For metals, literature explains that the valence band overlaps the conduction band. Im not clear how this happens, the way I see it, valence and conduction bands are concepts which arise only when there is a bandgap between them and electrons occupy partially one band (usually the conduction band due to thermal energy). If metals do not have a bandgap then why is it said that they both overlap ?
You argue with the density of s-state and d-states, The densities overlap,yes
But you should look for bands within the Brilloin zone, For noble metals you see the a free electron band, both below and above the Fermi level, and the d-band, which has little dispersion and lies below the Fermi level. Try to find Brilloin zone and band structure in Wikipedia.In semiconductors , there is a gap between the valence and the conduction bands. In this case also the density of states does not overlap ,
My argument goes in the same direction. Starting from the well-separated energy levels of single atoms, characterised by certain symmetries of the wave functions (as, e.g. being s-states or p-states or d-states ...), these levels split into bonding and antibonding states and those merge into bands when the bulk material is formed. Remarkably, the symmetry information is not fully lost, so by the link to where they came from, the resulting bands can still be identified as "valence" ("bonding") and "conduction" ("antibonding") bands. There is a nice illustration showing this for silicon (http://www.tf.uni-kiel.de/matwis/amat/mw2_ge/kap_4/illustr/t4_2_1.html); for other materials it is the same in principle.
Actually, energy band is not with fixed value of energy, it has a range of energy. In both energy bands of valence and conduction bands of metals, ranges of energies overlap and electrons are easily moved to conduction band from valence band while this overlaping does not exist in semiconductors and insulators.
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