Exe.py: Ab initio fine structure parameters for trigonal defect qubits within the E$\otimes$e Jahn-Teller case

Abstract

Trigonal solid-state defects are often subjects of spontaneous symmetry breaking driven by the $E\otimes e$ Jahn-Teller effect, reflecting strong electron-phonon coupling. These systems, particularly paramagnetic defect qubits in solids are central for quantum technology applications, where accurate knowledge of their fine-structure parameters $-$ shaped by the complex interplay of spin-orbit and electron-phonon interactions $-$ is essential. We introduce the Exe.py code part of the jahn-teller-dynamics package, a Python code that implements the first-principles approach of [Phys. Rev. X 8, 021063 (2018)] to accurately compute the spin-orbit-phonon entanglement in trigonal defects utilizing the output from density functional theory calculations (DFT). By employing $Δ$SCF calculations, the method extends naturally to excited states and predicts fine-structure parameters of zero-phonon lines (ZPLs), including Zeeman shifts under external magnetic fields. The approach is applicable not only to solid-state defects but also to Jahn-Teller active trigonal molecules such as the $X$CH$_3$ family. We demonstrate the capabilities of Exe.py through applications to negatively charged Group-IV$-$vacancy (G4V) defects in diamond: SiV$^-$, GeV$^-$, SnV$^-$, PbV$^-$ and the neutral N$_3$V$^0$ defect in diamond, and the CH$_3$O methoxy radical.

Figures (8)

• Figure 1: Relationship between the experimentally observed photoluminescence spectrum of $\text{SnV}^-$ and the theoretical value of spin-orbit coupling $\lambda^{\text{gnd, ex}}_{\text{expt, theory}}$ of the ground (gnd) and excited (ex) states. The data for the experimental spectrum is taken from Ref. Gorlitz_2020.
• Figure 2: This figure represents the physics behind the zero-phonon-line (ZPL) transitions in trigonal point defects with degenerate electronic ground and excited states. The spin-orbit electronic states of the ground and excited state are fourfold degenerate. Due to spin-orbit coupling they split into two $E_{\frac{1}{2}}$ and $E_{\frac{3}{2}}$ multiplets and the energy difference between them is $\lambda_{\text{DFT}}$. If we take into consideration atomic vibrations by utilizing the $E\otimes e$ Jahn-Teller effect case, they will split to two distinct two-fold degenerate energy levels $(\ket{\varepsilon_{1}}, \ket{\varepsilon_{2}})$ and $(\ket{\varepsilon_{3}}, \ket{\varepsilon_{4}})$. They are entangled states of phonons, orbitals and spin. Under a constant external magnetic field the system experiences total loss of degeneracy because of Zeeman-effect. A,B,C,D represents the transition types as defined in Ref. tin_fine_structure.
• Figure 3: (a) Adiabatic potential energy surface (APES) of the Jahn-Teller distortion. An one-dimensional cut through the APES surface is shown in (b). Panel (c) displays the atomic configuration of the SnV center in diamond, with the distortions indicated in the insets that of panel (b). By applying the same distortion in two opposite directions, the global energy minimum and saddle point configurations can be found.
• Figure 4: This figure illustrates a group-IV vacancy defect embedded in a diamond crystal. The crystallographic basis vectors($\vec{a}$, $\vec{b}$, $\vec{c}$) are shown at the bottom of the figure in light red, green, and blue, respectively. The basis vectors for defect’s spin ($\vec{S}_x$, $\vec{S}_y$, $\vec{S}_z$) are depicted in darker shades and needs to be defined in the configuration file. For G4V centers, the spin $z$-axis aligns with the [111] crystallographic direction, which corresponds to the $C_{3}$ rotation axis. The two $x$- and $y$-axes can be freely chosen freely. In panel (a), the [100] crystallographic direction points toward the viewer, while in panel (b), the defect’s $C_3$rotational axis points toward the viewer.
• Figure 5: Magnetic field strength dependence of ZPL fine strucure that of the $\text{SnV}^-$ defect in diamond. Experimental data points are taken from Ref. tin-vac_diamond are compared to the result of our $\mathtt{Exe.py}$ code for the negatively charged $\text{SnV}^-$ defect in diamond.