Authors
Christoph Freysoldt Björn Lange
Introduction
- Sketch of a LED. The enlargement shows a TEM picture of an inversion domain, which might be responsible for the p-doping limit observed in GaN.
Light-emitting devices (LEDs) are seeing a tremendous development from special-case applications in CD or DVD players to small and efficient light sources in cars, mobile phones, or other electronic devices. LEDs are now approaching the stage for general lighting and are expected to replace the good old light bulb in the coming years. Yet, several challenges have to be met. Notably, the brightness of the LEDs must be increased.
The principle setup of a LED is shown on the right, and corresponds to an extended diode. When a voltage is applied, the current drives electrons through the n-doped half and holes through the p-doped half. At the interface, the electrons and holes recombine under emission of light. In modern LEDs, the bare p-n interface is replaced by an active layer, which enhances and tunes the light emission.
In order to support higher current densities in the actual device, one key factor is the doping level. The higher the doping, the more charge carriers are available. However, the achievable active dopant concentration is limited by thermodynamic constraints (e.g. concurrent phases), and compensation by unwanted impurities. For example, in GaN, Mg is used as p-dopant. At very high concentrations, the Mg will form a Mg3N2 phase rather than stay in the GaN lattice. Co-doping by hydrogen enhances the solubility of Mg via the formation of a MgH complex, but this compensates the Mg acceptor. Fortunately, the hydrogen can be driven out at higher temperatures (~600 °C) because hydrogen defects are mobile at these temperature, while Mg is not. Nevertheless, going beyond a Mg concentration of ~1018 cm-3 (i.e., one in ~10000 atoms) proves difficult. This is the starting point for our study.
Methods
To investigate defects, a structural model must be defined, which captures the relevant physics, but is at the same time computationally feasible. A realistic description (one defect in thousands or even millions perfect lattice sites), is clearly out of scope. Therefore, we use the supercell approximation: the defect and a small amount of host material (a few ten to a few hundred atoms) is taken as structural building block (or supercell), which is then repeated periodically throughout space. This allows to employ the well-tested and efficient computer codes for crystalline solids. Our code is S/PHI/nX. We then use density functional theory (here: with the PBE exchange-correlation functional) to compute the relaxed ground-state structure, the formation energy, charge transition levels, and other properties that help us to understand the defect's chemistry and physics.
The supercell approach corresponds to replacing an isolated defect by a periodic array of defects with a very large density. Therefore, the calculation includes artificial interactions between the defects. Notably electrostatic interaction between charged defects are a problem. Since a non-zero net charge in the supercell yields an infinite repulsion energy, such calculations always include a constant neutralizing background. The attraction between the charged defect and the oppositely charged background usually dominates over the repulsive interaction between the defects, and leaves an artificial stabilizing contribution to the energy of finite supercells. Therefore, systematic convergence tests and extrapolation techniques are required to obtain the wanted isolated-defect formation energy. Recently, we have developed a correction scheme that reliably estimates these artifacts and allows to extract the isolated defect energies from rather small supercells. This greatly improves the efficiency of our approach, reducing the computational effort for achieving a sufficient accuracy by a factor of 10 or even 100 because the supercells can be much smaller (note: the computational effort increase between linearly and cubically with the number of atoms, while the charge error decays only with the third root!).
Modelling defect distribution
The defect formation energies and related thermodynamical data serve as input for modelling the behavior of macroscopic bulk samples under varying external conditions. This allows to develop promising material compositions in the computer without tedious experiments. Of course, the "optimal" materials must be produced in reality in the end, but "bad" choices can be omitted right away...
From the defect formation energies, we can compute the electrical behavior of defect-containing bulk materials using thermodynamics. The equilibrium concentration depends on the formation Ef energy via
c = N0 N exp (- Ef/kT)
where kT is the thermal energy, N is the number of equivalent configurations in the unit cell, and N0 the concentration of lattice sites (i.e. the inverse of the unit cell volume). The formation energy depends on the chemical potentials, and, for charged defects, also on the charge state and the Fermi level. The chemical potentials describe the environment during the growth process, and the Fermi level will attain to a value where overall charge neutrality is achieved. In practice, we have written a computer code that finds the equilibrium position of the Fermi level and the associated defect concentrations. Alternatively, the program seeks the chemical potential if the concentration of a certain chemical species in the material is known from experiment better than the exact composition of the environment (which is rather often the case).
Yet, to describe processes like the hydrogen drive-out, we have to take spatial variations in the material composition into account. This implies a number of additional effects. For example, space charge zones develop when the local Fermi levels (relative to the bulk band edges) differ. Moreover defect diffusion and chemical reactions among the defects must be included. For LEDs, we may exploit the layered structure and model only variations in the vertical direction. The computed 1D "concentration profiles" can then be compared to experiment, e.g. SIMS (secondary ion mass spectrometry). For such simulations, we have developed a computer tool that takes the parameters from the DFT calculations as input.
Mg<sub>3</sub>N<sub>2</sub>
- TEM of a pyramidal inversion domain in GaN. The magnesium nitride is believed to form in the boundary.
The attempt to increase the doping level by providing more and more Mg in the growth process can lead to unwanted results. Rather than incorporating the Mg into the GaN lattice, a separate phase forms - magnesium nitride (Mg3N2). The thermodynamical limit can be predicted from ab initio simulations, but the precise way how this happens depends on experimental details (for example, kinetic barriers can prevent the nucleation of the extra phase). Indeed, at high Mg concentrations inversion domains appear in transmission electron micrographs (TEM). It is nowadays believed that the boundaries consist of magnesium nitride. At the same time, the hydrogen distribution starts to deviate from the dilute-defect models described above. Is this pure coincidence? Or do the inversion domain boundaries trap the hydrogen? Since the amount of magnesium nitride is very small, this is difficult to determine experimentally.
Therefore, we investigate the interaction of hydrogen with magnesium nitride. Unfortunately, the precise atomic structure of the inversion domain boundary inclusion is not known. To get some initial insight, we study the perfect bulk material, which should reflect the main chemical trends. Since the stability of a defect strongly depends on the Fermi level, which in term is determined by the most abundant defects, we also have to include the relevant intrinsic defects, i.e. vacancies and interstitials.
