental sets of parameters such the bulk modulus Bo, the factor of anisotropy Ca, the shear modulus G, and the Young’s modulus Y along two main crystallographic directions 共关100兴 and 关111兴兲 can be deduced according to the following relations 共see Table IX兲:
Y 100 = 1 Y 111
=
Bo =
C11 + 2C12 , 3
共12a兲
Ca =
C11 − C12 , 2C44
共12b兲
G=
C11 − C12 , 2
共12c兲
共C11 + 2C12兲共C11 − C12兲 , C11 + C12 1 Y 100
+
C11 − C12 − 2C44 . 3C44共C11 − C12兲
共12d兲
共12e兲
One can first note that Bo calculated from a Murnaghan fit 共see Table II兲 and from elastic constants 共Table IX兲 are very close. The factor of anisotropy is close to 1 in clathrates, indicating that they can be viewed as strongly isotropic systems according to strains. The Poisson’s ratio , which quantifies the stability of a crystal against shear, are considerably smaller for carbon systems than for the other silicon/ germanium systems, which suggests that carbon clathrates should be relatively stable against shear. To conclude, one
found that the Young’s Modulus vary little in function to the crystallographic direction. These results could explain the good stability of clathrates under pressure. VI. CONCLUSION
We have reported first-principles study of free guest carbon clathrates properties C-34 and C-46, and compared them with silicon and germanium clathrates and diamond analogs. In spite of a similar sp3 hybridization for the nine systems, we have shown that the lack of p core states in carbon materials induces strong modification in vibrational properties comparatively to Si and Ge. The pentagonal rings provide additional strong changes in electronic and vibrational properties. The study of electronic states around the band gap shows a slight dependency on the element, which results in the same optical properties in silicon and carbon clathrates. We have shown that the vibrational properties of carbon phases differ from their Si/Ge analog structures, whereas vibrational properties of Si and Ge systems can be relied through their atomic weight ratio. Raman and infrared spectra are displayed for an upcoming suitable experimental analysis. Thermodynamical properties 共S, F, Cv兲 are revealing to be less altered by crystallographic effect and that for a given element it is difficult to separate behaviors. It is only at high pressure that the evolution of lattice parameters of clathrates differs from diamond. The phase diagram calculated is found to be compatible with previous estimations. We conclude the paper with a presentation of elastic properties computed from acoustic slopes. ACKNOWLEDGMENTS
Calculations have been performed at supercomputer facilities at Calmip CICT Toulouse 共France兲. D.C. is indebted to P. Demont and D. Monceau for a critical reading of the manuscript.
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*
[email protected] 1 J.
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