3.2.3 Hole Transport

Modeling the hole transport in GaN is hindered by the lack of high quality $p$-type material. Several dopants such as Mg, Zn, Cd, and Be [188,189,190,191] have been investigated. Of all those dopants Mg is known to have the lowest ionization energy [192]. The first $p$-type conduction in magnesium-doped GaN grown by MOCVD was realized by Amano et al. [188]. The as-grown high-resistivity GaN:Mg is converted into $p$-conductive material by hydrogen depassivation [193]. Due to the high activation energy of Mg of 150 to 250 meV [194,192] only a few percent are active at room temperature. The highest reported efficiency is around 10% [195]. In order to reach a useful hole concentration of 10$^{17}$-10$^{18}$ cm$^{-3}$ a Mg doping as high as 2$\times$10$^{19}$-10$^{20}$ cm$^{-3}$ is needed [196]. However, such high doping concentrations lead to a deterioration of the hole transport properties (Fig. 3.5). Kozodoy et al. [197] suggest that at very high doping levels the degree of compensation and self-compensation [198] increases, which suppresses the hole mobility. On the other hand the hole concentration is extremely dependent on the temperature as demonstrated in [199].

Figure 3.5: Low-field hole mobility as a function of carrier concentration in GaN.

One way of improving the performance is by using a $\delta$-doping as suggested by Nakamari et al. [200]. The dislocation density is significantly reduced, and a higher conductivity is achieved in both the lateral and vertical directions. The latter is an issue for Mg-doped heterostructures, where the super-lattice introduces also potential barriers in vertical direction. Such an approach was used by several groups [201,202,203,204]. The variation of the valence band energy caused by the modulation of chemical composition leads to a reduction of the acceptor activation energy. The polarization fields increase the band bending, and the hole concentration rises in addition [202]. The acceptors in the Al$_{x}$Ga$_{1-x}$N are energetically closer to the valence band edge and are therefore ionized easier [205].

Another problem, which bipolar GaN-based devices face, is the high resistivity of the p-type ohmic contacts. They are sometimes referred to as closer to leaky Schottky contacts in their characteristics [84]. Introduction of InGaN/AlGaN super-lattices greatly improves the contact sheet resistance [206], due to the large oscillations of the valence band.

Figure 3.6: Low-field hole mobility as a function of lattice temperature.

A parameter also crucial for the modeling of hole transport is the effective hole mass. Estimation of the hole effective masses and their anisotropy was the subject of numerous studies (a comprehensive review is found in [207,208,209,210]). Values ranging from 0.3m$_0$[211] to 2.2m$_0$[212] are reported. Kasic et al. [208] suggest that the effective hole mass depends on the hole concentration: 1.0m$_0$ for $p$=5$\times$10$^{16}$ cm$^{-3}$, and 1.4m$_0$ for $p$=8$\times$10$^{17}$ cm$^{-3}$ [213]. In this work a value of 1.4m$_0$ is assumed as reported in [208], lower than the one used in [199] (1.6m$_0$), and slightly higher than the one recommended by Vurgaftman et al. (1.0m$_0$) [209].

Fig. 3.7 shows the hole drift velocity as a function of the electric field as calculated by two groups. Rodrigues et al. [214] use a rather high hole mass (2.0m$_0$) and a doping concentration of 10$^{18}$ cm$^{-3}$. The calculation of Chen et al. [157] relies on a standard ensemble Monte Carlo approach and accounts for various scattering effects including impact ionization. Again the hole velocity is limited by the high density of states of the heavy band (1.8m$_0$).

Figure 3.7: Hole drift velocity versus electric field.