Formation of mid-gap states and ferromagnetism in semiconducting CaB$_6$

We present a consistent overall picture of the electronic structure and ferromagnetic interaction in CaB$_6$, based on our joint transport, optical, and tunneling measurements on high-quality {\em defect-controlled} single crystals. Pure CaB$_6$ single crystals, synthesized with 99.9999 %-pure boron, exhibited fully {\em semiconducting} characteristics, such as monotonic resistance for 2--300 K, a tunneling conductance gap, and an optical absorption threshold at 1.0 eV. {\em Boron-related defects} formed in CaB$_6$ single crystals synthesized with 99.9 %-pure boron induced {\em mid-gap states} 0.18 eV below the conduction band and extra free charge carriers, with the transport, optical, and tunneling properties substantially modified. Remarkably, no ferromagnetic signals were detected from single crystals made with 99.9999 %-pure boron, regardless of stoichiometry, whereas those made with 99.9 %-boron exhibited ferromagnetism within a finite range of carrier density. The possible surmise between the electronic state and magnetization will be discussed.