Atomic carbon and silicon

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The search for stable Si-C alloy structures

Silicon and carbon are arguably the two elements that have been the main driving forces behind human development. Carbon is not only the basis for all biological life on this planet but, along with silicon, has also been central to the technological advancements of the last century. Carbon has always been used as a traditional fuel source and was the driving force behind the industrial revolution of the 19th century. It is known to exist in many different allotropes ranging from extremely soft graphite to extremely hard diamond with the two most commonly known forms being graphite and diamond. Each allotrope has its own unique properties and possible applications. In diamond form, carbon has found application as an abrasive due to its extreme hardness. In graphitic form, it has made possible many advancements in technological areas from military applications to sports equipment because of its use as a strong, heat resistant fiber to create flexible composite materials.
In its recently researched forms of graphene, nanotubes and nano- strips, it promises to be a very important resource for future electronic technological advancements. Silicon is the second most abundant element in the earth’s crust comprising 27.7 % of all crustal rocks [1]; mostly in the form of SiO2 silica. The different known allotropes of silicon are Si-I cubic diamond (cd), Si-II beta-tin (β-Sn), Si-III 8-atom body-centered (BC8), Si-V simple hexagonal (sh), Si-VI Cmca, Si-VII hexagonal closed-packed (hcp), and Si-X face-centred cubic (f cc) with Si-I being the stable form at ambient conditions while the others are all high pressure phases. The use of diamond silicon as a substrate for semi-conductor devices has allowed for the integration of millions of circuit elements on a single chip and this has been the main driving force behind the transition from the largely industrial age to our new information age. Silicon and carbon both occupy group IV on the periodic table and therefore share some chemical characteristics. Both elements have a valency of four and can form sp3 covalent bonding with a coordination of four. For this reason, they both exist in the diamond structure although while this structure is the stable form for silicon at ambient conditions, it is a metastable form of carbon.
Both elements, however, have some significant differences. Whereas silicon prefers sp3 bonding under ambient conditions, carbon has the ability to form sp2 and sp hybridized bonding. Carbon is the smaller atom with closely bound valence electrons whereas silicon is larger with loosely bound electrons. Because of this, tetrahedrally bonded carbon atoms in diamond have a bond length approximately 2 3 of those in diamond silicon. Carbon is also more electronegative than silicon with the better ability to attract other electrons. Despite these differences, they co-exist in the stable 50:50 Si-C alloy, silicon carbide, with its many stacking polytypes. This stability is mainly due to the fact that all atoms are tetrahedrally bonded to each other with all bonds being homogeneous Si-C bonds of the same length. The question remains, can silicon and carbon exist in other structured alloys besides 50:50 silicon carbide? Experimental evidence shows that there is a low solubility of C in Si [2] and even though low concentrations of metastable C defects in Si have been created experimen- tally using molecular beam epitaxy[3,4] and chemical vapor deposition[5], off-50:50 alloys do not seem to readily exist. An extensive search of the literature shows a dearth of ordered, disordered, random, as well as amorphous and thin film structures for off-50:50 alloy concentrations. Clearly, the reasons for this must center more on the differences between Si and C rather than their similarities. Since SiC under ambient conditions exhibits tetrahedral coordination, this suggests that in an alloyed configuration under ambient conditions, sp3 bonding may also be preferred. Is it possible for off-50:50 alloys to be fabricated where the atoms maintain their tetrahedral coordination? The sizable difference in the bond lengths of these two elements in the diamond structure will make this difficult.
As mentioned, the stability of 50:50 SiC is largely due to the presence of single-length ho- mogeneous Si-C bonds. The presence of Si-Si, C-C and Si-C bonds in any single system will result in strains which can, in principle, only be relieved in complex geometries. It turns out that under conditions of high pressure, SiC transforms to the rock-salt (NaCl) structure, which gives the silicon and carbon atoms six-fold coordination with homogeneous Si-C bonds. Silicon also has high pressure phases which exhibit coordina- tion greater than four. Under increasing pressure it transforms from cd (coordination 4) → β-Sn structure (coordination 6) → sh (coordination 8) → hcp (coordination 12) → f cc (coordination 12). Perhaps pressure would be a useful tool to search for off- 50:50 alloys involving Si and C. Grumbach and Martin[6], in their theoretical work on high-pressure/high-temperature phases of C, discovered a dense liquid whose melting temperature decreased with pressure, which is a behavior that is very similar to that of Si and Ge at lower pressures. This liquid resembles the simple cubic structure with six-fold coordination, a structure similar to that of Si and Ge under pressure. This suggests that for off-50:50 alloys, simple high coordination ordered structures might form under conditions of high pressure with bonding that differs from simple sp3

Silicon and Carbon

Silicon has been much studied over the last few decades due to its importance in many technological applications. It was one of the first materials to be used as a prototype to validate the applicability of computational ab initio techniques for the study of the structural, electronic, lattice dynamic and phase transition properties of bulk materials. In 1980, Yin and Cohen [10] used a pseudopotential method within the local density formalism based on density functional theory to correctly calculate the ground state properties for cubic diamond silicon. By fitting calculated total energy versus volume values to an equation of state, they obtained values for the lattice constant, bulk mod- ulus and cohesive energy all to within 1% of their experimental values. It had also been known since 1962, using resistivity measurements, that diamond Si transforms into a more metallic phase under pressure[11], and that the structure for this new phase was identified through x-ray diffraction measurements in 1963 to be β-Sn[12]. Yin and Cohen theoretically validated this pressure-induced phase transition by identifying a 9.9 GPa tangent pressure line between the two equation of state curves for each phase.
This value is smaller than the 16 to 20 GPa values reported in the 1962 and 1963 findings but close to the value of 8.8 GPa reported later in 1983 by Olijnyk et al. [13] . Yin and Cohen also calculated phonon frequencies for LT O(Γ), T A(X), T O(X) and LOA(X) all within 3% of their observed values. These first tentative calculations gave credence to the suitability of these methods to study crystal solids. In 1982, they formalized these computational ab initio methods using Si and Ge as prototypes[14]. They studied the crystal stability for each element by considering cubic diamond (cd), simple hexagonal (sh), beta-tin (β-Sn), simple cubic (sc), body- centred cubic (bcc), hexagonal closed-packed (hcp) and face-centred cubic (f cc) phases. The calculations correctly predicted the stable ground states as being cubic diamond and they reproduced the expected cd to β-Sn transitions. Ground state structural properties based on equation of state fits were in good agreement with experimental values. The valence electron density contour plots for cd Si reproduced the same fea- tures as those found in plots synthesized from x-ray data. Contour plots for each phase showed that cd Si had the most covalent bonding character with the β-Sn phase show- ing a more metallic nature. They obtained good agreement between calculated x-ray structure factors and those derived from experiments. The calculated electronic band structure for both elements gave density of states peak positions that compare well to observed angle-integrated photo-emission data. Based on these results, the claim was made that these methods could be used to accurately describe static structural and electronic properties as well as crystal stability and pressure-induced phase transitions for solids. This laid the foundations for ab initio methods to be used to predict yet unobserved crystal phases. In 1963, it was already known that two new phases for Si exist[15]; a dense body- centered phase created by pressure reduction from the β-Sn phase and a simple hexag- onal (sh) phase created by heat treatments. In the 1980’s, the ab initio methods developed by Yin and Cohen [14] were now used to explain these transitions and predict possible new high pressure phases. In 1982, McMahan and Moriarty [16] used the generalized pseudopotential method (GPT)[17] and the linear muffin-tin orbital (LMTO) method[18] to predict the following possible phase transition sequence for Si cd → β-Sn → hcp → f cc → bcc with the hcp→f cc transition occurring at 76 GPa for LMTO and 80 GPa for GPT, and based on extrapolation of the results of Yin and Cohen, the β-Sn→hcp transition occurring at 41 GPa. This prediction was partially confirmed in 1983 when Olijnyk et al. [13] experimentally observed the following phase transitions: cd → β-Sn → sh → an unknown intermediate phase → hcp occurring at 8.8, 16, 35-40, and 40 GPa[13]. The β-Sn→sh transition was theoretically described in 1984 by Needs and Martin [19] and in 1986, Duclos et al. [20] observed the hcp→f cc transition at 78 GPa. The phase transition from β-Sn to the 8-atom body- centered (BC8) phase under pressure reduction was independently described by Biswas et al. [21] and Yin [22] .
Hu et al. [23], in 1986, compared their experimental results with these previous theoretical results which prompted them to state that “it is gratifying to note the good overall agreement between theory and experiment, and the ability of this theory to predict new phases”. In fact, during 1993-94 theoretical and experimental results were used to complement each other to explain a newly observed body-centered orthorhombic intermediate phase[24] between the β-Sn and sh structures[25,26] . The first two known forms of carbon were the stable graphite form produced by simple combustion and meta-stable diamond produced in the earth by high pressure and temperature. In 1967, Bundy and Kasper[27] produced a new meta-stable hexagonal form of C using the static pressure compression of graphite at high temperature. It was noted that this form of carbon is found in meteorite diamonds created by large impact collisions with the earth. Strel’nitskii et al. [28] reported in 1978, a novel new allotrope of carbon with body-center symmetry obtained by the deposition of carbon plasma onto a cooled substrate. Ab initio techniques were employed to find suitable structures for this new allotrope, these being the supercubane phase [29] and the now generally accepted 8 atom BC8 structure[30]. New methods such as shock-compression[31,32] were then used to create this and other new allotropes in the laboratory. Experimental evidence of new C allotropes spurred theoretical research into possible structures to match the observed x-ray diffraction patterns[33]. It also initiated the search for plausible but not yet observed new phases[34,35,36] . Since the advent of diamond-anvil cell [37] experiments to investigate pressure tran- sitions for other solids, it became important to know the upper pressure bound for the stability of diamond. After their seminal paper in 1982, Yin and Cohen [38] in 1983 studied the crystal stability of carbon considering β-Sn, sc, bcc, hcp and f cc phases and were able to predict a carbon pressure-induced phase transition from diamond to sc at 2300 GPa. A new upper limit was place on diamond when in 1984, Biswas et al. [21] and Yin [22] both predicted a cubic diamond (cd) to BC8 transition at 1200 GPa. It was later suggested in 1987 by Fahy and Louie [39] that the sc and BC8 carbon cannot be metastable phases at low pressures and that diamond first transforms into BC8 at 1.11 TPa.
It is known that carbon exists under conditions of high pressure and temperature inside planets and for this reason it was important to predict the phase diagram for carbon. Grumbach and Martin [40] in 1996 used first principles molecular dynamics to predict the pressure/temperature phase diagram for C. Their diagram showed that the sc phase is most probably a thermodynamically stable high pressure phase with BC8 existing between diamond and sc. Liquid C at high pressure was predicted to behave very much like lower pressure Si preferring six-fold coordination. Similar results were obtained by Wang et al. [41] in 2005 also using ab initio molecular dynamics.

Contents :

• Page
• List of Figures
• List of Tables
• Glossary
• 1 Introduction
  • 1.1 The search for stable Si-C alloy structures
  • 1.2 Aims and objectives
    • 1.2.1 Off-50:50 Si-C alloys
    • 1.2.2 Two dimensional Si, C and Si-C alloys
  • 1.3 Thesis structure
• 2 Review of previous work
  • 2.1 Silicon and Carbon
  • 2.2 Silicon Carbide
  • 2.3 Off-50:50 SiC alloys
  • 2.4 2D silicon-carbon systems
• 3 Theoretical framework
  • 3.1 The many-body problem
  • 3.2 The Hartree-Fock approximation
  • 3.3 Density functional theory
  • 3.4 The Kohn-Sham ansatz
    • 3.4.1 Local spin density approximation
    • 3.4.2 Generalized gradient approximation
  • 3.5 Algorithmic implementation of the KS equations
  • 3.5.1 Plane-waves
    • 3.5.2 Integrating over the Brillouin Zone: k-point grid sampling
    • 3.5.3 Pseudopotentials
    • 3.5.4 PAW method
    • 3.5.5 Atomic relaxations
  • 3.6 Bulk elasticity
    • 3.6.1 Strain
    • 3.6.2 Stress
    • 3.6.3 Crystal symmetry
    • 3.6.4 Calculating elastic constants
    • 3.6.5 Elastic moduli
    • 3.6.6 Bulk equation of state
  • 3.7 Lattice dynamics
  • 3.8 Software codes
• 4 Bulk elemental and 50:50 Si-C systems
  • 4.1 Atomic carbon and silicon
    • 4.1.1 Choice of k-point sampling and kinetic energy cut-off
    • 4.1.2 Chosen calculation parameters
  • 4.2 Silicon allotropes
    • 4.2.1 Bonding and electronic properties
    • 4.2.2 Equation of state
    • 4.2.3 Phase transitions
    • 4.2.4 Elastic properties and stability
  • 4.3 Carbon
    • 4.3.1 Bonding and electronic properties
    • 4.3.2 Diamond, BC8, Supercubane, C4, Glitter – Equation of state and pressure transition
    • 4.3.3 Diamond, BC8, supercubane, C4 – Elastic properties and stability
    • 4.3.4 Graphite – Equation of state and elastic constants
  • 4.4 Silicon carbide polytypes
    • 4.4.1 Bonding and electronic properties
    • 4.4.2 Equation of state and pressure transition
    • 4.4.3 Elasticity and stability
  • 4.5 Conclusions
• 5 Bulk off-50:50 Si-C alloys
  • 5.1 Perovskite structure
  • 5.2 Pyrite structure
    • 5.2.1 Pyrite-SiC
    • 5.2.2 Pyrite-Si2C
  • 5.3 Glitter structure
    • 5.3.1 Glitter-SiC
    • 5.3.2 Glitter-Si2C
  • 5.4 t-BC2 structure
    • 5.4.1 t-SiC
    • 5.4.2 t-Si2C
  • 5.5 Stable silicon dicarbide
    • 5.5.1 Pressure transitions and formation energies
    • 5.5.2 Comparison of elastic properties
  • 5.6 Conclusions
• 6 Two dimensional Si-C systems and the 2D equation of state
  • 6.1 Theoretical framework
    • 6.1.1 The two dimensional equation of state
    • 6.1.2 Elastic theory
    • 6.1.3 Computational details
  • 6.2 Results and discussion
    • 6.2.1 Structures considered
    • 6.2.2 Mechanical properties
    • 6.2.3 Elastic properties
    • 6.2.4 Intrinsic strength
  • 6.3 Conclusions
• 7 General conclusions
  • 7.1 Elemental and 50:50 systems
  • 7.2 Off-50:50 alloys
  • 7.3 2D systems and the novel EOS
  • 7.4 Conclusion
  • 7.5 Future investigations
  • A Derivation of the 2D EOS
  • References

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