Energy of electric field In a semi conductor, for an electron in valence band to transfer to conduction band, it needs energy which can be provided by applying an electric field such that a force is applied on it until it breaks free from its parent atom. Now till the time it is at rest no change in kinetic energy is there. Hence, the energy that it gained and transfered to conduction band is electric potential energy. But this electric potential energy is stored in the field configuration i.e. The number at every point in space. And when u change that number you have to give off energy. N when the number is reverted back. It gives back to you the same energy. 
Now my question is that this case of the electron breaking free requires that the electric potential  energy be localised at the electron like the electron's kinetic energy and not be stored in the field configuration. How is this happening? Is electric potential energy stored in electron or the electric field configuration at each point in space? And if in the field then how does the electron jump the energy gap and enter the conduction band, where does it get the energy from? Is kinetic energy locally stored in the particle or it is also a field configuration kind of thing?
Is there a field associated with kinetic energy ?
Or my understanding is wrong in which case please explain what is meant by electric potential energy is stored in electric field? And what is electric field energy density?
edit:  leaving the semiconductor example aside. Suppose i have an atom and i apply external electric field and that causes the atom to ionise. Where did the electron that flew away got the energy from?
 A: An analogy to your last example of ionizing the atom with an electric field is a stone dropped from a tower. If the 'gravitational-field is turned on' i.e. the stone is released from the top of the tower, the  stone gains kinetic energy from the gravitational field of the Earth and drops towards it. In the same way (and from the fundamental classical viewpoint) the electron in an atom gains kinetic energy when an external electric field is switched on and moves towards that field (of course the strength of the external field must be greater than that from the atomic nucleus subtracted by the screening effect from other electrons in the same atom). 
Now to answer the question: where does that kinetic energy comes from? One interpretation is from simple calculus and newton's laws: Recall that since force is defined as the gradient of potential energy the stone/electron has to travel to the bottom-most point of the potential plot to reach mechanical equilibrium (no force acting on it). According to newton's first law the stone/electron moves (gains kinetic energy) as long as gravity/electric force acts on it and does so until it reaches a point where no force act on it or all forces balance each other.  The stone on top of the tower or an electron in an outermost shell is initially in equilibrium and introducing an electric field or gravity only  shifts equilibrium point to another location and the stone/electron moves towards it.
Now coming to semiconductors, the conduction and valence bands are states of a defined energy (both kinetic plus potential) which are large number of splittings of energy levels of individual atoms in the crystal and do not directly correspond to the electric potential within the crystal lattice (even though the splittings are heavily influenced by the electric potential). From my understanding based on your question, this is where your confusion lies . Depending on the crystal material there may be some energy states that are 'forbidden' or not attainable for reasons (quantum mechanics) very similar to an atom having discrete energy levels.  The electrons are filled in the valence and conduction bands from lowest to highest. An electron can jump from the valance band to the conduction band by introducing thermal energy (which is kinetic energy on an atomic level) or shining light on it with an appropriate wavelength. Exciting an electron from a valence band to conduction band increases its total energy (KE + PE) and not necessarily just its KE or PE. 
Of course one can assign a 'crystal momentum' to the electron when treating them as quantum mechanical waves, but I believe this discussion is digression.
Sources:
inferred from any standard freshman/sophomore level physics text.
