forming a cathode conductive layer, an insulator layer, and a gate layer on a substrate sequentially;

forming an opening in the gate layer and the insulator layer to expose a portion of the cathode conductive layer;

forming a metal layer on the exposed portion of the cathode conductive layer in the opening;

conducting a first anodization to the metal layer so as to form a first metal anodization layer from a portion of the metal layer;

removing the first metal anodization layer for exposing the un-anodized metal layer;

conducting a second anodization to the un-anodized metal layer to form a second metal anodization layer, wherein a plurality of through pores are formed in the second metal anodization layer in the opening;

forming a catalyst layer in the through pores; and

forming a plurality of carbon nanotubes inside the through pores, wherein each of the carbon nanotubes grows from the cathode conductive layer in each of the through pores.