Pressure-dependent point-defect thermodynamics and dopant compensation in cubic boron nitride

Cubic boron nitride (c-BN) is an ultrawide-band-gap semiconductor whose bipolar doping underpins its proposed use in high-power electronics and deep-UV optoelectronics, yet the first-principles defect thermodynamics that governs its dopability — donor solubilities, deep transition levels, and compensation by the triply charged boron vacancy $V_B^{3-}$ — has been computed only at zero pressure, while useful doped c-BN is made by high-pressure, high-temperature (HPHT) synthesis or by strongly non-equilibrium thin-film growth. The open problem is a quantitative account of how applied pressure modifies the formation energies, charge-state transition levels, and relative stability of native and dopant point defects in c-BN, how this reshapes carrier compensation and the equilibrium limits on n-type doping, and whether non-equilibrium synthesis conditions can overcome those limits, including the enhancement of Si incorporation above its equilibrium solubility.

Background

Cubic boron nitride (c-BN) is the cubic, zincblende-structured polymorph of boron nitride, an ultrawide-band-gap semiconductor (experimental gap about 6.4 eV) that is nearly as hard as diamond, rivals it in thermal conductivity, and is more chemically stable at high temperature. Since Mishima and co-workers produced an ultraviolet-emitting p-n junction diode from high-pressure-synthesized c-BN crystals in 1988, the material has been pursued for high-power electronics and deep-UV optoelectronics; every such application requires controlled n- and p-type doping. In practice, bulk single crystals of c-BN can only be grown under high-pressure, high-temperature (HPHT) conditions — pressures of several gigapascals with catalysts — while thin films are grown metastably at low pressure by strongly non-equilibrium techniques such as ion-beam-assisted molecular beam epitaxy (MBE, growth from atomic beams in vacuum, with an ion beam supplying the energy that stabilizes the cubic phase).

Whether a dopant produces free carriers is governed by point-defect thermodynamics. In the standard first-principles formalism, the formation energy of a defect $D$ in charge state $q$ is $E_f(D^q)=E_{\rm tot}(D^q)-E_{\rm tot}(\mathrm{bulk})-\sum_i n_i\mu_i+q(E_F+E_{\rm VBM})$, where $\mu_i$ are chemical potentials (atomic reservoirs, bounded by the competing phases that could form instead) and $E_F$ is the Fermi level; a defect is abundant when its $E_f$ is low. Charge-state transition levels $\epsilon(q/q')$ measure how deep a donor or acceptor is. Native defects with low formation energies compensate intentional dopants: the triply negatively charged boron vacancy $V_B^{3-}$, for example, is a deep acceptor that traps electrons released by n-type dopants. Some defects are negative-$U$ centers, where electron-lattice coupling is so strong that an intermediate charge state is thermodynamically unstable; in ultrawide-band-gap nitrides, donors such as Si can also form DX centers — negatively charged two-electron states that self-compensate n-type doping. Density-functional theory (DFT) in the local density approximation (LDA) severely underestimates the band gap, so early defect studies applied rigid conduction-band shifts, while modern work uses hybrid functionals (mixing a fraction of exact exchange) for accurate gaps and defect levels.

Before 2011, Orellana and Chacham (2001) had computed native-defect stability in hexagonal and cubic BN using LDA without a band-gap correction. Gai and Tang (2011) recomputed formation energies and transition energies of all native defects and of the candidate dopants (Be and Zn on boron sites; O and S on nitrogen sites; Si on a boron site, $Si_B$) in c-BN using LDA plus a rigid 2.05 eV conduction-band shift, under boron-rich and nitrogen-rich reservoirs at zero pressure and standard temperature. They found every native-defect level deep, $V_B^{3-}$ to be the dominant compensating acceptor under n-type conditions, $V_N$ and $N_B$ to be negative-$U$, the shallowest donor to be $Si_B$ (transition level about 0.38 eV below the conduction band) but with a high formation energy (about 5.87 eV, limited by the stability of Si$_3$N$_4$), and $Be_B$ to be the best acceptor (about 0.30 eV above the valence band). Their concluding statement records that all of this was computed at equilibrium ambient conditions, whereas the experimental realization of both n- and p-type c-BN relies on HPHT synthesis, so that the influence of applied pressure on formation energies, transition levels, and relative defect stability, and the quantitative effect of non-equilibrium conditions on the equilibrium limitations on n-type doping (low donor solubility and the abundance of $V_B^{3-}$), remained unexplored, with Si-incorporation enhancement by non-equilibrium growth suggested but not quantitatively assessed. Under pressure, formation energies become formation enthalpies ($E+PV$), each defect carrying its own formation volume; band edges, charge-state orderings, and the competing phases that bound the chemical potentials (BN polymorphs, Si$_3$N$_4$, Be$_3$N$_2$) all shift with pressure, so the zero-pressure ranking of defects need not carry over to synthesis conditions.

Problem Statement

The problem is to give a quantitative account of point-defect thermodynamics in cubic boron nitride under the conditions at which doped c-BN is actually made:

1. Pressure-dependent defect thermodynamics. Determine how applied hydrostatic pressure over the HPHT synthesis range of several gigapascals modifies the formation energies (formation enthalpies), charge-state transition levels, and relative stability of the native point defects of c-BN (boron and nitrogen vacancies, antisites, interstitials) and of the substitutional dopants $Si_B$, $O_N$, $S_N$, $Be_B$, and $Zn_B$.

2. Carrier compensation under pressure. Determine how these pressure dependences change the predicted compensation balance — in particular the abundance of the triply charged boron vacancy $V_B^{3-}$ relative to donor incorporation — and hence the equilibrium limits on n-type (and p-type) doping of c-BN under HPHT synthesis conditions, compared with the established zero-pressure picture.

3. Non-equilibrium synthesis versus equilibrium limits. Assess quantitatively whether non-equilibrium synthesis conditions (metastable ion-assisted epitaxial growth) can overcome the equilibrium limitations on n-type doping — the low equilibrium solubility of donors and the compensating abundance of $V_B^{3-}$ — including the degree to which Si incorporation is enhanced above its equilibrium solubility.

These three parts are jointly mandatory: a resolution must address each of them, and no single part alone resolves the problem.

References

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