Final answer:
The expression ε_α(k_x,k_y) represents the energy of an electron within a specific atomic orbital in a square lattice. It depends on atomic orbital symmetries, the tight binding model used, and the material-specific parameters. A simple model might use a cosine-like dispersion relation with the electron's hopping energy and lattice constants as parameters.
Step-by-step explanation:
The expression of ε_α(k_x,k_y) in the intra-orbital tight binding Hamiltonian Hintra for a square lattice with one atom per unit cell involves the energy dispersion relation associated with the atomic orbitals s, px, and py. To find this expression, we need to consider the atomic orbital symmetry and how these orbitals overlap in the lattice to form bonding and antibonding molecular orbitals. Typically, for s orbitals, the overlap is direct, while for px and py orbitals, the overlap is directional and depends on the relative orientation of the orbitals in the lattice.
The energy ε_α(k_x,k_y) is a function of the lattice momentum components k_x and k_y, representing the electron's probable energy within a given orbital type α, corresponding to the Hamiltonian's diagonal terms. The exact functional form of ε_α would be specific to the material and based on empirical parameters or first-principles calculations, which would take into account factors like orbital overlap, the distance between atoms, and potential interaction terms. In a simplistic tight-binding model, the dispersion relation might take on a cosine-like form due to the periodic potential of the lattice, i.e., ε_s(k_x,k_y) = -2t(cos(k_xa)+cos(k_ya)), where t is the hopping parameter describing the energy associated with an electron moving from one atom to an adjacent atom, and a is the lattice constant. The px and py orbitals will have similar, but orientationally dependent, dispersion relations.