Up to now, many guest atoms/molecules/ions have been successfully synthesized into graphite to form the various compounds. For example, alkali-atom graphite intercalation compounds are verified to reveal the stage-n structures, including LiC6n and Li
M8n [M=K. Rb and Cs; n=1, 2, 3; 4]. On the other side, AlCl4-ion/molecule ones are examined to show stage-4 and stage-3 cases at room and lower temperatures, respectively. Stage-1 and stage-2 configurations, with the higher intercalant concentrations, are unable to synthesize in experimental laboratories. This might arise from the fact that it is quite difficult to build the periodical arrangements along the longitudinal z and transverse directions simultaneously for the large ions or molecules. Our works are mainly focused on stage-1 and stage-2 systems in terms of geometric and electronic properties. The critical features, being associated with the atom-dominated energy spectra and wave function within the specific energy ranges, the active multi-orbital hybridization in distinct chemical bonds, and atom- & orbital-decomposed van Hove singularities, will be thoroughly clarified by the delicate simulations and analyses.
We use first-principles calculation within the density functional theory (DFT) to explore the electronic properties on stage-1 Li- and Li+-graphite-intercalation compounds (GIC) for different concentrations, LiCx/Li+Cx with x= 6,12,18,24,32 and 36. T
he essential properties, e.g. geometric structures, band structures and spatial charge distributions are determined by the hybridization of orbitals, the main focus of our works. The band structures/density of states/spatial charge distribution display that the Li-GIC possesses blue shift of fermi energy and just like metals, but the Li+-GIC still preserves as original graphite or so-call semimetal possessing the same densities of free electrons and holes. According to these properties, we find that there exists weak but significant van der Waals interactions between interlayer of graphite, and 2s-2pz hybridization between Li and C. There scarcely exists strong interactions between Li+-C. The dominant interaction between the Li and C is 2s-2pz orbital-orbital couple; the orbital-orbital couple is not significant in Li+ and C case but the dipole-diploe couple.
The diversified essential properties of the stage-n graphite alkali-intercalation compounds are thoroughly explored by the first-principles calculations. According to their main features, the lithium and non-lithium materials might be quite different
from each other in stacking configurations, the intercalated alkali-atom concentrations, the free conduction electron densities, and the atom-dominated & (carbon, alkali)-co-dominated energy bands. The close relations between the alkali-doped metallic behaviors and the geometric symmetries will be clarified through the interlayer atomic interactions, in which the significant alkali-carbon chemical bondings are fully examined from the atom- and orbital-decomposed van Hove singularities. The blue shift of the Fermi level, the n-type doping, is clearly identified from the low-energy features of the density of states. This study is able to provide the partial information about anode of Li+-based battery. There are certain important differences between AC$_6$/AC$_8$ and Li$_8$Si$_4$O$_{12}$.
We study the number of dimer-monomers $M_d(n)$ on the Tower of Hanoi graphs $TH_d(n)$ at stage $n$ with dimension $d$ equal to 3 and 4. The entropy per site is defined as $z_{TH_d}=lim_{v to infty} ln M_d(n)/v$, where $v$ is the number of vertices on
$TH_d(n)$. We obtain the lower and upper bounds of the entropy per site, and the convergence of these bounds approaches to zero rapidly when the calculated stage increases. The numerical value of $z_{TH_d}$ is evaluated to more than a hundred digits correct. Using the results with $d$ less than or equal to 4, we predict the general form of the lower and upper bounds for $z_{TH_d}$ with arbitrary $d$.
The essential properties of graphite-based 3D systems are thoroughly investigated by the first-principles method. Such materials cover a simple hexagonal graphite, a Bernal graphite, and the stage-1 to stage-4 Li/Li$^+$ graphite intercalation compoun
ds. The delicate calculations and the detailed analyses are done for their optimal stacking configurations, bong lengths, interlayer distances, free electron $&$ hole densities, Fermi levels, transferred charges in chemical bondings, atom- or ion-dominated energy bands, spatial charge distributions and the significant variations after intercalation, Li-/Li$^+$- $&$ C-orbital-decomposed DOSs. The above-mentioned physical quantities are sufficient in determining the critical orbital hybridizations responsible for the unusual fundamental properties. How to dramatically alter the low-lying electronic structures by modulating the quest-atom/quest-ion concentration is one of focuses, e.g., the drastic changes on the Fermi level, band widths, and number of energy bands. The theoretical predictions on the stage-n-dependent band structures could be examined by the high-resolution angle-resolved photoemission spectroscopy (ARPES). Most important, the low-energy DOSs near the Fermi might provide the reliable data for estimating the free carrier density due to the interlayer atomic interactions or the quest-atom/quest-ion intercalation. The van Hove singularities, which mainly arise from the critical points in energy-wave-vector space, could be directly examined by the experimental measurements of scanning tunneling spectroscopy (STS). Their features should be very useful in distinguishing the important differences among the stage-$n$ graphite intercalation compounds, and the distinct effects due to the atom or ion decoration.
