Origin of Bright Quantum Emissions with High Debye-Waller factor in Silicon Nitride

TL;DR

The study identifies nitrogen-vacancy–antisite defects (N$_\text{Si}$V$_\text{N}$) in $\beta$-Si$_3$N$_4$ as the microscopic origin of bright, near-visible single-photon emission observed in silicon nitride. Using hybrid density functional theory and Huang–Rhys theory, the authors show that the NV$^{-}$ center in the $C_{1h}$ configuration produces a linearly polarized ZPL at $E_{ZPL}=2.46$ eV with a radiative lifetime of $\tau_{rad}=9.01$ ns and a Debye–Waller factor of about $33\%$, while a pseudo-Jahn–Teller distortion yields a symmetry-broken $C_{1h}$-PJT form with a second ZPL at $1.80$ eV, $\tau_{rad}=10.17$ ns, and $\text{DW}\approx41\%$. The moderate electron–phonon coupling, quantified by a small total Huang–Rhys factor, preserves substantial ZPL emission. These results provide a coherent, microscopic mechanism for visible quantum emissions in integrated silicon-nitride photonics and enable targeted, deterministic defect engineering for scalable quantum photonic devices.

Abstract

Silicon nitride has emerged as a promising photonic platform for integrated single-photon sources, yet the microscopic origin of the recently observed bright quantum emissions remains unclear. Using hybrid density functional theory, we show that the negatively charged N$_\text{Si}$V$_\text{N}$ center (NV$^{-}$) in the C$_{1h}$ configuration exhibits a linearly polarized zero-phonon line (ZPL) at 2.46 eV, with a radiative lifetime of 9.01 ns and a high Debye-Waller (DW) factor of 33%. We further find that the C$_{1h}$ configuration is prone to a pseudo-Jahn-Teller distortion, yielding two symmetrically equivalent defect structures that emit bright, linearly polarized ZPL at 1.80 eV with a lifetime of 10.17 ns and an increased DW factor of 41%. These nitrogen-vacancy-related defects explain the origins of visible quantum emissions, paving the way for deterministic and monolithically integrated silicon-nitride quantum photonics.

Figures (6)

• Figure 1: (a) Formation energy of nitrogen vacancy (V$_\text{N}$, grey), nitrogen antisite (N$_\text{Si}$, red), and nitrogen antisite-vacancy complex (N$_\text{Si}$V$_\text{N}$, green) under N-rich growth conditions. The slopes and kinks in the plot represent the charge states and charge transition levels, respectively. (b) Geometry of the N$_\text{Si}$V$_\text{N}$ complex, shown along the c-axis projection. Blue, silver, and red atoms represent silicon, nitrogen, and nitrogen-antisite, respectively.
• Figure 2: (a) Defect-level diagram of the NV$^{-}$ center in the C$_{1h}$ configuration (left) and the pseudo-Jahn-Teller (PJT) distorted configuration (C$_{1h}$-PJT, right). Energies are referenced to the valence band maximum (VBM). Red markers indicate Kohn-Sham levels, with a grey arrow denoting the transition from the occupied level [$a'$(1)] to the unoccupied level [$a'$(2)]. In the C$_{1h}$-PJT configuration, the defect levels are labeled $a(1)$ and $a(2)$ due to the absence of symmetry elements, and a blue arrow indicates the transition between them. Isosurfaces of the real part (30% of the maximum) of the wave functions corresponding to occupied and unoccupied levels in the minority-spin channel are shown. (b) Schematic potential energy surfaces (PES) for the ground state (GS, green) and excited state (ES, orange). The PES nodes represent the symmetric ground state (G1), distorted ground states (G2 and G3), and corresponding excited states (E1, E2, and E3). The phonon mode (red arrows) responsible for the distortion from C$_{1h}$ (G1) to C$_{1h}$-PJT configurations (G2 and G3) is illustrated. The energy differences between configurations are: $\Delta E_{\text{G1-G2}} = 1.00$ eV, $\Delta E_{\text{E1-E2}} = 1.66$ eV, $\Delta E_{\text{E1-G1}} = 2.46$ eV, and $\Delta E_{\text{E2-G2}} = 1.80$ eV. The changes in configuration coordinate are: $\Delta Q_{\text{G1-G2}} = 2.53$ amu$^{1/2}$Å, $\Delta Q_{\text{E1-E2}} = 2.63$ amu$^{1/2}$Å, $\Delta Q_{\text{E1-G1}} = 0.44$ amu$^{1/2}$Å, and $\Delta Q_{\text{E2-G2}} = 0.36$ amu$^{1/2}$Å.
• Figure 3: (a) Spectral function [$S(\hbar\omega)$] (black line, left axis) and partial Huang-Rhys factors (blue dots, right axis) with phonon energy, illustrating the energy-resolved electron-phonon coupling and dominant phonon modes associated with the $a(2) \rightarrow a(1)$ electronic transition in the C$_{1h}$-PJT configuration of the NV$^{-}$ defect. (b) Calculated photoluminescence (PL) spectrum showing a pronounced ZPL emission at 1.80 eV and the vibronic sideband features, highlighting the role of electron-phonon coupling in the optical emission profile. The inset illustrates the dominant phonon mode at 87 meV.
• Figure S1: Formation energy of nitrogen vacancy, nitrogen antisite, and nitrogen antisite-vacancy defects in Si-rich growth conditions.
• Figure S2: Localized states formed inside the bandgap of $\beta$-Si$_3$N$_4$. The left panel shows Kohn-Sham levels of neutral V$_\text{N}$ defects in both spin channels within the bandgap. The Fermi level is shown on the vertical axis with values referenced to the valence band maximum. The irreducible representations of localized KS levels are indicated according to the C$_{1h}$ point group symmetry of the defect. The right panel shows the real part of the wave function of the highest occupied (solid border) and lowest unoccupied (dashed border) localized levels. The red line represents the mirror plane symmetry element perpendicular to the c-axis.