A cathode structure suitable for a flat-panel display contains an emitter layer (213) divided into emitter lines, a plurality of electron emitters (229, 239, or 230) situated over the emitter lines, and a gate layer (215A) having an upper surface spaced largely above the electron emitters. The gate layer has a plurality of gate holes (215B) each corresponding to one of the electron emitters. The cathode structure further includes a carbon-containing layer (340, 240, or 241) coated over the electron emitters and directly on at least part of the upper surface of the gate layer such that at least part of the carbon-containing layer extending along and above the gate layer is exposed.