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I think :
first, the elctricity of the gate oxide is greater than field oxide. If the field oxide is thinner than gate oxide, the field oxide will be breakdown quicker than gate oxide while them on the same voltage.
second. if the field oxide is thinner than gate oxide, the vth of the field is lower than gate oxide. so the parasitic devices will much more, and the circuit will not on work.
The gate oxide and the field oxide are totally two different thing.
If you have some knowledge of semiconductor Device Phsics and semiconductor process,you will know the gate oxide is used to form the MOS(Metal_Oxide_Semiconductor) structure . To maintain the gate Voltage's control ove the conduction channel ,the gate oxide,more precisely the gate dielectric must not be too thick,in 130nm technology the gate oxide is 1.6nm thick,in 90 nm,it is about 1.2nm.
the field oxide is used for the purpose of isolation and insulation.Isolate each transistors from each other and allow interconnect wires run on it without short. To avoid forming parasitic MOS structures in field oxide region,the field xoide is much more thick than gate oxide.
Gate oxide is, and always will be thiner than field oxide...But why?? Well the answer rely on what are they used for.
Field oxide: Isolate, IC Robustness, etc...
Gate oxide: Makes part of the conception of a MOS transistor, will be thiner with the technology scalling for current drive issues...