Method of producing a display structure with triode control scheme comprising fabrication of anode structure made in the form of parallel discrete elements, fabrication on a dielectric substrate made from a high temperature material of the discrete parallel metallic elements of addressable field-emission cathode which elements are perpendicular to the said discrete elements of the anode structure and made from high temperature metal and provided with the contact pads, fabrication of a control grid placed between the addressable field-emission cathode and anode structure via deposition on the said discrete metallic elements of the addressable field-emission cathode, but excluding the contact pads, of a layer of dielectric and layer of a metal, opening the holes in the said layers of dielectric and above deposited metal in places of crossing of the discrete elements of the addressable field-emission cathode and anode structure, which holes are formed of the required shape and penetrate down to the discrete elements of the cathode, deposition of a carbon containing emissive layer, wherein on the dielectric layer a layer of metal is deposited which electrical field strength threshold for beginning of emission is higher than electrical field strength at which the required current is emitted by the cathode, and the carbon containing emissive layer is deposited on the said discrete elements of the addressable field-emission cathode via method of gas phase synthesis comprising heating of metallic filaments of the reactor and the dielectric substrate in the reactor in flow of hydrogen with admission of carbon containing gas into the said flow, conducting deposition through a protective meshed screen, and selecting deposition regime to provide growth rate of the carbon containing emissive layer on the dielectric substrate being substantially less than growth rate of the carbon containing emissive layer on the discrete metallic elements of the addressable auto-emission cathode.