The improvement of material quality^{[1, 2]} and the realization of p-type conduction^{[3, 4]} are regarded as great breakthroughs in realizing GaN-based devices. During the last 30 years, p-type GaN has been studied and widely used in the mass production of Ⅲ-nitride devices, such as light-emitting diodes and laser diodes. Metal-organic chemical vapor deposition (MOCVD) is widely used to grow Ⅲ-nitride materials and the growth conditions have great influences on the incorporation of dopants^[5] and contaminations^[6−8]. Mg is a routine p-type dopant in GaN but it can be easily passivated by forming Mg−H complexes during MOCVD growth, because hydrogen is usually used as a carrier gas and also exists in ammonia and metalorganics^[9]. In order to activate Mg acceptors, thermal annealing is generally used to break Mg−H bonds and hydrogen will diffuse out from bulk materials during this procedure^{[4, 10−12]}. Besides, electron irradiation^[3] and minority carrier injection^{[13, 14]} can also break Mg−H bonds and transfer H to a non-passivating state. Therefore, the electrical properties of p-GaN are determined by Mg doping level, residual donor concentration and the degree of H passivation.

Hole concentration (p) will increase while resistivity (ρ) and mobility (μ) will decrease after the activation of Mg. Those electrical properties will saturate to a certain level when annealing is sufficient. It is necessary to confirm this saturation level because activation will become more difficult when [Mg] is higher or p-GaN layer is thicker. Then, higher temperature or longer time is essential during annealing.

There are limited systematic studies on the relationship between a wide range of [Mg] and whole electrical properties (p, ρ, and μ). Kaufmann et al.^[15] reported the relationship between [Mg] and p and μ, other researchers like Iida et al.^[16] mainly reported the relationship between [Mg] and p.

In this paper, p-GaN samples with different [Mg] ranging from low 10¹⁸ to high 10¹⁹ cm⁻³ were annealed under different conditions. We confirmed the saturation level which corresponds to the highest p, the lowest ρ and μ for each p-GaN sample by Hall measurement at room temperature (RT). The experimental trends between [Mg] and electrical properties after sufficient annealing are presented in this paper. Theoretical curve relating [Mg] and p is derived by a charge neutrality equation and the ionized-acceptor-density dependent ionization energy of Mg acceptor.

We believe the relationship between [Mg] and p, ρ, μ after sufficient annealing and the residual H concentration obtained by secondary ion mass spectroscopy (SIMS) as described in this paper could supply good references to researchers in this field in order to judge the degree of Mg activation.

All the p-GaN samples were grown on 2-inch c-plane GaN-on-sapphire templates or free-standing GaN substrates at atmospheric pressure in Taiyo Nippon Sanso (TNSC) horizontal MOCVD reactor (SR-4000KS). Trimethylgallium (TMGa), trimethylaluminum (TMAl), trimethylindium (TMIn), ammonia (NH₃), and bis-cyclopentadienyl magnesium (Cp₂Mg) were used as the precursors for Ga, Al, In, N, and Mg, respectively.

These p-GaN samples can be categorized into two series. In the first series, mixture gas (H₂ : N₂ = 1 : 1) was used as a carrier gas during the growth. 1 μm thick undoped GaN was grown at 1000 °C, followed by 500−700 nm thick Mg-doped p-GaN grown at 960 °C. The TMGa molar flow rate was set at 190 μmol/min resulting in the growth rate of 0.14 nm/s, the Ⅴ/Ⅲ ratio was ~8500, and the total flow rate was fixed at 142 slm. These p-GaN layers were doped to a relatively higher range from 2 × 10¹⁹ to 9 × 10¹⁹ cm⁻³. For p-GaN with doping levels lower than 4 × 10¹⁹ cm⁻³, a 20 nm thick p⁺-GaN ([Mg] = 5 × 10¹⁹ cm⁻³) contact layer was added on top of p-GaN. In the second series, 1 μm thick undoped GaN was grown at 1000 °C, followed by 500−700 nm thick Mg-doped p-GaN and a 22 nm thick p⁺-GaN contact layer grown at 850−960 °C, and finally a 2 nm p-In_0.2Ga_0.8N was grown at 740 °C. The carrier gas was pure H₂ for GaN and pure N₂ for InGaN. The TMGa molar flow rate was set at 84 μmol/min (growth rate: 0.28 nm/s), Ⅴ/Ⅲ ratio was ~8500 during GaN growth, and the total flow rate was fixed at 72 slm. These p-GaN were doped to a relatively lower range from 0.3 × 10¹⁹ to 1 × 10¹⁹ cm⁻³. The reasons for these two different series are both the utilization of p-InGaN/p⁺-GaN contact layer^[17] and the increase of growth rate for faster feedback.

[Mg] of p-GaN was controlled by changing the Cp₂Mg molar flow rate relative to the TMGa molar flow rate. [Mg] in all p-GaN layers in the first series was measured by SIMS. In the second series, some samples were not measured by SIMS, however in this case, [Mg] were determined by the orange calibration line as shown in Fig. 1, where [Mg] measured by SIMS is plotted against the ratio of Cp₂Mg to TMGa multiplied by the density of Ga sites in the solid. The Mg doping level is close to the 1 : 1 ratio shown by the broken line in Fig. 1, but the slopes of the dependence appear somehow different between the two series, which could be ascribed to the different growth conditions described above.

After thermal annealing, samples were cut into 5 × 5 mm² squares. For the first series samples, Pd/Pt/Au (30/20/200 nm) deposited at ~1 × 10⁻⁵ Torr in Lab-18 by sputtering was used as electrodes, followed by alloying at 550 °C in air to obtain good ohmic contacts. While for the second series, just indium metal was used as electrodes conveniently, as described in detail in our previous paper^[17]. Hall-effect measurements with van der Pauw geometry were performed by Accent HL5500 at RT. All the samples showed ohmic properties during the measurements. The magnetic field was 0.388 T and the currents were 10−100 μA.

It is crucial to use optimal thermal annealing conditions in order to activate Mg acceptors sufficiently. Different annealing methods were chosen for p-GaN samples with different doping ranges of [Mg] in this study. Rapid thermal annealing (RTA) at relatively high temperatures in N₂ ambient was used for the samples of the first series with higher [Mg], and furnace annealing at relatively low temperatures in air was used for the samples in the second series with lower [Mg]. Two typical samples taken from different series were annealed under the different conditions to achieve sufficient activation. [Mg] is 2.7 × 10¹⁹ cm⁻³ for sample A in the first series. This [Mg] is chosen because the doping level is close to the highest acceptor concentration of ~3 × 10¹⁹ cm⁻³. When [Mg] exceeds 3 × 10¹⁹ cm⁻³, acceptor concentration tends to decrease due to the self-compensation^{[15, 16]} or the segregation of Mg at structural defects^[18]. [Mg] is 1.0 × 10¹⁹ cm⁻³ for sample B taken from the second series and it is nearly the highest concentration in the second series. If the annealing condition is sufficient for sample B, it should be also enough for other samples with lower [Mg].

First sample A was annealed at temperatures ranging from 700 to 950 °C with an interval of 50 °C, with a N₂ flow rate of 3 slm. After the electrical properties saturated at 950 °C, the sample was annealed at 950 °C with a higher N₂ flow rate of 10 slm. Hall results of samples A (circles) are shown in Fig. 2. The open circles in Fig. 2 show the relationship for p, ρ, and μ when N₂ flow rate was 3 slm. When the annealing temperature was raised, p increased and corresponding ρ and μ decreased. Then, these values saturated to a certain level, which is judged by the overlap of data points in Fig. 2. When the N₂ flow rate was increased from 3 to 10 slm, the saturation value moved further as marked by solid circles in Fig. 2. Consequently, the temperature of 950 °C and the flow rate of 10 slm were determined as the optimal annealing conditions for the first series. This annealing was also repeated a few times for selected samples with [Mg] higher than 3 × 10¹⁹ cm⁻³. The resultant electrical properties exhibited no significant change after extra annealing, indicating sufficient activation of Mg under these conditions. The change of the saturation level with increased flow rate can be explained by the better removal of H in the gas phase which minimizes the chances of repassivation and accelerates the diffusion of H from solid phase. Although not shown here, the flow rate of 10 slm was enough in this experiment because the saturation level did not change when the flow rate was further increased.

In the second series, sample B (squares in Fig. 2) was annealed at temperatures ranging from 440 to 520 °C with an interval of 20 °C, in air with a high flow rate of 10 slm. Hall results of sample B show the similar tendency to sample A and finally saturate. The temperature range could be reduced considerably compared to those in N₂ annealing due to the existence of oxygen in the ambient^[12]. The electrical properties were saturated around 510 °C, which was determined as the optimal temperature for the second series. However, the annealing temperature or time had to be increased for samples with higher [Mg] ( >2 × 10¹⁹ cm⁻³), resulting in worse contact and bulk properties mainly due to oxidation near the surface^[19]. Hence the annealing in air is thought to be suitable for samples with [Mg] less than 2 × 10¹⁹ cm⁻³.

Here, it is also found that the highest p always corresponds to the lowest ρ and μ for each p-GaN sample. This indicates that the mobility is also influenced by the ionized acceptor-related scattering, which suggests that the mobility value could be one of the measures to judge the degree of activation.

Residual H concentrations, [H] were measured by SIMS and plotted against [Mg] in Fig. 3 for selected p-GaN samples annealed under their optimal annealing conditions. Broken line in Fig. 3 refers to [Mg] = [H], which is generally observed for p-GaN samples before annealing^{[18, 20]}. Mg−H bond will be broken and H will diffuse out during annealing. These H atoms may be captured by Mg again or interact with V_N, V_Ga or H during the diffusion^[21]. The slope of the data points in Fig. 3 is ~1.8 and it is close to the slope of 2 for the interstitial H₂ relative to neutral H as indicated in Fig. 6 of the article reported by Myers et al^[22]. So, it is possible that those residual H remains mainly forming interstitial H₂ and becomes electrically neutral, although further study is needed to conclude explicitly on this point. When the Mg doping level is less than 2 × 10¹⁹ cm⁻³, the difference between the residual [H] and [Mg] becomes larger than one order of magnitude, indicating the influence of residual [H] becomes insignificant.

To describe the electrical properties of p-GaN, at least two of p, ρ, and μ are necessary, since there is a relationship among these properties, μ=1/qρp, where q is the electronic charge. Here all of three properties are shown as a function of [Mg], in order to clarify and discuss each trend. The measured p, ρ, and μ for p-GaN samples annealed under the optimal conditions described above were plotted against [Mg] as blue solid circles in Figs. 4(a)−4(c), respectively. These data points were average values among a few samples for the first series and several samples for the second series, although the variation was small, typically within ±5%. Also, the trends of two series can be smoothly connected to each other at [Mg] = 1.4 × 10¹⁹ cm⁻³, which is the doping range boundary between the two series. The highest p at some selected [Mg] levels together with ρ and μ reported for MOCVD-grown p-GaN in the literature^{[15, 23−26]} are also plotted for comparison as black open symbols in Fig. 4. Kaufman et al. reported values consistent with ours in the [Mg] < 2 × 10¹⁹ cm⁻³ range, but their p was much lower than ours in the higher [Mg] range, which could be ascribed to their relatively weak annealing conditions used: 600 °C, 10 min in N₂. The [Mg] reported by Kozodoy et al.^[25] appears to be too overestimated, so their [Mg] ~2 × 10²⁰ cm⁻³ is shifted in the figure to ~3 × 10¹⁹ cm⁻³, the common value exhibiting the highest acceptor concentration^{[18, 20]}. Tsuchiya et al.^[26] only reported the value of p with [Mg] and their highest value is added as a cross in Fig. 4(a). In general, the highest p (lowest ρ) collected from literatures are in good agreement with our data.

Looking at these figures, on the other hand, the trends can be categorized into 3 regions, namely, low, medium, and high doping regions. In the low doping range ([Mg] < 1.5 × 10¹⁹ cm⁻³), with the increase of [Mg], p increases gradually and μ decreases weakly, so ρ also decreases gradually. In the medium range (1.5 × 10¹⁹ cm⁻³ ≤ [Mg] ≤ 4 × 10¹⁹ cm⁻³), p continues to increase gradually with the increase of [Mg], but μ drops sharply, so ρ starts to increase. In the high doping range ([Mg] > 4 × 10¹⁹ cm⁻³), p drops sharply due to the compensation or segregation effect, so the number of ionized acceptors also decreases, and μ recovers slightly due to the decrease of ionized acceptors, so ρ increases more steeply than the medium range.

The maximum p achievable is thus 1.3 × 10¹⁸ cm⁻³ when [Mg] is ~4 × 10¹⁹ cm⁻³, and the minimum ρ is 0.8 Ω·cm when [Mg] is ~1.5 × 10¹⁹ cm⁻³. Mobility is mainly determined by deformation-potential acoustic phonon scattering and optical phonon scattering at RT^{[18, 27]}, but also influenced by the ionized impurity scattering. This is why the slopes of p and μ become opposite in sign in these figures. It was also confirmed (although not shown here) that the growth temperature has little influence on the electrical properties of p-GaN samples between 850 and 960 °C.

Theoretically, the hole concentration, p can be calculated by the following equation, which is a solution of charge neutrality equation:

1p=14(ND+β)2+β(NA−ND)−12(ND+β),β=NVgexp(−EAkT),

where N_A and N_D are the doped acceptor and compensating donor concentrations, respectively. N_V is the effective density of states in valence band, g is the acceptor degeneracy factor, E_A represents the acceptor ionization energy, k and T are Boltzmann constant and absolute temperature, respectively.

It should be noted, however, that the following three points are important to calculate p correctly from [Mg] using Eq. (1).

First, the total amount of residual donor concentration N_D must be assessed carefully. SIMS for possible donor impurities were performed for this purpose. Fig. 5 shows the SIMS result of a typical p-GaN sample grown at 850 °C after annealing. The concentrations of O and Si were near the detection limits and the C concentration [C] was 3.9 × 10¹⁶ cm⁻³. When growth temperature was raised to 960 °C, [C] decreased to 1.0 × 10¹⁶ cm⁻³. C is an amphoteric impurity in GaN and can occupy either Ga site or N site. Ga−C bond originating from TMGa is one route for the incorporation of C, and C will sit on N site acting as an acceptor in this case. When growth temperature is raised, [C] will be reduced by chemical reaction Ga−CH₃ + N−H → GaN + CH₄, because more N−H will generate due to higher cracking efficiency of NH₃ at higher growth temperatures. Besides, methyl radicals may be captured by the defects at the surface and incorporate into solid phase. When growth temperature is raised, the cracking of NH₃ will be enhanced and more N atoms will be fed to the growing surface, so the number of V_N in solid phase near the surface will decrease. This could also explain the decrease of C concentration at higher growth temperature. The effect of C is negligible in this experiment because the C concentration is by two orders of magnitude lower than Mg, and C is more likely to occupy N site to form C_N as a deep acceptor in GaN.

As for N_A, [Mg] is assumed to be equal to N_A for the range [Mg] < 4 × 10¹⁹ cm⁻³ in this paper, and N_A starts to decrease rapidly with increasing [Mg] beyond this concentration due to the self-compensation and segregation effects. This assumption will be checked below in Fig. 6 by comparing with the case when N_A = [Mg] − [H] is assumed for the range [Mg] < 4 × 10¹⁹ cm⁻³.

Second, appropriate materials/band parameters, such as the density-of-state effective mass of valence band, mhdos, the crystal field and spin-orbit splitting energy, Δ_cr and Δ_so, must be used to derive the figures of N_V and g in Eq. (1). The parameter values of wurtzite GaN calculated by the empirical pseudopotential method reported by Yeo et al.^[28] are used in this paper, since the quantum levels of InGaN/GaN quantum wells calculated based on their values agreed well with our experimentally observed quantum levels^[29]. The resulting mhdos = 2.13 m₀, N_V = 7.78 × 10¹⁹ cm⁻³, and g = 4.62. The details of these derivations considering the three valence bands are written in the appendix, and the mhdos determined here is in good agreement with the value reported by Im et al.^[30].

Third, the ionized-acceptor-density dependent ionization energy of acceptor should be used for E_A, because E_A are known to decrease with increasing the acceptor density in semiconductors, known as degenerate phenomena. Actually, Mott et al. showed, as early as more than 60 years ago, that the ionization energy of impurity decreases with increasing impurity density and it effectively vanishes when the average distance of ionized impurity approaches to about three times of Bohr radius of the acceptor bound hole/donor bound electron^[31]. This distance of ionized impurity is defined by the radius r_A of the sphere volume, which one ionized impurity occupies on the average in a unit volume of solid; rA=(4πNA−/3)−13. Meanwhile the Bohr radius r_B can be calculated by rB=ϵrϵ0m0mh*×0.0529nm, where ε_r and ε₀ are the relative and vacuum dielectric constants, respectively, m_h^* and m₀ are the effective mass of hole and the mass of free electron, respectively. The NA− dependence of E_A then may be expressed, as usual, by the equation, EA=EA,0−α0(NA−)13, where EA,0 is the ionization energy of acceptor at the low concentration limit and α0 is a constant including a geometric factor which can be written by the following equation^{[25, 32]}.

2α0=Γ(23)(4π3)1/3×q24πϵrϵ0,

where q is the electron charge.

When EA,0 is calculated by a H-like model for an isolated acceptor in GaN taking ϵr=10.28ϵ0^[33] and mh*=1.65m0, of which derivation is also shown in the appendix, EA,0 becomes mh*m0(ϵ0ϵr)2×13.6eV=212meV. And α0=3.06×10−5 meV·cm. Using these values, the ionized acceptor density at which E_A becomes zero can be determined as (EA,0/α0)3=3.33×1020cm−3 for GaN, leading to the r_A value of 0.895 nm. Since the value of r_B is 0.330 nm, the r_A/r_B ratio becomes 2.71, which is in good agreement with the Mott’s condition. The ratio of r_A/r_B was also confirmed for Si : B, GaAs : Zn, GaInP : Zn using E_A reported experimentally in the literature^[34−36]_, assuming the effective mass of 0.43 m₀, 0.42 m₀, 0.63 m₀, respectively. All the values were confirmed to fall into the similar value of 2.78±0.07. However, the actual ionization energy of shallow acceptors in various semiconductors generally deviates from the calculated ionization energy of the H-like model within ±30% depending on the specific impurity^[37]. Therefore, the following equation for ionized-acceptor-density dependent ionization energy of Mg acceptor (EAMg) should be used to keep satisfying the Mott’s condition for Mg-doped GaN.

3EAMg=EA.0Mg−EA.0Mg212×3.06×10−5×(NA−)13meV,

where EA.0Mg is the ionization energy of Mg acceptor at the low concentration limit and NA− equals to p+ND from the charge neutrality. This equation together with Eq. (1) are the two basic equations to calculate p in p-GaN, and need to be solved self-consistently using an iteration procedure.

Now, the theoretical p curve is fitted to the experimentally obtained Hall data of p with only one fitting parameter, EA,0Mg, in Fig. 6, under the assumptions, N_A = [Mg], N_D = 0, and NA− = p, based on the above arguments. The red curve shows the fitting curve when 184 meV is taken for EA,0Mg, indicating that the Hall data can be reproduced satisfactorily with EAMg=184−2.66×10−5×(NA−)13meV. The dotted curve shows the case when the total N_D = 1 × 10¹⁷ cm⁻³, NA−=p+ND, and the same EAMg with the red curve is assumed. From the figure, it is also concluded that the total amount of compensating donors in our p-GaN samples should be negligibly small compared with the amount of acceptors. Finally, the effect of the residual hydrogen is inspected with the broken curve, which is the curve obtained assuming N_A = [Mg] − [H], with the same other parameters as the red curve. The residual [H] is taken from Fig. 3 (red line in Fig. 3). When compared with the simple case, N_A = [Mg], the fitting becomes worse, although the difference appears to be small. Especially it should be pointed out that p exceeding 1 × 10¹⁸ cm⁻³ for the [Mg] level of (3−4) × 10¹⁹ cm⁻³ becomes difficult when N_A = [Mg] − [H] is assumed, conflicting with the reported p of more than 1 × 10¹⁸ cm⁻³ by several groups^{[24, 26, 38, 39]} including us. Therefore, our assumption that [Mg] is equal to N_A for the [Mg] range of lower than 4 × 10¹⁹ cm⁻³ should be acceptable practically.

We optimized the annealing conditions for p-GaN samples with different Mg concentration in the range of 10¹⁸−10¹⁹ cm⁻³ grown by MOCVD. Sufficient annealing was confirmed by obtaining the saturation level of p, ρ, and μ for each p-GaN sample with different [Mg]. The lowest ρ among those samples was 0.8 Ω·cm when [Mg] = 1.4 × 10¹⁹ cm⁻³, and the highest p was 1.3 × 10¹⁸ cm⁻³ when [Mg] = 3.6 × 10¹⁹ cm⁻³. The relationship between p and [Mg] was carefully analyzed by considering residual donor concentration, appropriate materials/band parameters and the ionized-acceptor-density dependent ionization energy of acceptor. The theoretical curve between p and [Mg] was thus given with the Mg ionization energy of EAMg = 184 − 2.66 × 10⁻⁵ × [NA−]^1/3 meV at RT. These values of p, ρ, and μ and residual [H] for different Mg concentrations shown in this paper could supply good references for sufficient activation of p-GaN.