Fundamentals of Metal Contact to p-Type GaN—A New Multilayer Energy-Saving Design

Abstract

The electrical properties of contacts to p-type nitride semiconductor devices, based on gallium nitride, were simulated by ab initio and drift-diffusion calculations. The electrical properties of the contact are shown to be dominated by the electron-transfer process from the metal to GaN, which is related to the Fermi-level difference, as determined by both ab initio and model calculations. The results indicate a high potential barrier for holes, leading to the non-Ohmic character of the contact. The electrical nature of the Ni–Au contact formed by annealing in an oxygen atmosphere was elucidated. The influence of doping on the potential profile of p-type GaN was calculated using the drift-diffusion model. The energy-barrier height and width for hole transport were determined. Based on these results, a new type of contact is proposed. The contact is created by employing multiple-layer implantation of deep acceptors. The implementation of such a design promises to attain superior characteristics (resistance) compared with other contacts used in bipolar nitride semiconductor devices. The development of such contacts will remove one of the main obstacles in the development of highly efficient nitride optoelectronic devices, both LEDs and LDs: energy loss and excessive heat production close to the multiple-quantum-well system.

1. Introduction

Despite the successful development of nitride optoelectronic devices, such as light-emitting diodes (LEDs), for which the Nobel Prize was awarded in 2014 [1], and laser diodes (LDs) [2], both visible and UV, several formidable obstacles still remain to the considerable improvement of these devices, which remains badly needed. The most harmful problems are related to the p-type properties of nitrides, namely the high resistivity of both the plain bulk and the contact. The first problem is related to the absence of a shallow acceptor, as the only reliable defect is the Ga-substitutional Mg atom, which creates a deep acceptor with an ionization energy of approximately 170 meV [3]. Therefore, at room temperature, approximately 1% of Mg is ionized, and p-type conductivity suffers from low density and mobility. Nevertheless, an effective method to overcome the first obstacle was developed, first by the concept of p-type doping [4] and subsequently by the discovery of a method to obtain mobile hole charge [5]. This methodology promises to achieve temperature-independent hole density, which is applicable to UV devices [6,7]. As the above solution has been proven effective, progress in this respect is largely a matter of technological advancements.

The second problem has not yet been resolved. The solution is the subject of this study. The high contact resistance is related to the wide bandgap of nitrides, which necessarily locates their valence bands (VB) much below the Fermi energy of any metal candidate for contact. Typically, work functions of the metal are close to 5 eV; the highest value is reported for gold, which is ${\varphi }_{Au}=5.10 ÷5.47 \mathrm{e}\mathrm{V}$ [8]. The data span is related to the work-function dependence on the configuration of atoms at the surface of the material. The other metals have lower work-function values, typically in the range ${\varphi }_{Au}=4.0 ÷5.0 \mathrm{e}\mathrm{V}$ [8]. Therefore, the most frequently used metal contacts are based on a combination of Au and other metals such as Au/Ni [9,10,11], Pt/Ni/Au [12], Ti/Pt/Au [13], and Pd/Au [14].

The work-function difference, or equivalently, the Fermi-level difference between the metal and the semiconductor, is the energy cost at which the holes from the metal can be transferred to the semiconductor at the Fermi level. In the case of p-type materials, this is close to the energy of the valence-band maximum (VBM), that is, the ionization potential I. Aluminum and gallium nitrides are characterized by strong bonds and a low energy of valence-band states. Therefore, the ionization potentials I calculated for metal-terminated clean AlN and GaN are $I_{AlN}=9.13 \mathrm{e}\mathrm{V}$ and $I_{GaN}=7.54 \mathrm{e}\mathrm{V}$ [15]. The ionization potential can be affected by band bending; nevertheless, the magnitude of the bending rarely exceeds 1 eV. Therefore, most of this energy difference remains, creating an energy barrier $∆E$ for holes in excess of $∆V_o \cong 2 \mathrm{e}\mathrm{V}$. Hence, the direct deposition of metals on p-type GaN cannot be used for the construction of any working nitride optoelectronic device. To alleviate this problem, sophisticated contact-formation procedures have been developed, such as deposition of Ni and Au layers and annealing in an oxygen atmosphere. This procedure is used in the majority of present-day nitride diodes [9,10,11,16,17,18,19,20].

The current-voltage (I-V) electric measurement results obtained for LEDs and LDs at high currents, controlled mostly by the contact resistance, exhibit nonlinear behavior, indicating a non-Ohmic type of resistance. Thus, the contact is characterized by a Schottky energy barrier [21]. Detailed investigations have revealed the nature of Ni/Au contact formation on the metal side [9,10,11,16,17,18,19,20]. During the annealing process, nickel atoms diffused across the gold layer and reacted with oxygen, creating a NiO layer on the external side. Thus, the chemical potential difference that drives the Ni current to the surface is maintained. Consequently, the Ni layer underneath becomes very thin and nonuniform. It is speculated that some Ga atoms diffused from the GaN layer to the Ni layer. As a result, the contact has an optimally acceptable resistance on the order of $\rho ~ {10}^{-4 }\Omega {\mathrm{c}\mathrm{m}}^2$ at a current density $j=4000 \mathrm{A} {\mathrm{c}\mathrm{m}}^{-2}$. The claimed minimal voltage at the contact is $\Delta V=0.4 \mathrm{V}$ [10,22]. Unfortunately, this is not standard, and the results show considerable scattering as a function of weakly controlled parameters, such as surface morphology or dislocation density, which most likely induce strain and instability in the structure. In general, the voltage is much higher, with the probable value being $\Delta V \cong 1 ÷3 \mathrm{V}$.

Apart from annealed metal contacts, the other ways to deal with the aspect of hole injection to the p-type GaN, which have been published recently, are the InGaN cap layers [23] or Schottky barrier contacts [24] based on graphene. However, these types of structures are not within the scope of this article. This study focuses on the analysis of the electrical properties of metal-p-type GaN contacts. It has been proven that any contact will lead to an energy barrier and a non-Ohmic character of the contact. Then, ab initio calculations were used to obtain the basic properties of bulk GaN and Au, followed by the combined GaN–Au slab, which is a small model of the contact. Finally, drift-diffusion analysis was performed to obtain the potential profile of Mg-doped GaN with Au metal. This is finalized by a multilayer contact structure based on implantation by a set of deep-level acceptors, which serve as a path for the low-energy barrier transfer of charge across the contact. In principle, the proposed structure can considerably lower the electrical resistivity of the contact.

2. The Model

The electrical stability of the metal and nitride heterointerface leads to the attainment of the common Fermi energy level [25]. This occurs via an increase in the energy of the quantum states in the nitride bulk owing to the creation of the electric dipole layer. As shown in Figure 1, the dipole negative part is due to electron transfer to the nitride, and the opposite positive part appears in the metal due to the absence of electrons. The screening length in the metal is approximately $0.01 Å$, therefore, the effect on the potential energy is negligible [26]. At the surface of the contact, during the charge transfer, the difference in the energy of the quantum states of the semiconductor and the metal is preserved, which allows the estimated magnitude of the band bending. Band bending is equivalent to the barrier for the introduction of holes into the semiconductor, $E_{bar}$:

$$E_{bar}\cong E_F\left(\mathrm{m}\right)-E_F\left(\mathrm{G}\mathrm{a}\mathrm{N}\right)$$

(1)

where $E_F\left(\mathrm{m}\right)$ and $E_F\left(\mathrm{G}\mathrm{a}\mathrm{N}\right)$ are the Fermi energies of the metal and p-type GaN bulk, respectively.

The large magnitude of this barrier explains the high resistance of the as-deposited metal contacts, including gold.

These profiles provide an explanation of the relative success of the oxygen-atmosphere-annealed Ni–Au contact. In parallel to the metal-side formation, the outdiffusion of Ga atoms occurs, creating a layer of high-density Ga vacancies ($V_{Ga}$). Gallium vacancies are multiple-ionized deep acceptors with three ionization energies ($E_A\left(V_{Ga}^{1-}\right)\cong 2.35 \mathrm{e}\mathrm{V}$, $E_A\left(V_{Ga}^{2-}\right)=1.02 ÷ 0.62 \mathrm{e}\mathrm{V}$, and $E_A\left(V_{Ga}^{3-}\right)\cong 0.48 \mathrm{e}\mathrm{V}$) [27,28]. Moreover, nickel atoms may diffuse over the lattice of gallium vacancies, creating substitutional ${Ni}_{Ga}$ atom defects, also good candidates for deep acceptors ($E_A\left(Ni\right) \cong 1.6÷1.5 \mathrm{e}\mathrm{V}$) [29,30]. The mixture of these states serves as a tunneling channel for holes. It is also likely that the arrangement of the Ni layer under the Au layer may result in a structure favorable for the incorporation of Ga atoms, thus facilitating the creation of such a defect layer.

3. Calculation Procedure

A combination of various approaches was used to determine the properties of the composite Au–GaN system. First, ab initio simulations were used to determine the basic electronic properties of the separate Au and GaN and the connected Au–GaN system. For these simulations, the DFT-based SIESTA package was used. The SIESTA shareware package was developed by the Spanish Initiative for Electronic Simulations with Thousands of Atoms (SIESTA) [31]. The fundamental routine iteratively solves a set of nonlinear Kohn–Sham equations to obtain a set of eigenfunctions and real eigenvalues [31]. The eigenfunctions are a linear combination of the radial numeric atomic orbitals of finite extent, multiplied by spherical harmonics, and effectively limited to s, p, and d polynomials of angular sine and cosine functions [31,32]. The s and p orbitals centered on both Ga and N atoms were expressed by triple zeta functions. The Gallium d-shell electrons were incorporated into the valence set. Ga d orbitals are single-zeta functions. The Au bases include 6s, 6p—triple zeta function, and 5d, 5f—double zeta [33,34]. The eigenfunctional set is limited to a single period of these orbitals using Troullier–Martins atomic pseudopotentials [35,36]. K-space integration is approximated by summation over the grid of Monkhorst–Pack points, the number of which depends on the system considered, $\left(7\times 7\times 7\right)$ and $\left(5\times 5\times 1\right)$, for Au and Ga bulk and for the Au–GaN slab, respectively [37]. The GGA-PBE (PBEJsJrLO) functional was parameterized using the β, µ, and κ values set by the jellium surface (Js), jellium response (Jr), and Lieb–Oxford bound (LO) criteria [38,39]. The iterative SCF loop was terminated when the difference for any element of the density matrix in two consecutive iterations was less than 10⁻⁴.

The ab initio calculation for the bulk hexagonal wurtzite GaN gave its lattice parameters as follows: $a_{GaN}^{DFT}=3.21 Å$ and $c_{GaN}^{DFT}=5.23 Å$, close to the experimental values: $a_{GaN}^{exp}=3.189 Å$ and $c_{GaN}^{exp}=5.186 Å$ [40]. The DFT-obtained lattice parameter of the bulk Au fcc cubic lattice was $a_{Au}^{DFT}=4.14 Å$, which is quite close to the experimental value, $a_{Au}^{exp}=4.0782 Å$ [41]. The quantum states’ energies were corrected using the Ferreira et al. approach, denoted as GGA-1/2, giving reasonably good band parameters [42,43], e.g., GaN bandgap $E_g^{DFT}\left(GaN\right)=3.47 \mathrm{e}\mathrm{V}$, identical to the value from low-temperature measurement: $E_g^{exp}\left(GaN\right)=3.47 \mathrm{e}\mathrm{V}$ [44,45]. The Born–Oppenheimer approximation was applied to all atomic position relaxations. The atom position relaxation was terminated when the values of the forces acting on any atom decreased below 0.005 eV/Å.

4. The Calculation Results

4.1. Doping in the GaN Bulk

The creation of such a layer is weakly controlled, which explains the scattering and unreliability of the contact-formation procedure. Naturally, some modifications have been introduced, such as the In-rich GaInN contact layer. This may be beneficial because of the much lower bandgap, so the barrier energy may be smaller. Nevertheless, the basic electrical structures remain the same. In all these implementations, p-type doping is imposed with a high concentration of Mg acceptors, which have a level located at approximately $E_A=180 \mathrm{m}\mathrm{e}\mathrm{V}$ above the VBM [3]. Mg doping can be controlled during the growth process. Typically, this affects the position of the Fermi level, as only a part of the acceptors are ionized. The changes in the Fermi level and concentration of holes in the pure GaN contact are presented in Figure 2.

The charge-neutrality Fermi level corresponding to the acceptor level $E_A=180 \mathrm{m}\mathrm{e}\mathrm{V}$ also depends on the compensation. The position of the Fermi level at the acceptor energy is attained for $N_A=1.15 \times {10}^{17} {\mathrm{c}\mathrm{m}}^{-3}$ in the absence of compensation. This corresponds to the density of charged acceptors $N_A^{-}=3.81 \times {10}^{16} {\mathrm{c}\mathrm{m}}^{-3}$ (in this case, this is also the density of holes). For compensation at 10% and 30%, this condition is attained for $N_A=1.62 \times {10}^{17} {\mathrm{c}\mathrm{m}}^{-3}$ and $N_A=1.15 \times {10}^{18} {\mathrm{c}\mathrm{m}}^{-3}$, respectively. This is scaled with respect to the $N_A- N_A$ difference. At lower and higher Mg concentrations, the Fermi level was above and below the acceptor energy, respectively. This has a drastic influence on the concentration of holes, and accordingly, for charged acceptors $N_A^{-}$, thus affecting the overall functionality of the p-type contact, as shown below.

4.2. Ab Initio Data

Ab initio simulations cannot be used for direct calculation of semiconductor–metal heterojunctions because the number of atoms needed for simulations would be prohibitively large, reaching tens of thousands of atoms. Hence, simulations of some features were implemented. Following this route, the electronic properties of the bulk GaN and Au systems were separately obtained from the DFT procedure. The simulation cells and results are shown in Figure 3.

The calculated electronic properties of both bulk systems are shown in Figure 3. The panel arrangement was adjusted to facilitate direct comparison of these properties. In fact, these properties have been calculated and presented in many publications, for example, GaN [46,47,48] or Au [49,50]. The obtained bandgap for GaN is direct, equal to $E_g^{DFT}\left(GaN\right)=3.47 \mathrm{e}\mathrm{V}$. In addition, the density of states (DOS) of both systems was plotted in the same energy range. The gold DOS has a strong maximum at $E_{DOS}^{DFT}\left(Au\right)= -6.26 \mathrm{e}\mathrm{V}$, associated with several band maxima at this energy. A similar occurrence was identified for GaN at the valence band maximum (VBM) located at $E_{DOS}^{DFT}\left(GaN\right)= -8.15 \mathrm{e}\mathrm{V}$. An important parameter is the Fermi level position, which is affected in the case of GaN by the incorporation of the acceptor density in the SIESTA background simulation procedure to $N_A= {10}^{19} {\mathrm{c}\mathrm{m}}^{-3}$ [31]. From the obtained simulation data, it follows that the Fermi level for gold is $E_F\left(Au\right)= -4.520 \mathrm{e}\mathrm{V}$ and for p-type GaN, it is $E_F\left(p-GaN\right)= -7.997 \mathrm{e}\mathrm{V}.$ Therefore, the Fermi energy difference is $∆E_F=3.477 \mathrm{e}\mathrm{V}.$ The creation of a heterostructure leads to electron flow from Au to GaN, a downward shift in the energy level in Au, and an upward shift in the energy level in GaN. This can be partially observed in the ab initio calculations of the GaN–Au slab containing a finite number of atomic layers of both crystals. A simulation slab representing the Au–GaN heterointerface is presented in Figure 4. The slab consisted of two atomic layers of Au and eight double Ga–N atomic layers. At the bottom, it was terminated by hydrogen pseudoatoms with a fractional $\left(Z= 3/4\right)$ charge.

As shown, the contact Au atom layer was distorted to adjust to the GaN lattice. The distortion of the topmost Ga–N layer was very small. Nevertheless, these diagrams indicate that a connection is created, and Au is wetting the Ga-terminated GaN surface. This confirmed the good properties of the GaN–Au contact observed in the experiment.

The electronic properties of the Au–GaN heterointerface obtained by ab initio calculations are shown in Figure 5. The properties of the system are determined by electron transfer from the metal to the semiconductor. The Fermi level of the composite-calculated system was determined by the balance of electrons in Au and p-type GaN. The doping level was selected to be relatively high, approximately $N_A= {3 \times 10}^{20} {\mathrm{c}\mathrm{m}}^{-3}$, which generates a large number of empty states in the GaN valence band. Therefore, the electrons can be shifted relatively easily from the metal, creating a potential difference. This is evident in the average potential profile shown in Figure 5.

In this case, electrons can be shifted to this region, as shown in the diagram. The electron energy profiles show a minimum in the gold region, which is related to the electron shift to GaN and the net positive charge of the Au ions. This leads to the emergence of an electric dipole layer in which the GaN states are shifted upward and the Au states are shifted downward. This is evident in the PDOS and COHP diagrams. The PDOS panel presents the Au states in both layers. They were essentially identical, confirming the efficient screening of the field in the metal. The other two lines in the PDOS panel represent the states of the two Ga–N atom pairs. The first pair, Ga–N1, represents the DOS of the pair of Ga–N atoms in the second layer from the top. These atoms bond only to other Ga and N atoms; therefore, their states are not affected by bonding to Au. The second pair, Ga–N2, was located in the second layer from the bottom. These states are bound to other Ga and N atoms. Thus, their states are not affected by bonding to hydrogen pseudoatoms. The observed energy difference of approximately 1.2 eV was related to the potential profile within the GaN slab. This shift was confirmed by the COHP plots. The COHP plots showed peaks related to the positive (i.e., bonding) overlap of the Au and Ga interface atoms located at the Fermi energy. This confirms the good wetting of Au and GaN, which is related to the formation of bonds between these layers.

The magnitude of the shift, that is, the electric potential profile difference $∆V$, may be estimated using the energy DOS maxima in Au and GaN separately and in the heterostructure. In separate systems, $E_{DOS}^{DFT}\left(Au\right)= -6.26 \mathrm{e}\mathrm{V}$ and $E_{DOS}^{DFT}\left(GaN\right)= -8.15 \mathrm{e}\mathrm{V}$. The heterostructure PDOS maximum for GaN is diffuse, owing to the potential slope in the GaN region. Nevertheless, the steep increase in the DOS in GaN starts at an energy $∆E \cong 0.4 \mathrm{e}\mathrm{V}$ higher than that for Au. From these data, we obtained the potential difference to

$$q∆V=E_{DOS}^{DFT}\left(Au\right)-E_{DOS}^{DFT}\left(GaN\right)-∆E\cong -2.29 \mathrm{e}\mathrm{V}\approx -2.3 \mathrm{e}\mathrm{V}$$

(2)

Thus, the potential difference was approximately $∆V=2.3 \mathrm{V}$. This difference is partially attributed to the Fermi level in the system, which is located about 0.6 eV above the VBM. In Mg-doped p-type GaN, the Fermi level is located about 0.18 eV above the VBM. This adds an additional contribution to the potential difference, which can be estimated to be $∆V\approx 2.6 \mathrm{V}$. The potential difference ${∆V}_0= 2.0 \mathrm{V}$ was used as a benchmark for further investigations.

It should be noted that this potential difference may be partially affected by doping in p-type GaN. Similar simulations for other doping levels show the Fermi level located at different energies: (i) in the upper part of the bandgap for $N_A= {10}^{19} {\mathrm{c}\mathrm{m}}^{-3}$, (ii) at the VBM for $N_A= {10}^{21} {\mathrm{c}\mathrm{m}}^{-3}$. Naturally, the Fermi level position is affected by the volume of the bulk Au and p-GaN. A further increase in the Au volume leads to an increase in Fermi energy, whereas a higher p-type doping leads to a downward shift. The obtained estimate and detailed ab initio picture confirm the basic assumptions of the drift-diffusion heterojunction model that will be used below for the design of a new type of contact.

4.3. Heterojunction Potential Profiles

Three possible metal–semiconductor heterojunction types have been identified [12]. They were proposed by Schottky and Mott [49,50,51,52,53,54], Bardeen and Heine [55,56], and Tersoff [57]. The models differ based on the assumptions of the influence of interface states, which may pin the Fermi level. Because such states were not confirmed, we adopted the simplest assumption, reducing it to the Schottky–Mott model, given by Equation (1). In this study, we propose using this model to create a controlled defect structure at the contact via multiple implantations of nitride surface sublayers. These sublayers should be implanted across the entire band-end region. First, high-density Mg acceptors were introduced. Based on the high-concentration Mg doping during growth, the Mg acceptor density $N_A=5 \times {10}^{19} {\mathrm{c}\mathrm{m}}^{-3}$ is used. A higher density may lead to the creation of Mg–Mg pairs, which is detrimental to the p-type doping.

The potential profiles are obtained from the uniaxial Poisson equation for the electric potential V(z):

$${\displaystyle \frac{{\mathrm{d}}^2\mathrm{V}}{\mathrm{d}{\mathrm{z}}^2}}={\displaystyle \frac{\mathrm{e}\left[{\mathrm{N}}_{\mathrm{A}}^{-}-{\mathrm{N}}_{\mathrm{D}}^{+}+\mathrm{n}-\mathrm{p}\right]}{{\epsilon }_{\mathrm{G}\mathrm{a}\mathrm{N}}{\epsilon }_{\mathrm{o}}}}$$

(3)

where e is the elementary charge, ${\epsilon }_{GaN}$ is the GaN dielectric constant, ${\epsilon }_o$ is the vacuum permittivity, and $N_A^{-},{ N}_D^{+}, n, p$ are the densities of the charged acceptors, donors, electrons, and holes, respectively. The neutral defects did not contribute to this equation. The simplified Schottky plot considers only fully ionized acceptors, i.e., $N_A^{-}=const$ and $N_D^{+}=n=p=0$. Then, the solution is a simple parabolic dependence, $\left(0 \le z \le z_o\right)$

$$\mathrm{V}\left(\mathrm{z}\right)={\mathrm{V}}_{\mathrm{o}}+{\displaystyle \frac{\mathrm{e}{\mathrm{N}}_{\mathrm{A}}^{-}{\left[\mathrm{z}-{\mathrm{z}}_{\mathrm{o}}\right]}^2}{2{\epsilon }_{\mathrm{G}\mathrm{a}\mathrm{N}}{\epsilon }_{\mathrm{o}}}}$$

(4)

where ${\mathrm{L}}_{\mathrm{S}\mathrm{c}\mathrm{h}}={\mathrm{z}}_{\mathrm{o}}$ is the Schottky width of the charged layer, and ${\mathrm{V}}_{\mathrm{o}}$ is the far-distance potential value. From the potential deviation (jump) amplitude $∆{\mathrm{V}}_{\mathrm{o}}$, the Schottky width can be obtained as ${\mathrm{L}}_{\mathrm{S}\mathrm{c}\mathrm{h}}=\sqrt{{\displaystyle \frac{2{\epsilon }_{\mathrm{G}\mathrm{a}\mathrm{N}}{\epsilon }_{\mathrm{o}}∆{\mathrm{V}}_{\mathrm{o}}}{\mathrm{e}{\mathrm{N}}_{\mathrm{A}}}}}$ [26]. The specified acceptor density ${\mathrm{N}}_{\mathrm{A}}^{-}={10}^{19} {\mathrm{c}\mathrm{m}}^{-3}$ and the potential difference $∆{\mathrm{V}}_{\mathrm{o}}\cong 2 \mathrm{e}\mathrm{V}$ give the width ${\mathrm{L}}_{\mathrm{S}\mathrm{c}\mathrm{h}}=4.70\times {10}^{-7} \mathrm{m}=4.70\times {10}^3 \mathrm{\AA }$. This width cannot be overcome using direct tunneling.

A more precise description was obtained by considering the presence of defects and mobile carriers, that is, solving Equation (3) reformulated as:

$${\displaystyle \frac{{\mathrm{d}}^2\nu \left(\mathrm{u}\right)}{\mathrm{d}{\mathrm{u}}^2}}={\displaystyle \frac{\left[{\mathrm{N}}_{\mathrm{A}}^{-}\left(\mathrm{u}\right)-{\mathrm{N}}_{\mathrm{A}}^{+}\left(\mathrm{u}\right)+\mathrm{n}\left(\mathrm{u}\right)-\mathrm{p}\left(\mathrm{u}\right)\right]}{\left({\mathrm{p}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{b}}\right)}}$$

(5)

using dimensionless quantities: thermal energy scaled potential $\nu \equiv {\displaystyle \frac{\mathrm{e}\left(\mathrm{V}-{\mathrm{V}}_{\mathrm{o}}\right)}{\mathrm{k}\mathrm{T}}}$ (k—Boltzmann constant) and length $\mathrm{u}\equiv {\displaystyle \frac{\mathrm{z}}{{\mathrm{L}}_{\mathrm{D}}}}$, where the Debye–Hückel screening length is ${\mathrm{L}}_{\mathrm{D}}\equiv {\left[{\displaystyle \frac{\mathrm{k}\mathrm{T}{\epsilon }_{\mathrm{o}}{\epsilon }_{\mathrm{G}\mathrm{a}\mathrm{N}}}{{\mathrm{e}}^2\left({\mathrm{p}}_{\mathrm{b}}+{\mathrm{n}}_{\mathrm{b}}\right)}}\right]}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$ [26]. For the benchmark temperature, $\mathrm{T}=300 \mathrm{K}$, the screening length extends from ${\mathrm{L}}_{\mathrm{D}}=3.82\times {10}^{-6} \mathrm{m}$ to ${\mathrm{L}}_{\mathrm{D}}=1.21\times {10}^{-7} \mathrm{m}$, respectively. This is wider than the Schottky length $\left({\mathrm{L}}_{\mathrm{D}}>{\mathrm{L}}_{\mathrm{S}\mathrm{c}\mathrm{h}}\right)$ which is understandable because the latter assumes complete ionization of acceptors. Because the charge is screened by electrons and holes in the bulk of the density ${\mathrm{n}}_{\mathrm{b}}$ and ${\mathrm{p}}_{\mathrm{b}}$, respectively, the far-distance limit is $\underset{\mathrm{u}\to \mathrm{\infty }}{\lim }\nu =0$. The far distance values were used as the reference energies; therefore, the energy of the conduction/valence band was shifted by the potential as ${\mathrm{E}}_{\mathrm{C},\mathrm{V}}\left(\mathrm{z}\right)={\mathrm{E}}_{\mathrm{C},\mathrm{V}}\left(\mathrm{\infty }\right)-\mathrm{e}\mathrm{V}\left(\mathrm{z}\right)$. Inside the layer, the carrier density is position-dependent, which can be expressed using the scaled potential as

$$\mathrm{n}\left(\mathrm{u}\right)={\displaystyle \frac{2{\mathrm{N}}_{\mathrm{C}}}{\sqrt{\mathsf{\pi }}}}{\mathrm{F}}_{1/2}\left({\mathsf{\eta }}_{\mathrm{F}}-{\mathsf{\eta }}_{\mathrm{C}}\left(\mathrm{u}\right)\right)={\displaystyle \frac{2{\mathrm{N}}_{\mathrm{C}}}{\sqrt{\mathsf{\pi }}}}{\mathrm{F}}_{1/2}\left({\mathsf{\eta }}_{\mathrm{F}}-{\mathsf{\eta }}_{\mathrm{C}}+\nu \right)$$

(6a)

$$\mathrm{p}\left(\mathrm{u}\right)={\displaystyle \frac{2{\mathrm{N}}_{\mathrm{V}}}{\sqrt{\mathsf{\pi }}}}{\mathrm{F}}_{1/2}\left({\mathsf{\eta }}_{\mathrm{V}}\left(\mathrm{u}\right)-{\mathsf{\eta }}_{\mathrm{F}}\right)={\displaystyle \frac{2{\mathrm{N}}_{\mathrm{V}}}{\sqrt{\mathsf{\pi }}}}{\mathrm{F}}_{1/2}\left({\mathsf{\eta }}_{\mathrm{V}}-\nu -{\mathsf{\eta }}_{\mathrm{F}}\right)$$

(6b)

where ${\mathrm{F}}_{\mathrm{j}}\left(\mathrm{x}\right)\equiv {\int }_0^{\mathrm{\infty }}{\displaystyle \frac{{\mathrm{y}}^{\mathrm{j}}\mathrm{d}\mathrm{y}}{1+\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{y}-\mathrm{x}\right)}}$ in the Fermi integral and ${\mathrm{N}}_{\mathrm{V},\mathrm{C}}=2{\mathrm{M}}_{\mathrm{C},\mathrm{V}}{\left({\displaystyle \frac{2\mathsf{\pi }{\mathrm{m}}_{\mathrm{e},\mathrm{h}}^{\ast }\mathrm{k}\mathrm{T}}{{\mathrm{h}}^2}}\right)}^{3/2}$ are the effective densities of states in the conduction (C) and valence (V) bands, respectively. The factor is introduced to consider the value of the Fermi integral at zero ${\mathrm{F}}_{1/2}\left(0\right)={\int }_0^{\mathrm{\infty }}{\displaystyle \frac{{\mathrm{y}}^{1/2}\mathrm{d}\mathrm{y}}{1+\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{y}\right)}}$ = ${\displaystyle \frac{\sqrt{\mathsf{\pi }}}{2}}$, thus $\mathrm{n}\left(0\right)={\mathrm{N}}_{\mathrm{C}}$ and $\mathrm{p}\left(0\right)={\mathrm{N}}_{\mathrm{V}}$. The ${\mathrm{M}}_{\mathrm{C},\mathrm{V}}$ coefficients denote the number of branches in the conduction and valence bands, respectively, that effectively participate in the screening, that is, ${\mathrm{M}}_{\mathrm{C}}=1$ and ${\mathrm{M}}_{\mathrm{V}}=3$. The other symbols are the standard: ${\mathrm{m}}_{\mathrm{e},\mathrm{h}}^{\ast }$ is the electron/hole effective mass, and h is the Planck constant. The dimensionless energies (at a far distance) are denoted as ${\mathsf{\eta }}_{\mathrm{F}}={\displaystyle \frac{{\mathrm{E}}_{\mathrm{F}}}{\mathrm{k}\mathrm{T}}}$—Fermi energy, ${\mathsf{\eta }}_{\mathrm{C}}={\displaystyle \frac{{\mathrm{E}}_{\mathrm{C}}}{\mathrm{k}\mathrm{T}}}$—conduction band minimum (CBM), ${\mathsf{\eta }}_{\mathrm{V}}={\displaystyle \frac{{\mathrm{E}}_{\mathrm{V}}}{\mathrm{k}\mathrm{T}}}$ is the valence band maximum (VBM).

Typically, the energy of the defect states (donors and acceptors) is shifted by the electric potential in the same way as the change in the band energy, that is, $E_{D,A}\left(z\right)= E_{D,A}\left(\infty \right)-eV\left(z\right)$. The charged defects contribute to screening; thus, their densities are

$${\mathrm{N}}_{\mathrm{D}}^{+}\left(\mathrm{u}\right)={\displaystyle \frac{{\mathrm{N}}_{\mathrm{D}}}{1+\mathrm{e}\mathrm{x}\mathrm{p}\left[{\mathsf{\eta }}_{\mathrm{F}}-{\mathsf{\eta }}_{\mathrm{D}}\left(\mathrm{u}\right)\right]}}={\displaystyle \frac{{\mathrm{N}}_{\mathrm{D}}}{1+\mathrm{e}\mathrm{x}\mathrm{p}\left[{\mathsf{\eta }}_{\mathrm{F}}-{\mathsf{\eta }}_{\mathrm{D}}+\nu \left(\mathrm{u}\right)\right]}}$$

(7a)

$${\mathrm{N}}_{\mathrm{A}}^{-}\left(\mathrm{u}\right)={\displaystyle \frac{{\mathrm{N}}_{\mathrm{A}}}{1+\mathrm{e}\mathrm{x}\mathrm{p}\left[{\mathsf{\eta }}_{\mathrm{A}}\left(\mathrm{u}\right)-{\mathsf{\eta }}_{\mathrm{F}}\right]}}={\displaystyle \frac{{\mathrm{N}}_{\mathrm{A}}}{1+\mathrm{e}\mathrm{x}\mathrm{p}\left[{\mathsf{\eta }}_{\mathrm{A}}-\nu \left(\mathrm{u}\right)-{\mathsf{\eta }}_{\mathrm{F}}\right]}}$$

(7b)

where $N_{A,D}$-are the total densities of donors/acceptors, respectively.

The fundamental equation for the potential in the charged layer in dimensionless form is [58,59]:

$${\displaystyle \frac{{\mathrm{d}}^2\nu \left(\mathrm{u}\right)}{\mathrm{d}{\mathrm{u}}^2}}={\displaystyle \frac{1}{\left({\mathrm{n}}_{\mathrm{b}}+{\mathrm{p}}_{\mathrm{b}}\right)}}\left\{\begin{array}{c}{\displaystyle \frac{{\mathrm{N}}_{\mathrm{A}}}{\left[1+2\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathsf{\eta }}_{\mathrm{A}}-{\mathsf{\eta }}_{\mathrm{F}}-\nu \right)\right]}}-{\displaystyle \frac{{\mathrm{N}}_{\mathrm{D}}}{\left[1+2\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathsf{\eta }}_{\mathrm{F}}-{\mathsf{\eta }}_{\mathrm{D}}+\nu \right)\right]}}\\ +n\left({\mathsf{\eta }}_{\mathrm{C}}-{\mathsf{\eta }}_{\mathrm{F}}-\nu \right)-p\left({\mathsf{\eta }}_{\mathrm{F}}-{\mathsf{\eta }}_{\mathrm{V}}+\nu \right)\end{array}\right\}$$

(8)

which can be reformulated into the form suitable for wide bandgap semiconductors:

$${\displaystyle \frac{{\mathrm{d}}^2\nu \left(\mathrm{u}\right)}{\mathrm{d}{\mathrm{u}}^2}}={\displaystyle \frac{1}{\left({\mathrm{n}}_{\mathrm{b}}+{\mathrm{p}}_{\mathrm{b}}\right)}}\left\{\begin{array}{c}{\displaystyle \frac{{\mathrm{N}}_{\mathrm{A}}}{\left[1+2\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathsf{\eta }}_{\mathrm{A}}-{\mathsf{\eta }}_{\mathrm{F}}-\nu \right)\right]}}-{\displaystyle \frac{{\mathrm{N}}_{\mathrm{D}}}{\left[1+2\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathsf{\eta }}_{\mathrm{F}}-{\mathsf{\eta }}_{\mathrm{A}}+\nu \right)\right]}}\\ +{\displaystyle \frac{2}{\sqrt{\mathsf{\pi }}}}\left[{\mathrm{N}}_{\mathrm{C}}{\mathrm{F}}_{1/2}\left({\mathsf{\eta }}_{\mathrm{F}}-{\mathsf{\eta }}_{\mathrm{C}}+\nu \right)-{\mathrm{N}}_{\mathrm{V}}{\mathrm{F}}_{1/2}\left({\mathsf{\eta }}_{\mathrm{V}}-{\mathsf{\eta }}_{\mathrm{F}}-\nu \right)\right]\end{array}\right\}$$

(9)

Equation (9) can be integrated directly by multiplication by ${\displaystyle \frac{\mathrm{d}\nu \left(\mathrm{u}\right)}{\mathrm{d}\mathrm{u}}}$ and straightforward integration to determine the magnitude of the electric field $\left|{\overrightarrow{\mathrm{E}}}_{\mathrm{z}}\right|$ [58,59]:

$${\displaystyle \frac{\mathrm{k}\mathrm{T}\left|{\overrightarrow{\mathrm{E}}}_{\mathrm{z}}\right|}{\mathrm{e}}}={\displaystyle \frac{\mathrm{d}\nu \left(\mathrm{u}\right)}{\mathrm{d}\mathrm{u}}}=2{\left\{\begin{array}{c}{\displaystyle \frac{{\mathrm{N}}_{\mathrm{A}}}{\mathrm{n}+\mathrm{p}}}\ln \left[{\displaystyle \frac{2+\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathsf{\eta }}_{\mathrm{F}}-{\mathsf{\eta }}_{\mathrm{A}}\right)}{2+\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathsf{\eta }}_{\mathrm{F}}-{\mathsf{\eta }}_{\mathrm{A}}-\nu \right)}}\right]+{\displaystyle \frac{{\mathrm{N}}_{\mathrm{D}}}{\mathrm{n}+\mathrm{p}}}\ln \left[{\displaystyle \frac{2+\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathsf{\eta }}_{\mathrm{D}}-{\mathsf{\eta }}_{\mathrm{F}}+\nu \right)}{2+\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathsf{\eta }}_{\mathrm{D}}-{\mathsf{\eta }}_{\mathrm{F}}\right)}}\right]\\ +{\displaystyle \frac{4{\mathrm{N}}_{\mathrm{C}}}{3\left(\mathrm{n}+\mathrm{p}\right)\sqrt{\mathsf{\pi }}}}\left[{\mathrm{F}}_{3/2}\left({\mathsf{\eta }}_{\mathrm{F}}-{\mathsf{\eta }}_{\mathrm{C}}+\nu \right)-{\mathrm{F}}_{3/2}\left({\mathsf{\eta }}_{\mathrm{F}}-{\mathsf{\eta }}_{\mathrm{C}}\right)\right]\\ +{\displaystyle \frac{4{\mathrm{N}}_{\mathrm{V}}}{3\left(\mathrm{n}+\mathrm{p}\right)\sqrt{\mathsf{\pi }}}}\left[{\mathrm{F}}_{3/2}\left({\mathsf{\eta }}_{\mathrm{V}}-{\mathsf{\eta }}_{\mathrm{F}}-\nu \right)-{\mathrm{F}}_{3/2}\left({\mathsf{\eta }}_{\mathrm{V}}-{\mathsf{\eta }}_{\mathrm{F}}\right)\right]\end{array}\right\}}^{1/2}$$

(10)

This expression is not amenable to an effective analysis. Therefore, the presence of minority carriers could be neglected because, at $\mathrm{T}=300 \mathrm{K}$ the carrier density $\mathrm{n}={\displaystyle \frac{{2\mathrm{N}}_{\mathrm{C}}\left(\mathrm{T}\right)}{\sqrt{\mathsf{\pi }}}}{\mathrm{F}}_{1/2}\left(-{\displaystyle \frac{{\mathrm{E}}_{\mathrm{C}}-{\mathrm{E}}_{\mathrm{F}}}{\mathrm{k}\mathrm{T}}}\right)=1.42\times {10}^{-37} {\mathrm{c}\mathrm{m}}^{-3}$ i.e., that is, $\mathrm{n}\approx 0.$ Thus, a simplified expression for single-mobile-carrier screening is

$${\displaystyle \frac{\mathrm{k}\mathrm{T}\left|{\overrightarrow{\mathrm{E}}}_{\mathrm{z}}\right|}{\mathrm{e}}}={\displaystyle \frac{\mathrm{d}\nu \left(\mathrm{u}\right)}{\mathrm{d}\mathrm{u}}}=2{\left\{\begin{array}{c}{\displaystyle \frac{{\mathrm{N}}_{\mathrm{A}}}{\mathrm{p}}}\ln \left[{\displaystyle \frac{2+\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathsf{\eta }}_{\mathrm{F}}-{\mathsf{\eta }}_{\mathrm{A}}\right)}{2+\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathsf{\eta }}_{\mathrm{F}}-{\mathsf{\eta }}_{\mathrm{A}}-\nu \right)}}\right]+{\displaystyle \frac{{\mathrm{N}}_{\mathrm{D}}}{\mathrm{p}}}\ln \left[{\displaystyle \frac{2+\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathsf{\eta }}_{\mathrm{D}}-{\mathsf{\eta }}_{\mathrm{F}}+\nu \right)}{2+\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathsf{\eta }}_{\mathrm{D}}-{\mathsf{\eta }}_{\mathrm{F}}\right)}}\right]\\ +{\displaystyle \frac{4{\mathrm{N}}_{\mathrm{V}}}{3\mathrm{p}\sqrt{\mathsf{\pi }}}}\left[{\mathrm{F}}_{3/2}\left({\mathsf{\eta }}_{\mathrm{V}}-{\mathsf{\eta }}_{\mathrm{F}}-\nu \right)-{\mathrm{F}}_{3/2}\left({\mathsf{\eta }}_{\mathrm{V}}-{\mathsf{\eta }}_{\mathrm{F}}\right)\right]\end{array}\right\}}^{1/2}$$

(11)

To obtain the electric potential, this expression should be integrated, which is impossible analytically. It should be noted that the incorporation of several types of defects is possible. Because the charge at these defects is additive, it requires a mere summation over all defects in the system.

To obtain insight into the change in effective potential, numerical integration was applied. In order to do so, the numerical values of the parameters at $\mathrm{T}=300 \mathrm{K}\left(\mathrm{k}\mathrm{T}=25.85 \mathrm{m}\mathrm{e}\mathrm{V}\right)$ were used as follows: ${\mathrm{N}}_{\mathrm{C}}\left(300 \mathrm{K}\right)=2.21\times {10}^{18} {\mathrm{c}\mathrm{m}}^{-3}$, ${\mathrm{N}}_{\mathrm{C}}\left(300 \mathrm{K}\right)=1.33\times {10}^{19} {\mathrm{c}\mathrm{m}}^{-3}$. For p-type doping, Mg acceptors were used in concentrations in the experimentally available ranges ${\mathrm{N}}_{\mathrm{A}}\in \left[{10}^{15} {\mathrm{c}\mathrm{m}}^{-3},{1.15\times 10}^{17} {\mathrm{c}\mathrm{m}}^{-3},{10}^{19} {\mathrm{c}\mathrm{m}}^{-3}\right]$. The potential profiles were obtained for three landmark acceptor concentrations: ${\mathrm{N}}_{\mathrm{A}}={10}^{15} {\mathrm{c}\mathrm{m}}^{-3}$, ${\mathrm{N}}_{\mathrm{A}}=1.15\times {10}^{17} {\mathrm{c}\mathrm{m}}^{-3}$ and ${\mathrm{N}}_{\mathrm{A}}={10}^{19} {\mathrm{c}\mathrm{m}}^{-3}$. For the calculations, a Schottky approximation was used, assuming the total occupation of acceptors, in which the potential profile is readily obtained:

$$\mathrm{V}\left(\mathrm{z}\right)={\displaystyle \frac{{\mathrm{N}}_{\mathrm{A}}}{2{\epsilon }_{\mathrm{G}\mathrm{a}\mathrm{N}}{\epsilon }_{\mathrm{o}}}}{\left(\mathrm{z}-{\mathrm{L}}_{\mathrm{S}\mathrm{c}\mathrm{h}}\right)}^2$$

(12)

The second diagram in Figure 6 presents the full dependence obtained by the numerical integration of the electric field obtained in Equation (11).

These data indicate that the Schottky approximation provides an incorrect dependence, especially for higher concentrations of acceptors such as $N_A={10}^{19} {\mathrm{c}\mathrm{m}}^{-3}$. This can be understood from Figure 2, as the Fermi level position is higher for lower acceptor concentrations. For concentrations above $N_A={10}^{17} {\mathrm{c}\mathrm{m}}^{-3}$ the level falls below the acceptor level; thus, its occupation is considerably below unity, and the Schottky approximation gives a large error.

To investigate the compensation, the Si donor concentration was used with three compensation levels: $N_D/N_A=0.0$, $N_D/N_A=0.1$, and $N_D/N_A=0.5$. The GaN bandgap was assumed to be $E_g\left(GaN\right)=3.47 \mathrm{e}\mathrm{V}$ [45,46]. The bonding energy of the shallow silicon donor is assumed to be $20 \mathrm{m}\mathrm{e}\mathrm{V}$, i.e., $E_C- E_D=0.02 \mathrm{e}\mathrm{V}$ [45]. As shown in Figure 2, a relatively small compensation affects the Fermi level at $N_A={10}^{18} {\mathrm{c}\mathrm{m}}^{-3}$. Accordingly, the Fermi level is close to the acceptor level. Therefore, the density of holes is stationary, $p\cong {2 \times 10}^{16} {\mathrm{c}\mathrm{m}}^{-3}$ and $p\cong { 10}^{17} {\mathrm{c}\mathrm{m}}^{-3}$ for compensation of 10% and 30%, respectively. This value is significantly lower than the number of charged acceptors. To estimate the technically important role of compensation, the potential profile obtained for the upper concentration of acceptors $N_A={10}^{19} {\mathrm{c}\mathrm{m}}^{-3}$ was plotted in Figure 7.

As the influence of the compensation is essentially negligible, it only extends the width of the potential well by a few percent. Thus, the limited presence of donors can be neglected in the design of p-type contacts.

The results of these ab initio and drift-diffusion calculations were verified by direct measurements. The potential well can be measured using phase-space ab initio direct and reverse ballistic electron emission spectroscopy [60]. Moreover, the combined steady-state photocapacitance (SSPC) and deep-level transient spectroscopy methods open the possibility of direct determination of deep-level trap energies and charge states [61]. For these measurements, a Ni electrode could be used, opening the possibility of experimental confirmation of the obtained results.

4.4. Tunneling Path—Doping in the Potential Well

The extension of the potential well at the upper Mg acceptor density $N_A={10}^{19} {\mathrm{c}\mathrm{m}}^{-3}$ could be estimated to be $L_{well}\approx 2.5\times {10}^{-8} m= 250 \AA$. This distance is approximately two orders of magnitude greater than the noticeable extent of the wavefunctions of the deep acceptor level. Note that shallow acceptor levels are not known for gallium nitride, and they cannot be used. Thus, direct tunneling is not a technically viable method for implementing the contact. This was confirmed by the use of Au as the deposited contact. The annealing of Ni–Au contacts in an oxygen atmosphere has been investigated from the metal side [10,11]. However, the semiconductor side has not been precisely investigated. Nevertheless, it was suspected that the outdiffusion of Ga may be beneficial for contact performance. Typically, Ni defects may play the role of deep acceptors with an acceptor energy level of approximately 1.5–1.6 eV above the valence band maximum (VBM) [28,29]. Thus, these defects can also play a role in electron tunneling.

The outdiffusion of Ga atoms is supported by the motion of Ga vacancies. For annealing temperatures in the range $T_{anneal} \cong 400 ÷500 °\mathrm{C}$, the vacancy mechanism is the only possible mechanism. Ga vacancies were investigated in GaN by positron annihilation [62,63]. From these investigations, it follows that the concentration of Ga vacancies is high in n-type GaN, while it is significantly reduced in p-type materials [64,65]. In contrast, ab initio investigations of Ga diffusion showed a relatively high energy barrier [66,67]. From these data, it follows that the creation of a Ga vacancy layer is difficult; nevertheless, some vacancies may be created. Therefore, it can be assumed that Ga vacancies could be created on the surface. From ab initio investigations published by Van de Walle and Neugebauer, Ga vacancies are deep triple acceptors [28,64]. Gallium vacancies are deep, multi-ionized acceptors, of three ionization energies ($E_A\left(V_{Ga}^{1-}\right)\cong 2.35 \mathrm{e}\mathrm{V}$, $E_A\left(V_{Ga}^{2-}\right)=1.02 ÷ 0.62 \mathrm{e}\mathrm{V}$, and $E_A\left(V_{Ga}^{3-}\right)\cong 0.48 \mathrm{e}\mathrm{V}$) [27,28]. Thus, the uppermost level was the initial step in the path. The two others are located lower, and thus they could create a ladder with the barrier energy of the order of ${∆E}_{bar} \approx 1.0 \mathrm{e}\mathrm{V}$, i.e., considerably lower than the entire barrier. These Ga vacancies may serve as paths for electron tunneling. Even if the location and concentration are not precisely controlled, the conduction effect may be considerable.

From the above results, it follows that the creation of a defect layer in a controlled manner is indispensable for achieving good conductivity contact to p-type GaN. The solution is to use acceptors with energy levels above the VBM in the following range: $E_{A } \in \left[0.2 \mathrm{e}\mathrm{V}, 2 \mathrm{e}\mathrm{V}\right]$. Fortunately, extensive data on deep acceptors are available. The deepest state may be a Ga vacancy with the energy $E_A\left(V_{Ga}\right) \cong 2.3÷2.0 \mathrm{e}\mathrm{V}$. The next in the ladder may be the already used Ni acceptor with the energy $E_A\left(Ni\right) \cong 1.6÷1.5 \mathrm{e}\mathrm{V}$ [40,41]. Next in the ladder may be Mn located at the Ga site with the energy level at $E_A\left({Mn}_{Ga}\right) \cong 1.42 \mathrm{e}\mathrm{V}$ [65,66,67]. Carbon on the nitrogen site has an ionization energy, indicating that it could be used as the next in the ladder $E_A\left(C_{Ga}\right)\cong 0.9 ÷1.1 \mathrm{e}\mathrm{V}$ [68]. This acceptor may be followed by mercury with $E_A\left({Hg}_{Ga}\right)\cong 0.77 \mathrm{e}\mathrm{V}$ and cadmium, which has an estimated ionization energy of about $E_A\left({Cd}_{Ga}\right)\cong 0.55 \mathrm{e}\mathrm{V}$ [69,70]. Finally, beryllium could be used with much smaller ionization energy, at $E_A\left({Be}_{Ga}\right)\cong 0.35 \mathrm{e}\mathrm{V}$ [70,71] towards substitutional Mg at $E_A\left({Mg}_{Ga}\right)\cong 0.18 \mathrm{e}\mathrm{V}$ [3,70]. In addition, iron can be used in the first stage as it is frequently incorporated during growth as unintentional doping with an energy level at $E_A\left({Mg}_{Ga}\right)\cong 2.6 \mathrm{e}\mathrm{V}$ above the VBM [68,69]. The design of a new low-resistance contact to p-type GaN can be constructed using these data. The design is shown in Figure 8.

From this design, it follows that the closest part of the multilayer contact is similar to a naturally formed contact. The additional part must be added by the implantation of additional atoms. The depth of the layer can be determined from this diagram.

i/VGa	$z\in \left(0,2.5 \mathrm{n}\mathrm{m}\right)$
ii/Ni	$z \in \left(2.0 \mathrm{n}\mathrm{m},6.0 \mathrm{n}\mathrm{m}\right)$
iii/Mn	$z\in \left(5.0 \mathrm{n}\mathrm{m},9.0 \mathrm{n}\mathrm{m}\right)$
iv/C	$z\in \left(8.0 \mathrm{n}\mathrm{m},12.0 \mathrm{n}\mathrm{m}\right)$
v/Hg	$z\in \left(11.0 \mathrm{n}\mathrm{m},15.0 \mathrm{n}\mathrm{m}\right)$
vi/Cd	$z\in \left(14.0 \mathrm{n}\mathrm{m},17.0 \mathrm{n}\mathrm{m}\right)$
vii/Be	$z \in \left(16.0 \mathrm{n}\mathrm{m},25.0 \mathrm{n}\mathrm{m}\right)$

The implanted layers must be annealed before the deposition of the Ni contact. The Ni and V_Ga layers may be added by the standard formation of the Ni/Au contact, that is, annealing in an oxygen atmosphere. If possible, an alternative procedure for the creation of these layers may be used; for example, iron may be added depending on the type of metal used. It should be mentioned that this is a rough estimate. In fact, a much wider region could be used, or even the entire well thickness could be implanted by all dopants. This has to be used with some caution, as ionized acceptors may impede electron tunneling. However, this is technically much easier.

It should be noted that these energies are burdened by some errors; therefore, the precise implementation of these contacts will require a series of experiments. In addition, the discovery of additional acceptors may be useful for fine-tuning the contact. On the other hand, the stability of such a structure may be enhanced owing to the absence of standard contact formation when an alternative method is devised. It would also be useful to substitute Ga vacancies with other deep substitutional acceptors because such structures would be much more stable.

Finally, the design shown in Figure 8 is presented as a general explanation of the idea. Naturally, these layers can overlap in a unique manner. In fact, an acceptor with higher activation energy may be implanted at a lower depth, that is, further away from the metal; for example, nickel can be implanted over the width of the entire contact barrier. On the contrary, those that are closer to the surface will be occupied, so they would play the role of a deep acceptor, that is, the ${\mathrm{B}\mathrm{e}}^{-}$ charge will add to the ${\mathrm{M}\mathrm{g}}^{-}$ charge in limiting the width of the well. They should not be located too close to the metal, as they contribute to lower-energy states that capture the carriers; therefore, the location of the deep acceptors should be avoided. Conversely, e.g., implanting Ni far away from the heterointerface is not useful, but it is not drastically harmful. These defect states will be far above the Fermi level, so they will not be occupied and do not contribute to the contact conductivity. Therefore, the deep implantation of high-energy defects is not precise.

5. Summary

The results obtained in this study can be assessed best by representing (i) the state of the art before publication, (ii) a summary of the results in the paper, and (iii) the state of the art after publication.

Accordingly, the state of the art before publication was as follows:

(i)

Any contact to p-type GaN has a high resistance, which seriously hampers the development of devices.

(ii)

The best contact is the deposited Ni/Au double layer, formed by short-time formation in an oxygen atmosphere at temperatures above $400 °\mathrm{C}$ $\left(T>400 °\mathrm{C}\right)$.

(iii)

The formation leads to the diffusion of Ni across the Au layer to create NiO.

(iv)

The influence of the contact formation on the semiconductor part is not known.

The summary of the results obtained within this publication:

(i)

Metal-p-type GaN contacts lead to the transfer of electrons from the metal to the semiconductor part, the creation of a dipole layer, and the equalisation of the Fermi level in the system.

(ii)

Ab initio investigation of the Au–GaN heterostructure band diagram confirmed electron transfer from Au to p-type GaN.

(iii)

The potential profile obtained from ab initio investigation of the Au–GaN heterostructure indicates that the depth of the potential well is higher than 2 V,

(iv)

The thickness of the potential well decreases rapidly for higher concentrations of Mg acceptors; for a concentration density $N_A={10}^{19} {\mathrm{c}\mathrm{m}}^{-3}$ it could be estimated to be $L_{well}\approx 2.5\times {10}^{-8} \mathrm{m}= 250 \AA$,

(v)

The thickness of the potential well does not depend on donor compensation up to a relatively high (30%) compensation level.

(vi)

The thermal annealing of the Ni/Au contact leads to the outdiffusion of Ga and influx of Ni, thus creating defects $V_{Ga}$ and Ni that provide quantum states necessary for electron tunneling, the main mechanism of hole influx into p-type GaN.

(vii)

The multilayer contact structure, created by the controlled implantation of a well-defined set of deep acceptors, could create an effective tunneling path, which is necessary for low-resistance contact.

The state of the art after the publication:

(i)

The semiconductor part of the metal-p-type GaN Ni/Au contact structure was elucidated.

(ii)

Electron transfer from the metal to p-type GaN is an essential part of the Au–GaN contact, as confirmed by ab initio calculations.

(iii)

The dependence of the potential well width on the acceptor doping level was established.

(iv)

Compensation (up to 30%) plays negligible role in the contact properties.

(v)

The design of a new, possibly low-resistance multilayer contact for p-type GaN was proposed.

These results pave the way for the rapid development of a wide class of nitride optoelectronic devices, both LEDs and LDs.