The alumina substrate

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An alumina substrate on which an AlN layer is formed and that causes less warping, and a substrate material strong enough to withstand normal handling when an AlN crystal is grown upon it, and prevents cracking and fracturing of a grown crystal when stress is applied during growing or cooling. The substrate has a gap and a rare earth element-containing region inside the AlN layer or at the interface between the AlN layer and the alumina substrate. Warping of the AlN layer can be reduced by lattice-mismatch stress being concentrated at the region and releasing of stress by the gap. The region having a concentrating of stress, and the gap having a low mechanical strength, can induce crackings and fracturings. As a result, contamination of crackings and fracturings into the crystal grown on the substrate can be prevented. The region can ensure a level of mechanical strength sufficient for handling.