Interaction between hydrogen and gallium vacancies in β-Ga2O3

In this paper, the revised Heyd-Scuseria-Ernzerhof screened hybrid functional (HSE06) is used to investigate the interaction between hydrogen with different concentrations and gallium vacancies in β-Ga2O3. The hydrogen can compensate a gallium vacancy by forming hydrogen-vacancy complex. A gallium vacancy can bind up to four hydrogen atoms, and formation energies decrease as the number of hydrogen atoms increases. Hydrogen prefers to bind with three coordinated oxygen. The bonding energy and annealing temperatures of complexes containing more than two hydrogen atoms are computed, and show relatively high stability. In addition, vacancy concentrations increase with the increasing vapor pressures. This paper can effectively explain the hydrogen impact in β-Ga2O3.

Scientific REPORTS | (2018) 8:10142 | DOI: 10.1038/s41598-018-28461-3 where, E defect (q) is the total energy of a relaxed supercell containing vacancies with charge state q, E perfect (0) is the total energy of the pure host crystal. E F is the Fermi level in the band gap with respect to the valence band maximum E v . It is necessary to adapt a potential-alignment with a correction term ΔV to correct the image charge.
The energy is corrected with the method given by S. Lany and A. Zunger method 18,19 . Equation (1) shows that defect formation energy depends on the chemical potential of the associated atomic species. The chemical potential is related to processing conditions, such as temperature and pressure, and this quantity is a variable in the formalism. The Ga atom out of the crystal lattice is placed in a reservoir with energy μ Ga , for which we make reference to the energy per Ga atom in the bulk. The chemical potential μ Ga can vary to represent experimental conditions during the crystal growth or annealing, ranging from O-rich (Ga-poor) to Ga-rich (O-poor) conditions, which are bounds set by the formation enthalpy of β-Ga 2 O 3 : Ga O f 2 3 In order to prevent formation of bulk Ga phases, O 2 and H 2 , the chemical potentials are bounded as follows: , , 0 Ga O H µ µ µ ≤ Similarly, the growth conditions reflect the range set by the formation enthalpy of each compound. To prevent formation of secondary H 2 O phases, the following condition is required:  (2) and (4) gives H f Ga f 2 3 2 The inequalities in Eqs (3) and (5) allow us to describe the region of chemical potentials in the μ Ga -μ H plane for which Ga-O-H is stable.
The charge state transition level ε(q/q′), describing the E F at which the formation energy of two states with different charges is equal, is given by 20 The formation energy is used to calculated defect concentrations. As entropy has an important effect on gibbs energy under high temperatures, it is needed to be considered. Therefore, in the strong dilution limit the concentration of oxygen vacancy is expressed as 4

Results and Discussion
Bulk properties of β-Ga 2 O 3 . According to the above computational approach, the lattice parameters of β-Ga 2 O 3 are given in Table 1, which are in a good agreement with calculated and experimental results in the references 7,8,[21][22][23][24] . The Hartree-Fock mixing parameter is set to 39%, which reproduces the band gap in Fig. 1 Moreover, complex vacancy containing more than four hydrogen atoms has no transition level in band gap which acts as shallow donor and cannot affect carrier recombination and electronic property. Therefore, we only consider the vacancy with no more than four hydrogen atoms. Considering Eqs (2)-(5), the region of chemical potentials in the μ Ga -μ H -μ O plane for which Ga-O-H is shown in Fig. 3. From Fig. 3, it can be seen that potentials of hydrogen and gallium decrease when that of oxygen increases. Potentials can be obtained from intersections between potentials of O-rich or Ga-rich and their compound lines. Potentials of gallium, oxygen, and hydrogen are −3.78 eV, −16.58 eV, and −2.64 eV.   Previous density-functional-theory studies have examined the atomic structure, formation energy, and defect energy levels of V Ga H in β-Ga 2 O 3 7 . In this paper, formation energy with different hydrogen numbers [n = 1, 2, 3,4] is listed in Table 2 together with ref. 7 . According to Table 2, it can be seen that the formation energy decreases when the number of hydrogen increases. This is because the addictive hydrogen can effectively reduce dangling bonds resulting in lower the formation energy. And the number of electron the vacancy keeps decreasing, as the number of hydrogen increases. When it gets four hydrogen atoms, only the singly positive charged vacancy is stable within the band gap. There is not much available room for extra electrons, when hydrogen has dangling bonds. From Eq. (1), it can be acknowledge that the formation energy is closely connect with chemical potential reference. Gallium has many kinds of structures. Van de Walle uses chemical potential references coming from the α-phase of bulk Ga, while ones coming from the orthorhombic-phase of bulk Ga is used in this paper. The reason why choose this structure is that it has lower formation energy, which means a higher tendency to from. Meanwhile, the amount of exact exchange in the revised Heyd-Scuseria-Ernzerhof screened hybrid functional can affect the energy slightly. Therefore, reasons all above lead to differences of formation energy.
According to Eq. (6), the transformation levels with different number of hydrogen are shown in Fig. 4. The transformation levels of V Ga nH are all deep acceptors. But it can be seen vacancy related defect levels within the band gap is gradually removed as the number of hydrogen increases. For Ga(II), ε (−1/−2) is removed when H increase to 2 H. And ε (+1/−1) changes from 1.818 to 1.773 which is also removed till the number of hydrogen increases to 3. All the transition levels are all removed when a gallium vacancy can bind up to four hydrogen atoms. For Ga(I) being contrary to Ga(II), ε (+1/0) and ε (0/−1) change from 1.382 to 1.545 and 2.264 to 2.063, respectively. ε (−1/−2) is removed when V Ga H becomes V Ga 2H. When the hydrogen number increases to 3, only ε (+1/0) is left. For V Ga 4H, all defect levels are removed. According to Peter Déak' results, it is found that the chosen epsilon can effectively affect transition levels 4 . However, in order to understand the interaction between hydrogen and gallium vacancy, and compare with C G Van deWalle, the static dielectric constant is used. The acceptor levels are effectively passive, when hydrogen atoms are added into the vacancy. However, contrary to C G Van deWalle 9 , the ε (+1/−1) replace ε (+1/0) and ε (0/−1) within the band gap for Ga(II).
Atomic structure of hydrogen-vacancy complexes. There are six oxygen atoms surrounding a vacancy site and hydrogen will likely passivate their dangling bonds. Half of coordinated oxygen atoms are three coordinated. We generated initial V Ga nH structures by placing hydrogen atoms in the vacancy site, and then relaxed these structures. After atomic structure relaxations, the corresponding stables with different numbers of hydrogen are shown in Fig. 5. By differentiating the connecting atom, hydrogen is more prior to bind to O(I) and O(II) than O(III). When gallium vacancy generates, three coordinated oxygen is the relatively less stable than four coordinated ones. Therefore, hydrogen prefers to bind to O(I) and O(II). Although hydrogen prefers to bind to three coordinated oxygen, there are only half of hydrogen atoms binding to them when hydrogen numbers increase up to four.   The distance between hydrogen and oxygen nearby is shown in Table 3. All distances are around 1 Å. And it can be seen distances increase when the bulk is with more electrons, meaning a relatively weaker bind between hydrogen and oxygen. Meanwhile, when more hydrogen is added, distances of earlier added hydrogen with oxygen in the same charged state increase. It is because hydrogen added later can effectively reduce the interaction between hydrogen added former and oxygen nearby. And when we compare octahedral gallium with tetrahedral ones, it is found tetrahedral ones are easier to bind with O(I) and O(II).

Defect reactions of hydrogen-vacancy complexes.
The actual defects present in a real system will depend on the processing, electron source, applied voltage, and temperature of that system. However, assuming an initial distribution of defects in different charged states and electron transfer between defects, we can combine the obtained information about the various defects with different charged states to make some predictions about which defect combinations are energetically more favorable.
Various reactions and their energies are presented in Table 4. These energies have been calculated as differences in total energies of pairs of individual defects and each pair has the same total charge state and number of atoms, where the binding energy is obtained from 25 , and E f (V Ga nH) is the formation energy of the complex. When n − N = 0, V Ga is the formation energy of the isolated vacancy.
Positive energies indicate that a reaction in the direction of the arrow is energetically favorable. Note that we do not consider any reactions that include total energies with delocalized states. The energies presented in Table 4 also do not include the interaction between defects, which can be strong especially in close charged defect pairs. The formation energy of interstitial hydrogen atom and hydrogen is −2.94 eV and −0.43 eV, respectively. The activation energy E a for dissociation of a defect complex can be estimated by the sum of binding and migration barrier of interstitial hydrogen 7 . We estimate the dissociation temperature based on an activated process with a hopping rate of the form Γ = Γ 0 exp(−Ea/k B T) 9 . A typical vibrational frequency Γ 0 is estimated by ν = (2E a /ml 2 ) 1/2 26,27 . According to the formula, different charged states cannot enormously affect the vibrational frequency, therefore, neutral states are used to estimate vibrational frequency. Meanwhile, a distance around 3.1 Å is used as it is the distance of equivalent atom. As for Γ, it is not very sensitive and 1 min −1 is an appropriate value to evaluate temperature, which has been proved for many materials 12 . Therefore, we assume that dissociation starts occurring once the rate Γ reaches a value of 1 min −1 , and Γ 0 = 100 THz based on typical vibrational frequencies. Using this expression, we obtain an estimated dissociation temperature. Annealing temperatures of different reactions are also shown in Table 4.   As Table 4 shows, most of reactions are hard to diffuse at low temperature, due to its relatively high E b . Products composing of V Ga nH, or (V Ga nH) − with + H i have relatively lower annealing temperatures such as No. 1 and No. 5. However, (V Ga nH) − is hard to anneal at this temperature. And we can see E b and annealing temperatures increase with the number of + H i . Annealing temperatures of No. 11,No. 12,No. 20,and No. 26 are much higher than those of No. 6,No. 9,No. 17,and No. 24, which means the more hardness to form hydrogen gas than interstitial hydrogen. Annealing temperatures increase, when more hydrogen atoms are separated from complex vacancy. Moreover, interstitial H 2 is harder to separate from complex vacancy than interstitial H.

Defect concentrations of hydrogen-vacancy complexes.
From the combination of the individual defect formation energy for HSE06, we have also calculated Schottky, and Frenkel energies for T = 0 K (all per created defect). The corresponding reaction equations for Frenkel and Schottky disorders are following.
Schottky equilibrium: For both two kinds of gallium vacancies, the Frenkel disorder energies are close and distinctly lower than the Schottky energies. It seems that Frenkel disorder is dominant.
Next, vacancy concentrations under different temperature and pressure are considered. The chemical potentials used for the calculation of the formation energy were constrained to lie within the stability field of β-Ga 2 O 3 in Fig. 2. However, these values are only valid for T = 0 K. As we want to extend our analysis of the defect properties of gallium oxide to the case of T > 0 K, we also have to include the temperature dependence of the chemical potentials in our phase stability considerations. First, we take the T = 0 K total energy of the hydrogen molecule E tot (H 2 ) to be approximately the enthalpy under reference conditions, i.e., temperature T 0 = 298.15 K and pressure P 0 = 1 bar. The remaining temperature and pressure dependence are taken from thermochemical tables,  where, ΔH(T) = C p (T − T 0 ) and ΔS(T) = C p ln(T/T 0 ) are the enthalpy and entropy changes, and H 0 = 8.7 kJ mol −1 and entropy S 0 = 205 J mol −1 K −1 . And C p is the constant-pressure heat capacity, equating to 29.4 J/(mol · K −1 ) 30 .
We also chose the entropy of gallium vacancy for calculating the gibbs energy in Eq. 4, as it is very dilute in β-Ga 2 O 3 . And the entropy is −1.61 eV by the research of T. Zacherle 28 . In Fig. 6, we display the defect concentration of complex vacancy against the hydrogen partial pressure for T = 1273 K. The calculated stability range is from pH 2 = 0.1 bar to pH 2 ≈ 10 −21 bar. At high pressure, it is found that the concentration of positive charged V Ga nH is much higher than other charged states. All negative charged vacancies have much lower concentrations. Combining with the annealing temperature, the positive charged vacancies cannot be a major factor affecting the characteristic of devices, due lower concentrations and annealing temperatures. With the pressure decreasing, all vacancy concentrations change a lot and decrease to less than 1 cm −3 . And concentrations of tetrahedral gallium are larger than octahedral ones.

Conclusion
The stability of hydrogen -gallium complex vacancy in β-Ga 2 O 3 is systematically discussed. It is found that gallium vacancy can bind up to four hydrogen atoms and the formation energy decreases and transformation levels are gradually disappeared with the hydrogen number increases. Moreover, V Ga 3H and V Ga 4H are predicted to be unstable in n-type β-Ga 2 O 3 , precluding complete passivation of gallium vacancies in n-type material. Hydrogen can either compensate a vacancy by donating an electron to a vacancy acceptor level, or passive the vacancy by forming a hydrogen-vacancy complex. The added electron can reduce the bond interaction between hydrogen and oxygen nearby. Hydrogen atoms prefer to bind with three coordinated oxygen atoms, then begin to bind with four coordinated oxygen when the number of hydrogen is more than one. By calculating bind energies, it is found that the complex vacancy with more than two hydrogen atoms is stable, which also has relatively high annealing temperatures. With more hydrogen atoms are separated from complex vacancy, annealing temperatures increase. Compared with interstitial hydrogen atom, interstitial H 2 is harder to separate from complex vacancy. All vacancy concentrations decrease with the pressure decreasing. The vacancy filled with more hydrogen atoms has higher concentration. The gallium vacancy containing four hydrogen atoms has largest concentration among all kinds of vacancies. This paper can effectively explain the hydrogen impact in β-Ga 2 O 3 .