TECHNOLOGY FOCUS - INORGANIC CHEMISTRY - Preferred Method: The method further includes: controlling the oxidation of the graphene sheet to maintain the oxygen content to a value of less than 10%; uniformly pre-coating the precursor particles either on a rolled graphene sheet or on a planar graphene sheet; and forming the graphene sheet to a degree of oxidation and controlling the reaction temperature of the re-crystallizing of the nanoparticles, to control the morphology of the single-crystal structure to set capacitance and discharge current density values of the electrode. In the method, step (b) includes: diffusing the nanoparticles across a graphitic lattice of the graphene sheet and re-crystallizing the nanoparticles into single-crystalline structures/single-crystalline nanoplate, and further includes forming many single-crystalline nanoplates and stacking the nanoplates to form an electrochemical pseudo-capacitor electrode; hydrothermally treating the nanoparticles and forming at least one of hydroxide, oxide, sulfide and selenide nanocrystals, respectively of at least one of manganese(II,III) oxide nanoparticles and iron family elements comprising nickel, cobalt, manganese, molybdenum, ruthenium or iron; and forming nickel hydroxide nanoplates attached to the graphene sheet at (001) crystallographic planes. The precursor particles are composed of a material used to form the nanoparticles, and the step (a) includes pre-coating precursor particles on a graphene sheet having an oxygen content of less than 10%, preferably less than 5%. Preferred Components: In the apparatus, the single-crystal structure and graphene sheet are coupled to an electrode, and the single-crystal structure is configured and arranged to facilitate the exchange of charge carriers with the electrode. In the apparatus, the single-crystal structure includes: at least one of manganese(II,III) oxide and iron family elements comprising nickel, cobalt, manganese, molybdenum, ruthenium or iron; and a nickel hydroxide nanoplate attached to the graphene sheet at (001) crystallographic planes. In the supercapacitor: the single-crystal structure is a pseudo-capacitive material configured and arranged with the graphene sheet to exhibit specific capacitance and energy density that is higher than a specific capacitance and energy capacity of the graphene sheet; the single-crystal structure includes a nickel hydroxide; the first electrode includes many graphene sheets with a single-crystal structure grown upon each sheet, where many graphene sheets overlap one another to form a three-dimensional conducting network that is configured and arranged to transfer electrons between the graphene sheet and the separator; and the second electrode includes a ruthenium(IV) oxide graphene hybrid material including ruthenium(IV) oxide grown on a graphene sheet. In the battery either: the first electrode is an anode and the single-crystal active material includes single-crystal manganese(II,III) oxide configured and arranged to interact with lithium ions for passing charge carriers between the first and second electrodes; or the first electrode is a cathode and the single-crystal active material includes lithium-manganese-iron phosphate (LiMn(1-x)FexPO4) nanorods (1520) configured and arranged to facilitate diffusion of lithium charge carriers between the first and second electrodes, where x is not defined.    TECHNOLOGY FOCUS - INSTRUMENTATION AND TESTING - Preferred Components: The apparatus further includes many graphene sheets with a single-crystal structure grown upon it and stacked in layers that form an electrochemical pseudo-capacitor electrode. EXAMPLE - No suitable example given.