Auger recombination in Si and GaAs semiconductors: Ab initio results

A detailed description, at the atomistic scale, of the dynamics of excess electrons and holes is fundamental in order to improve the performance of many optoelectronic devices. Among all recombination processes, nonradiative decay paths play a fundamental role in most semiconductor devices, such as optoelectronic devices and solar cells, limiting their efficiency. In this work, a precise ab initio analysis of the direct Auger recombination processes in both n- and p-type Si and GaAs crystals is presented. Our simulations of minority carrier Auger lifetimes rely on an accurate electronic band structure, calculated using density functional theory with the inclusion of quasiparticle corrections. The results obtained are in good agreement with experimental data for both n-Si and p-GaAs, proving the importance of the direct Auger recombination mechanism in such systems. On the contrary, we show that different nonradiative recombination paths are necessary to explain the experimental results for both p-Si and n-GaAs.

Figure 1

Schematic representation of radiative (left) and nonradiative direct Auger (right) band-to-band recombinations.

Figure 2

Top: Diagrammatic representation of the direct and exchange processes. Bottom: Schematic representation of Dyson’s equation for the screened Coulomb interaction $W$ and its solution. The RPA has been used to express the irreducible density-density response function ${\Pi }_{\mathit{RPA}}$. $V$ denotes the bare Coulomb interaction.

Figure 3

QP self-energy corrections as a function of the DFT-LDA KS energy of an ($11\times 11\times 11$) k mesh. Each point represents a KS state. Results obtained emphasize that the ${\text{G}}_0{\text{W}}_0$ corrections are approximatively constant within $2\text{eV}$ below the valence band edge and about $2.5$ eV above the conduction band edge. This result justifies the SO approach.

Figure 4

Calculated band structure of the Si bulk system, where the conduction band has been rigidly shifted according to the SO approach. The origin of the energy scale has been set at half-gap, which corresponds to the position of the Fermi level in the absence of doping. Here, ${\text{E}}_{\text{X}}=1.16\text{eV}$, ${\text{E}}_{\Gamma }=3.20\text{eV}$, and ${\text{E}}_{\text{L}}=2.18\text{eV}$. Colored pockets represent energy zones to which the states $n_2{\mathbf{k}}_2$, $n_3{\mathbf{k}}_3$, and $n_4{\mathbf{k}}_4$ belong, in accordance with the adopted statistical distribution. Gray windows represent energy zones where the final state $n_1{\mathbf{k}}_1$ has a non-negligible probability of being detected (lower window for $ehh$ processes, upper window for $ehh$ processes).

Figure 5

Fermi energy levels calculated at $300\text{K}$ for different majority carrier concentrations by using different k-point meshes. Dashed lines denote the conduction band edge (top) or the valence band edge (bottom). The origin of the energy scale has been set at half-gap.

Figure 6

Calculated Auger lifetimes for n-Si as a function of the k grid at a fixed doping carrier density ($1\times {10}^{19}{\text{cm}}^{-3}$). Different implementations of the energy $\delta$ (rectangular, Lorentzian, and Gaussian functions) are compared.

Figure 7

Auger lifetimes calculated at $300\text{K}$ for n-Si and p-Si as a function of the majority carrier density vs. experimental data. A good agreement between the theoretical and the experimental data is achieved in the case of n-Si (solid line and red symbols, respectively). On the contrary, a bad agreement exists in the case of p-Si (dashed line and blue symbols). Results were obtained using Eq. (12) with a rectangular $\delta$ function and a ($35\times 35\times 35$) mesh. Experimental results from Beck et al.[44] (B73), Iles et al.[45] (I75), Dziewior et al.[46] (D77), Mertens et al.[47] (M80), Wieder[48] (W80), Vaľtkus et al.[49] (V81), Passari et al.[50] (P83), Burk et al.[51] (B84), Swirhun et al.[52] (S85), Häcker et al.[53] (H94), Ciszek et al.[54] (C89), Schöfthaler et al.[55] (S94), and Schmidt et al.[56] (S97). The dotted orange line represents the radiative lifetime.[57]

Figure 8

Auger energy distribution $P\left(E\right)$ for n-Si and p-GaAs bulks. $P\left(E\right)dE$ represents the probability of finding the final state $n_1{\mathbf{k}}_1$ with an energy $E$. ${\text{E}}_{\text{CB}}$ and ${\text{E}}_{\text{VB}}$ are the conduction band edge energy and the valence band edge energy, respectively. The $P\left(E\right)$ distribution is normalized to unity in both cases. Energetic thresholds (approximated to ${\text{E}}_{\text{gap}}$) are depicted by vertical dashed lines for both considered recombination processes.

Figure 9

Number of permitted purely collisional Auger transitions for both n- and p-type Si (left) and GaAs (right) bulks as a function of the k-grid sampling (label $11,$ e.g., refers to an ($11\times 11\times 11$) uniform grid of k vectors).

Figure 10

Calculated band structure of the bulk-GaAs system, where the conduction band has been rigidly shifted according to the SO approach. The origin of the energy scale has been set at half-gap, which corresponds to the position of the Fermi level in the absence of doping. Here, ${\text{E}}_{\text{X}}=2.09\text{eV}$, ${\text{E}}_{\Gamma }=1.35\text{eV}$, and ${\text{E}}_{\text{L}}=1.79\text{eV}$. Colored pockets represent the energy zones to which states $n_2{\mathbf{k}}_2$, $n_3{\mathbf{k}}_3$, and $n_4{\mathbf{k}}_4$ belong, according to the adopted statistical distribution. Gray windows represent energy zones where the final state $n_1{\mathbf{k}}_1$ has a non-negligible probability of being detected (lower window for $ehh$ processes, upper window for $ehh$ processes).

Figure 11

Fermi energy levels calculated at $300\text{K}$ for different majority carrier concentrations by using different k-point meshes. Dashed lines denote the conduction band edge (top) or the valence band edge (bottom). The origin of the energy scale has been set at half-gap.

Figure 12

Calculated Auger lifetimes at $300\text{K}$ for n-doped (dashed line) and p-doped (solid line) GaAs as a function of the majority carrier density. Differently from Si, we do not report experimental values of the Auger lifetimes.