Crystals having a modified regular tetrahedron shape are provided. Crystals preferably have four substantially identical triangular faces that define four truncated vertices and six chamfered edges. The six chamfered edges can have an average length of l, and an average width of w, and 8≦l/w≦9.5.

Crystals having a modified regular tetrahedron shape are provided. Crystals preferably have four substantially identical triangular faces that define four truncated vertices and six chamfered edges. The six chamfered edges can have an average length of l, and an average width of w, and 8≦l/w≦9.5.

A biomedical implant (16, 18) is formed from magnesium (Mg) single crystal (10). The biomedical implant (16, 18) may be biodegradable. The biomedical implant (16, 18) may be post treated to control the mechanical properties and/or corrosion rate thereof said Mg single crystal (10) without changing the chemical composition thereof. A method of making a Mg single crystal (10) for biomedical applications includes filling a single crucible (12) with more than one chamber with polycrystalline Mg, melting at least a portion of said polycrystalline Mg, and forming more than one Mg single crystal (10) using directional solidification.

A RAMO.sub.4 substrate includes a single crystal represented by a formula of RAMO.sub.4 (in the formula, R indicates one or a plurality of trivalent elements selected from a group consisting of Sc, In, Y, and a lanthanoid element, A indicates one or a plurality of trivalent elements selected from a group consisting of Fe(III), Ga, and Al, and M indicates one or a plurality of bivalent elements selected from a group consisting of Mg, Mn, Fe(II), Co, Cu, Zn, and Cd). An epitaxially-grown surface is provided on at least one surface of the RAMO.sub.4 substrate. An unevenness having a height of 500 nm or more is not provided on the epitaxially-grown surface.

A RAMO.sub.4 substrate includes a single crystal represented by a formula of RAMO.sub.4 (in the formula, R indicates one or a plurality of trivalent elements selected from a group consisting of Sc, In, Y, and a lanthanoid element, A indicates one or a plurality of trivalent elements selected from a group consisting of Fe(III), Ga, and Al, and M indicates one or a plurality of bivalent elements selected from a group consisting of Mg, Mn, Fe(II), Co, Cu, Zn, and Cd). An epitaxially-grown surface is provided on at least one surface of the RAMO.sub.4 substrate. An unevenness having a height of 500 nm or more is not provided on the epitaxially-grown surface.

A RAMO.sub.4 substrate is formed from single crystal represented by a formula of RAMO.sub.4 (in the formula, R indicates one or a plurality of trivalent elements selected from a group consisting of Sc, In, Y, and a lanthanoid element, A indicates one or a plurality of trivalent elements selected from a group consisting of Fe(III), Ga, and Al, and M indicates one or a plurality of bivalent elements selected form a group consisting of Hg, Mn, Fe(II), Co, Cu, Zn, and Cd). An epitaxially-grown surface is provided on at least one surface of the RAMO.sub.4 substrate. The epitaxially-grown surface includes a plurality of cleavage surfaces which are regularly distributed, and are separated from each other.

A RAMO.sub.4 substrate is formed from single crystal represented by a formula of RAMO.sub.4 (in the formula, R indicates one or a plurality of trivalent elements selected from a group consisting of Sc, In, Y, and a lanthanoid element, A indicates one or a plurality of trivalent elements selected from a group consisting of Fe(III), Ga, and Al, and M indicates one or a plurality of bivalent elements selected form a group consisting of Hg, Mn, Fe(II), Co, Cu, Zn, and Cd). An epitaxially-grown surface is provided on at least one surface of the RAMO.sub.4 substrate. The epitaxially-grown surface includes a plurality of cleavage surfaces which are regularly distributed, and are separated from each other.

A RAMO.sub.4 substrate includes a single crystal represented by a formula of RAMO.sub.4 (in the formula, R indicates one or a plurality of trivalent elements selected from a group consisting of Sc, In, Y, and a lanthanoid element, A indicates one or a plurality of trivalent elements selected from a group consisting of Fe(III), Ga, and Al, and M indicates one or a plurality of bivalent elements selected from a group consisting of Mg, Mn, Fe(II), Co, Cu, Zn, and Cd). An epitaxially-grown surface is provided on one surface of the RAMO.sub.4 substrate, a satin-finish surface is provided on another surface. The satin-finish surface has surface roughness which is larger than that of the epitaxially-grown surface.

A RAMO.sub.4 substrate includes a single crystal represented by a formula of RAMO.sub.4 (in the formula, R indicates one or a plurality of trivalent elements selected from a group consisting of Sc, In, Y, and a lanthanoid element, A indicates one or a plurality of trivalent elements selected from a group consisting of Fe(III), Ga, and Al, and M indicates one or a plurality of bivalent elements selected from a group consisting of Mg, Mn, Fe(II), Co, Cu, Zn, and Cd). An epitaxially-grown surface is provided on one surface of the RAMO.sub.4 substrate, a satin-finish surface is provided on another surface. The satin-finish surface has surface roughness which is larger than that of the epitaxially-grown surface.

Provided is a Ga.sub.2O.sub.3-based single crystal substrate capable of achieving a high processing yield. A Ga.sub.2O.sub.3-based single crystal substrate having a crack density of less than 0.05 cracks/cm can be obtained that has as a principal surface thereof a surface rotated 10-150° from the (100) plane, when a rotation direction from the (100) plane to the (001) plane via the (101) plane is defined as positive, having the [010] axis as the rotation axis.