The electron thermal conductance, κ, of a dot has been calculated in the regime of weak coupling with two electrode leads within a linear response theory. We discuss the effect of the interplay between the charging energy, the thermal energy, and the confinement in the Coulomb oscillations of κ. Hence, we consider three energy regions: the quantum limit, where quantum confinement dominates over the thermal energy; the classical regime, where the discreteness of the energy spectrum is screened by the thermal energy; and the intermediate energy region. In the quantum limit, the periodicity of the oscillations of the electron thermal conductance is the same as the Coulomb-blockade oscillations of the conductance, G. Analytical expressions have been obtained for κ and G in the cases of nondegenerate and for doubly degenerate energy spectrum. The obtained dependence of κ on the energy level spacing and the thermal energy explicitly shows that quantum confinement is responsible for the fast decrease of the electron thermal conductance of a dot. It is found that degeneracies in the energy spectrum of a dot are opposed to the decrease of the electron thermal conduction due to quantum confinement. It is shown that an external field that raises the degeneracies causes a considerable enhancement in κ. In the classical and in the intermediate regimes, the electron thermal conductance shows distinct behavior at low and high temperatures. In the classical regime, Coulomb blockade oscillations are shown at low temperatures and simple formulas are obtained for κ and G. The Wiedermann-Franz law holds at the peaks of κ and G. The temperature dependence of κ and G has been calculated up to the limit where transport occurs through two isolated barriers. The relation between κ and G with increasing thermal energy is discussed.
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