Prospects and limitations for $p$-type doping in boron nitride polymorphs

Using first-principles calculations, we examine the potential for $p$-type doping of BN polymorphs via substitutional impurities. Based on density functional theory with a hybrid functional, our calculations reveal that group-IV elements (C, Si) substituting at the N site result in acceptor levels that are more than 1 eV above the valence-band maximum in all of the BN polymorphs, and hence far too deep to allow for $p$-type doping. On the other hand, group-II elements (Be, Mg) substituting at the B site lead to shallower acceptor levels. However, for the ground-state hexagonal phase ($h$-BN), we show that $p$-type doping at the B site is inhibited by the formation of hole polarons. Our calculations reveal that hole localization is intrinsic to $sp^2$ bonded $h$-BN, and this places fundamental limits on hole conduction in this material. In contrast, the $sp^3$ bonded wurtzite ($w$-BN) and cubic ($c$-BN) polymorphs are capable of forming shallow acceptor levels. For Be dopants, the acceptor ionization energies are 0.31 eV and 0.24 eV for $w$-BN and $c$-BN, respectively; these values are only slightly larger than the ionization energy of the Mg acceptor in GaN.