Origin of donor compensation in monoclinic (Al$_x$Ga$_{1{ m -}x})_2$O$_3$ alloys

(Al$x$Ga${1{\rm -}x})_2$O$_3$ alloys are frequently used in heterostructures with monoclinic Ga$_2$O$_3$, resulting in a large conduction-band offset, which leads to charge carrier confinement, a property that is desirable for device applications. However, when (Al$x$Ga${1{\rm -}x})_2$O$_3$ alloys are $n$-type doped with Si, the most efficient shallow donor, there is a significant reduction in the number of charge carriers when the Al content of the alloys is greater than 26%, rendering intentional doping ineffective. Here we show that this compensation is due to cation vacancies forming in response to donor doping. We use hybrid density functional theory to study cation vacancies in monoclinic AlGaO$_3$ and monoclinic Al$_2$O$_3$. We find that vacancies prefer to occupy split-vacancy configurations, similar to vacancies in Ga$_2$O$_3$. Furthermore, by comparing the formation energy of the vacancy with the formation energy of Si donors, we show that vacancies are lower in energy than Si donors, independent of the Fermi level, as soon as the alloys contain more than 16% Al. Therefore, cation vacancies will compensate the donor doping, explaining experimental observations.

💡 Research Summary

**
The paper addresses a long‑standing discrepancy between experimental observations and theoretical predictions for n‑type doping of monoclinic (AlₓGa₁₋ₓ)₂O₃ alloys. While silicon is the most efficient shallow donor in β‑Ga₂O₃ and is expected to remain shallow up to very high Al contents (≈85 % according to previous DFT studies), experiments show that the free‑carrier concentration drops dramatically when the Al fraction exceeds roughly 20–26 %. The authors propose that this “compensation” originates from the formation of cation vacancies (V_Al or V_Ga) that act as deep acceptors and bind up to three electrons.

Using the Vienna Ab‑initio Simulation Package (VASP) with the HSE06 hybrid functional (32 % exact exchange), the authors model 120‑atom supercells of monoclinic Ga₂O₃, Al₂O₃, and intermediate AlGaO₃ alloys. They examine five possible vacancy configurations: tetrahedral (I), octahedral (II), and three split‑vacancy complexes (ia, ib, ic) in which a neighboring cation moves into an interstitial site, creating two half‑vacancies. Formation energies are calculated under both O‑rich (μ_O = 0) and O‑poor (μ_Ga = 0) limits, with Si chemical potentials referenced to SiO₂. Charge‑state corrections are applied via the sxdefectalign code.

The key findings are:

1. In all three materials the lowest‑energy vacancy is the split‑vacancy “ic” configuration, consistent with earlier work on β‑Ga₂O₃. In AlGaO₃ the same ordering holds: V_ic (Al‑ or Ga‑based) is most stable, followed by V_ib and the two ia variants. Removing an octahedral Al atom (V_AlII) is energetically comparable to the ia split‑vacancies and significantly more favorable than removing a tetrahedral Ga atom.

2. Si donors (charge +1) have formation energies that increase with Al content because the conduction‑band minimum (CBM) moves upward, widening the gap between the donor level and the VBM. Conversely, cation vacancies in the –3 charge state become increasingly favorable as the Fermi level approaches the CBM.

3. By comparing the formation energy of the lowest‑energy vacancy (V⁻³) with that of Si⁺¹ at the VBM, the authors define a “cross‑over” Fermi level ε(V/Si). This level falls within the band gap once the alloy contains ≈16 % Al, regardless of growth conditions. Above this Al fraction the vacancy formation energy is lower than that of the Si donor, implying that vacancies will form preferentially and compensate the donors.

4. The cross‑over point is shifted deeper into the gap under O‑rich conditions, meaning that O‑rich growth exacerbates compensation. O‑poor (oxygen‑deficient) environments raise the vacancy formation energy, partially mitigating the effect.

5. The authors extrapolate these results to the full composition range, showing that for Al > 16 % the vacancy concentration will increase dramatically, eventually neutralizing all Si donors. This explains why experimental Si‑doped AlGaO₃ heterostructures only show effective n‑type behavior up to ≈26 % Al.

The paper concludes that cation vacancies are the dominant compensating defects in Al‑rich (AlₓGa₁₋ₓ)₂O₃ alloys. To achieve reliable n‑type doping, growth should be performed under oxygen‑deficient conditions, and Al content should be kept low enough (≤ 20 %) or alternative dopants (e.g., Sn, Ge) should be explored. The work provides a quantitative thermodynamic framework for defect engineering in ultra‑wide‑bandgap oxide semiconductors, guiding the design of high‑performance power devices and solar‑blind photodetectors based on β‑Ga₂O₃ and its alloys.