  We have studied the origin of zero volume expansion below the Curie
temperature (Tc), variable range hopping (VRH) behaviour using structural,
magnetic, transport and thermal studies on the oxygen deficient double
perovskite NdBaCo_2O_{5+{\delta}} ({\delta} = 0.65). The valence state of Co
ions and the possible properties exhibited by such compound were studied using
electronic structure calculations for {\delta} = 0.75. Careful investigation of
structure shows that the compound stabilizes in tetragonal structure (P4/mmm)
having 2a_p X 2a_p X 2a_p (222) superstructure, where a_p is the cubic
perovskite lattice parameter. The compound exhibits a minimum in resistivity,
ferromagnetic and ferrimagnetic transitions around 375 K, 120 K (T_c) and 60 K,
respectively with signature of Griffiths phase above T_c. Our detailed
structural analysis suggests signature of the onset of the above magnetic
transitions at temperatures well above its stabilisation at long range level
thereby leading to VRH behaviour. The observed zero thermal expansion in volume
below Tc appears to be due to competing magnetic interactions within and
between the magnetic sublattices. Our electronic structure calculations show
(a) the importance of electron-electron correlation in Nd 4f and Co 3d states
(b) Co ions stabilize in intermediate spin (IS) state, having oxidation state
less than +3 (c) half metallicity. Our results show the possibility of coupling
between magnetism and ferroelectricity. We believe that our results especially
on the valence state of the Co ion, zero thermal expansion in volume, short
range magnetic orderings and the connection between different degrees of
freedom will be helpful in clearing the ambiguities existing in literature on
the nature of magnetism and thereby aiding in designing new functionalities.
