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This study investigates the vertex corrections of the optical conductivity in high-$T_c$ superconductors for hole-spin wave scattering in the superconducting state. The Feynman diagram technique is used to determine the mathematical expressions of the ladder diagrams in the calculation of the current–current correlation function. The sum of these vertex terms leads to a self-consistent form of the vertex function. By evaluating the frequency summation through contour integration, a general expression for the current–current correlation function is obtained, which allows us to determine the zeroth-order term and the first-order correction of the optical conductivity explicitly. Our findings for the dc conductivity in a clean superconductor indicate that the zeroth-order term is zero, a result attributed to the conservation of momentum and energy. Meanwhile, the first-vertex correction reveals the Drude behavior in the superconducting state at zero frequency.

We study the energetic stability of the self-trapped magnetic polaron in an antiferromagnetic lattice of localized spins, including the effects of the intra-sublattice hopping which allows electron transfer within the same magnetic sublattice. The model consists of an itinerant electron moving in a one-dimensional antiferromagnetic lattice of localized spins. In addition to the first and the second neighbor electron hopping, terms describing the superexchange between the localized spins as well as the Hund's-rule coupling between the electron and the localized spins are included in the model Hamiltonian. The ground-state energy of the self-trapped polaron with the resulting ferromagnetic core region is compared with the energy of the propagating state in the antiferromagnetic lattice. We find the magnetic polaron to be self-trapped for all values of the Hamiltonian parameters, although the intra-sublattice hopping substantially reduces the polaron binding energy.

Using a variational approach, the binding energy of shallow hydrogenic impurities in a semimagnetic parabolic quantum dot is calculated within the effective mass approximation. The binding energy is computed for Cd_1-xinMn_xinTe/Cd_1-xoutMn_xoutTe structures as a function of the dot size in an external magnetic field. The results show that the impurity binding energy (i) increases with the reduction in dot sizes (ii) decreases when the magnetic field is increased for a given dot and (iii) increases to a maximum value at 100 Å and then decreases as the size of the dot increases beyond 100 Å for a realistic model. Spin polaronic shifts are estimated using a mean field theory. These results are compared with the existing literatures.

The Linked Cluster Theory in the spin polaron formulation is investigated in this paper using the finite temperature (Matsubara) Green’s function method. We use a representation where holes and spins are described as spinless fermions and normal bosons, respectively. An analytic expression of the thermodynamic potential of high-temperature superconductors in the superconducting state was obtained using the Linked Cluster theory and an expression for its normal state by setting the energy gap to zero. The expression of the thermodynamic potential enabled us to calculate two bulk thermodynamic properties of high-temperature superconductors, entropy and specific heat, both in the normal and superconducting states.

Theoretical investigations of spin polaron in a quantum dot in the "spin doping" superlattice systems Cd_{1 - Xin}Mn_XinTe/Cd_{1 - Xout}Mn_XoutTe are presented. We estimated the donor ionization energy by variational technique and the spin polaronic shift using the mean field theory with modified Brillouin function. Our results show that significant enhancement of donor ionization energy and spin polaronic shift occur in a dot when compared to a quantum well, wire and bulk systems.

The binding energies of shallow acceptors in Cd_1-xinMn_xinTe/ Cd_1-xoutMn_xoutTe quantum dots are calculated in the presence of external magnetic fields. Variational calculations are performed within effective mass approximation. The results show that the impurity binding energy (i) increases with the reduction in dot sizes, (ii) decreases with the magnetic field is increased for a given dot, and (iii) reaches a peak value as the dot radius decreases and then diminishes to a limiting value corresponding to the radius for which there are no bound states in the quantum dot. Spin polaronic shifts are estimated with the acceptor envelope function using a mean field theory. These results are compared with the existing literatures.