In the present work, we have studied the thermodynamics and electronic structure of few confined systems, namely, melting behavior of clusters and electronic structure calculations of graphene, hydrogenated graphene (graphane), graphene nanoribbons and carbon nanotubes. All the calculations are carried out using Density Functional Theory (DFT).

The first chapter is an introductory chapter. We have discussed the interesting properties, possible applications of the confined systems under study. We also take a brief review of related experimental and theoretical work pertaining to the topic. The melting in clusters has also been addressed elaborately.

In the second chapter, the detail discussion on density functional theory is carried out, as it is the key tool for all the calculations. It is followed by the discussion on molecular dynamics. Various data analysis tools are noted thereafter along with the error analysis.

In the first problem (chapter three) we have carried out extensive first principal thermodynamic simulations of carbon doped Al₁₃ and Ga₁₃ clusters. The doping with an impurity is an effective way to tune the melting temperature of the host cluster. In the present case, doping Al₁₃ and_{Ga 13} with a tetrahedral impurity like carbon, makes these 40 electron shell closed system under jellium approximation having enhanced stability. The host clusters are known to have distorted icosahedra and decahedral geometry respectively. Interestingly, the doped clusters Al₁₂C and Ga₁₂C exhibit a perfect icosahedral structure with carbon atom at the center. The bond length calculation between the surface and the central atom in all four clusters reveals that the presence of carbon shortens the bond lengths between central carbon and outer surface atoms. Examination of the various isosurfaces of total charge densities brings out the difference in the nature of bonding between the host and the doped clusters. Al₁₃ shows a delocalized, well spread charge density while Ga₁₃ shows typical covalent bonding. Upon doing, significant changes are seen. For both the doped clusters, most of the charge is around the central carbon atom and is spherically symmetric. Evidently there is a charge transfer from all surface atoms towards the central carbon. This establishes a partial ionic bond between central carbon and surface atoms and the size of cluster shrinks. This also results in weakening of the bonds on the surface atoms. As a result the melting temperature of the doped clusters is lowered than the host clusters. The calculated melting temperatures for Al₁₃and Ga₁₃are 1800 K and 1200 K respectively while the doped clusters melt at around 800-900 K. These results are supported by the mean square displacement and root mean square bond length fluctuation.

Next, we have investigated the finite temperature behavior of smallest gold cages namely Au₁₆ and Au₁₇ using ab initio method (chapter four). The nano-gold has wide applications as catalysis, medicine, electronic circuits etc. Gold clusters are known to exhibit caged structures for n=16, 17 as well as for higher number of atom clusters. The stability of these cages at finite temperature is an important issue for the application point of view. Here, we have examined the melting, geometry and various isomers of the smallest gold cages Au₁₆ and Au₁₇. Au₁₆ is known to be a flat cage viiwhile Au₁₇ is a hollow one. We have analyzed almost 50 isomers for both the clusters even 0.4 eV above the ground state and we have demonstrated a close relation between the isomer energy distribution and melting of the cluster. Our results show that Au₁₆ shows rather a broad specific heat curve ranging from 600 K to 1000 K while Au₁₇ exhibits a noticeable peak at 900 K which is identified as a melting peak. The analysis of the ionic trajectories and other phase change indicators such as mean square displacement and radial distribution function, clearly indicate that Au₁₇ is very stable and retains the shape upto 1000 K. Au₁₆ on the other hand, distorts significantly. The diffusive motion of the atoms begins at 600 K resulting in isomerization and the open cages structures are seen above 1000 K. According to Bixon and Jortner, the continuous isomer energy distribution leads to a broad peak in specific heat curve while branch-like distribution gives a peak. Our results show that Au₁₆ has a continuous isomer energy distribution while that of Au₁₇ is step-like. This leads to variation in their behavior at finite temperature. Specific heat curve for Au₁₆ has a very broad peak ranging from 600 K to 1000 K whereas Au₁₇ has a relatively sharper peak.

The third problem (chapter five) deals with the electronic structure calculation of graphene and graphane via partial hydrogenation. Graphene is a 2D one atom thick sheet of Au² bonded carbon atoms with zero band gap. Although, graphene exhibits many novel properties such as 'linear dispersion at Fermi level, anomalous quantum hall effect, Klein paradox etc.', due to the absence of band gap, it has limitations in the use of semiconducting field. Amongst the various ways proposed to open the gap in graphene, complete hydrogenation is the effective way. The fully hydrogenated structure is called as 'graphane' and has a DFT predicted band gap of 3.5 eV. In this work, we have probed the transition from zero gap graphene to graphane via successive hydrogenation. We have analyzed about 18 systems from 2% hydrogenated graphene to 90%H via DFT simulations. The first interesting issue that is addressed is the minimum energy configuration of the hydrogen atoms to decorate the graphene lattice. Our extensive simulations for different configurations (namely random placement of H, 2 islands, different edge patterns etc) of hydrogenated graphene show that hydrogens prefer to form a single compact island. We carried out such calculations for systems upto 50%H and with 2-3 different unit cells. Next, the analysis of the density of states (DOS) close to Fermi level brings out interesting features. For low hydrogen concentration, the V-shape DOS in pure graphene is viiidisturbed slightly due to loss of symmetry. As the hydrogen coverage increases there is a significant increase in the value of DOS at the Fermi level. The hydrogenated carbon atoms are now moved out of the graphene plane, in turn the lattice is distorted and the symmetry is broken. As a consequence, more and more k points in the Brillion zone contribute to the DOS near Fermi level. The region ranging from 30% coverage to about 70% coverage is characterized by the finite DOS of the order of 2.5 near the Fermi energy. Above 80% or so, there are too few bare carbon atoms available for the formation of delocalized π bonds. The value of DOS approaches zero and a gap is established with a few midgap states. It may be emphasized that the presence of states around the Fermi level giving finite DOS does not guarantee that the system is metallic unless we examine the nature of localization of the individual states. Therefore we have examined the energy resolved charge densities of the states near the Fermi level. A particularly striking feature is the formation of two spatially separated regions. The hydrogenated carbon atoms do not contribute to the charge density giving rise to the insulating regions while the neighboring bare carbon atoms form conducting regions via π bonding. This feature is prominent in the region from 30%-70% hydrogenated cases giving rise to the channels of delocalized bare carbon atoms. Since above 75%H, there are insufficient number of bare carbon atoms to form contagious channels, the mid gaps occur. To summarize, as the hydrogen coverage increases, graphene with a semi-metallic character turns first into a metal and then to an insulator. The metallic phase has some unusual characteristics: the sheet shows two distinct regions, a conducting region formed by bare carbon atoms and embedded into this region are the non-conducting islands formed by the hydrogenated carbon atoms. However it should be noted that hydrogenated systems chosen in this work are such that the energy is always minimum. These are the naturally preferred arrangements of the hydrogen atoms decorating graphene. The specially designed patterns of hydrogenated graphene can yield various band gaps. We have demonstrated such designed channels of hydrogenated graphene giving zigzag and armchair edge patterns and their effect on the band gap modulation.

In the last problem (chapter six), we have investigated the stability and confinement effects of graphene and H-graphene nanoribbon (GNR) encapsulating inside carbon nanotubes (CNT). GNR are of particular interest because they are known to exhibit width dependant band gaps. The recent calculations by two groups show that GNR can be stabilized by encapsulating in CNT. In the present work, we ixhave inserted the smallest (20 carbon atom unit cell) graphene and H-graphene ribbons inside the carbon nanotubes of diameters ranging from 8Å to 17Å. (The confinement effect on the geometry and the stability of the ribbons has been studied.) Firstly, the geometries of pure GNR are sensitive to the CNT radii. There is a tendency to break the vertical bonds as radius is increased. We observe the formation of 2 chains in the largest diameter CNT. The observed DOS for all the structures show that there is a substantial enhancement at Fermi energy in all the cases which mainly arises from p_z and p_y of GNR atoms. In the largest tube, the 2 chains exhibit delocalized charge density and solely contribute to Fermi level. These findings are confirmed by the analysis of site projected DOS and partial charge density counters. The structure and DOS pattern do not vary significantly for semiconducting CNT except for the states at Fermi occur in the gap of the tube. For hydrogenated GNR, we have studied 50%H and fully hydrogenated GNR encapsulated in CNT. For 50%H case, it has been observed that the geometries inside CNT are sensitive to the placement of the hydrogen atoms. The systematically placed hydrogens (on GNR) go over in-plane positions and resulting geometries are independent of the diameter considered. The structure is planer leading to hydrogen terminated GNR. On the other hand, the randomly placed hydrogens show diameter dependant structures retaining the tendency to form parallel chains by breaking the vertical bonds in large diameter tubes; however structures are modulated by strong C-H bond leading to displacements of carbons away from planer or linear shape. Lastly, we have examined fully hydrogenated GNR. Remarkably the optimized geometries are not sensitive to the diameter at all. The final structure remains the same, namely two parallel chains in all types of CNT. In the present case the carbon atoms show zigzag arrangement, as each atom is pulled by the attached hydrogen. The two chains are symmetrically placed with respect to the tube axis due to stronger confinement. The DOS for both types of tubes show enhancement at Fermi level. The isosurfaces of partial charges densities clearly show delocalized nature along the two chains which arises from the p_z orbitals of GNR carbons atoms. The contribution of both the chains is equal unlike in the case of pure GNR. In semiconducting tube, we get two stable conducting channels. Our results bring out the possibility of tuning the geometries of GNR and H-GNR inside CNT of different diameters to obtain one dimensional or two dimensional structures.