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In this paper, we show theoretically and experimentally that the large-area planar plasma with high density and good uniformity can be sustained by a surface microwave when the electron density is over-dense. From the experimental results we find that the nonuniformities in azimuthal plasma density and electron temperature have been greatly improved and in particular the nonuniformity is less than 10% when the gas pressure is 30 Pa. By improving the antenna shape, enhancing the microwave power and choosing the appropriate gas pressure, the large area planar plasma with high density can be produced.