| Summary: | The adsorbed atoms exhibit tendency to occupy a triangular lattice formed by periodic potential of the underlying crystal surface. Such a lattice is formed by, e.g., a single layer of graphane or the graphite surfaces as well as (111) surface of face-cubic center crystals. In the present work, an extension of the lattice gas model to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> fermionic particles on the two-dimensional triangular (hexagonal) lattice is analyzed. In such a model, each lattice site can be occupied not by only one particle, but by two particles, which interact with each other by onsite <i>U</i> and intersite <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mn>2</mn></msub></semantics></math></inline-formula> (nearest and next-nearest-neighbor, respectively) density-density interaction. The investigated hamiltonian has a form of the extended Hubbard model in the atomic limit (i.e., the zero-bandwidth limit). In the analysis of the phase diagrams and thermodynamic properties of this model with repulsive <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mn>1</mn></msub><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, the variational approach is used, which treats the onsite interaction term exactly and the intersite interactions within the mean-field approximation. The ground state (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>) diagram for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mn>2</mn></msub><mo>≤</mo><mn>0</mn></mrow></semantics></math></inline-formula> as well as finite temperature (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>) phase diagrams for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> are presented. Two different types of charge order within <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msqrt><mn>3</mn></msqrt><mo>×</mo><msqrt><mn>3</mn></msqrt></mrow></semantics></math></inline-formula> unit cell can occur. At <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> phase separated states are degenerated with homogeneous phases (but <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> removes this degeneration), whereas attractive <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mn>2</mn></msub><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula> stabilizes phase separation at incommensurate fillings. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>/</mo><msub><mi>W</mi><mn>1</mn></msub><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>/</mo><msub><mi>W</mi><mn>1</mn></msub><mo>></mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> only the phase with two different concentrations occurs (together with two different phase separated states occurring), whereas for small repulsive <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>U</mi><mo>/</mo><msub><mi>W</mi><mn>1</mn></msub><mo><</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> the other ordered phase also appears (with tree different concentrations in sublattices). The qualitative differences with the model considered on hypercubic lattices are also discussed.
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