First-principles calculations of internal conversion processes in spin defects

This paper introduces a predictive first-principles framework that combines multi-configurational TDDFT and analytical non-adiabatic couplings to accurately compute internal conversion rates in optically active spin defects, successfully resolving long-standing discrepancies with experimental data for diamond NV$^-$ centers and SiC divacancies.

The Gist

Imagine a tiny, glowing defect inside a crystal (like a diamond or silicon carbide) acting as a microscopic quantum computer. These defects are like tiny stage actors. When you shine a laser on them, they get excited and jump to a higher energy level (the "stage"). To get back to their resting state, they have to choose a path: they can either glow brightly (radiative decay) or quietly slip back down without making a sound (non-radiative decay).

For a long time, scientists trying to predict how fast these actors "slip back down" (a process called Internal Conversion) were using a very rough map. Their calculations were like trying to predict traffic by only looking at one car on a single-lane road. They kept guessing the speed was incredibly slow, but in reality, the traffic was moving fast. Their predictions were off by huge margins—sometimes a thousand times too slow.

This paper introduces a new, high-definition GPS system to fix those predictions. Here is how the authors did it, using simple analogies:

1. The "Many-Body" Problem: Seeing the Whole Orchestra

Previous methods looked at the electrons in the defect as if they were solo musicians playing a single note. But in reality, these electrons are a complex jazz band, all improvising and reacting to each other simultaneously.

• The Old Way: Ignoring the band's interaction, treating the electrons as if they were just one person.
• The New Way: The authors used a sophisticated method (TDDFT with hybrid functionals) to listen to the entire orchestra. By accounting for how all the electrons dance together (multi-configurational effects), they could finally hear the true complexity of the energy levels.

2. The "Vibration" Problem: Counting Every Step

When an electron drops down an energy level, it doesn't just fall; it has to dump its extra energy into the crystal's atoms, making them vibrate. Think of the crystal as a giant trampoline made of millions of springs.

• The Old Way: Scientists used to pretend the trampoline only had one spring, or maybe a few "main" springs, to save time. They calculated the energy dump based on just those few.
• The New Way: The authors realized that every single spring in the trampoline contributes to the fall. They developed a way to calculate the interaction with all the vibrating atoms at once, not just the ones closest to the defect. They did this by calculating "non-adiabatic couplings" (a fancy way of measuring how strongly the electron pushes the atoms) analytically, which is like having a mathematical formula for the push instead of guessing it by trial and error.

The Results: Fixing the Map

The authors tested their new GPS on two famous "actors":

1. The Diamond Actor (NV- center):

   • The Mystery: Scientists knew this actor had a very short life in a specific excited state, but old calculations said it should live much longer.
   • The Fix: The new method calculated the "slip-down" speed and found it was incredibly fast (about 100 billion times per second). This matched perfectly with recent, ultra-fast experimental measurements. It confirmed that the "slip-down" is the main reason this actor doesn't stay excited long.

2. The Silicon Carbide Actor (Divacancy center):

   • The Mystery: For this actor, old calculations said it should stay excited for about 37 nanoseconds (based only on glowing). But experiments showed it only lasts about 15 nanoseconds. Something was missing.
   • The Fix: The new method found a "hidden door" that scientists had missed. They discovered a significant, previously overlooked "slip-down" path (non-radiative channel) that speeds up the decay. When they added this hidden path to their math, the prediction finally matched the experiment (15 nanoseconds).

Why This Matters

The paper doesn't just fix a math problem; it provides a universal toolkit.

• It proves that ignoring the "whole orchestra" (electron interactions) or "all the springs" (vibrations) leads to wildly wrong answers.
• It allows scientists to predict exactly how these quantum defects behave without needing to guess or run expensive experiments first.
• It sets the stage for designing better quantum computers by accurately knowing how long these tiny "qubits" (the magnetic states of the defects) will last before they lose their energy.

In short, the authors built a microscope that sees both the complex dance of electrons and the vibration of every single atom, finally allowing us to accurately predict how fast these quantum defects "turn off."

Technical Summary

Technical Summary: First-principles calculations of internal conversion processes in spin defects

Problem Statement
Optically active spin defects, such as the negatively charged nitrogen-vacancy ($NV^-$) center in diamond and the neutral divacancy ($VV^0$) center in silicon carbide (SiC), are critical platforms for quantum technologies. While radiative transitions and inter-system crossing (ISC) rates in these systems have been extensively studied using first-principles frameworks, internal conversion (IC) processes—non-radiative transitions between electronic states of the same spin driven by electron-phonon coupling—remain poorly described. Common approximations, such as single-mode or accepting-mode approaches based on Kohn-Sham orbitals, have historically underestimated IC rates by orders of magnitude. This discrepancy hinders a complete first-principles description of the optical cycle, limiting the ability to predict excited-state lifetimes, temperature effects, and optically detected magnetic resonance (ODMR) spectra accurately.

Methodology
The authors propose a broad, predictive first-principles framework to compute non-radiative transition rates (NRTRs) for IC processes. The methodology integrates two critical advancements:

1. Many-Body Electronic Structure: The framework utilizes linear-response time-dependent density functional theory (LR-TDDFT) with hybrid functionals (specifically the DDH functional). This approach captures the multi-configurational character of excited states, which is often missing in single-orbital Kohn-Sham descriptions. The authors employ a spin-flip TDDFT framework to handle specific excited states.
2. Analytical Non-Adiabatic Couplings (NACs) and All-Phonon Modes: Instead of relying on finite-difference calculations for wavefunction derivatives, the authors implement an analytical formulation of NAC vectors ($d_{IF,A}$) based on an extended Lagrangian approach within the WEST code. This allows for the efficient computation of electron-phonon matrix elements ($M^k_{IF}$) for all phonon modes in large supercells (hundreds of atoms), avoiding the bottlenecks of previous methods.

The NRTR, $\Gamma_{NR,IF}(T)$, is calculated using Fermi's golden rule, summing contributions from all $N_{modes}$ phonon modes. The calculation employs a generating function approach to evaluate the phonon-relaxation contribution ($X^k_{IF}$) and the electron-phonon spectral density function, $D(E)$, accounting for multi-phonon processes.

Key Results
The framework was applied to the optical cycles of the $NV^-$ center in diamond and the $VV^0$ center in 4H-SiC:

• $NV^-$ Center (Diamond):

  • Singlet States ($^1A_1 \to ^1E$): The calculated NRTRs are on the order of $\approx 10^1$ GHz. The results show excellent quantitative agreement with femtosecond transient absorption experiments (e.g., $\tau_{^1A_1} \approx 117.4$ ps at 78 K vs. experimental values of $\sim 92-102$ ps). This confirms that IC is the dominant decay mechanism for the upper singlet state.
  • Triplet States ($^3E \to ^3A_2$): The calculated NRTR is negligible ($\lesssim 5$ MHz) compared to the radiative rate ($\approx 83.3$ MHz), consistent with the understanding that this transition is predominantly radiative.
  • Mode Analysis: The study reveals that while a localized vibrational mode at $\sim 170$ meV has strong non-adiabatic character, it accounts for only roughly half of the total rate. Including all phonon modes is essential for accuracy; single-mode approximations fail to capture the full spectral density.

• $VV^0$ Center (4H-SiC):

  • Triplet States ($^3E \to ^3A_2$): Previous first-principles estimates considering only radiative contributions overestimated the experimental lifetime ($\sim 15$ ns). By including IC processes, the authors resolve this long-standing discrepancy, showing that non-radiative decay significantly reduces the lifetime to match experimental observations.
  • Singlet States ($^1A_1 \to ^1E$): In the absence of experimental data, the framework predicts NRTRs of $\approx 10^1$ GHz, indicating that this transition is also dominated by IC processes.

Significance and Claims
The paper claims to provide the first efficient and accurate ab-initio framework capable of computing IC rates in spin defects with quantitative agreement to experiment. The authors emphasize two primary contributions:

1. Resolution of Discrepancies: The framework resolves orders-of-magnitude discrepancies between previous theoretical estimates and experimental lifetimes by correctly capturing the many-body nature of electronic states and the contribution of all phonon modes.
2. Methodological Advancement: By implementing analytical NACs, the work eliminates the computational bottleneck of finite-difference derivatives, enabling the inclusion of all phonon modes in systems with hundreds of atoms.

The authors conclude that effective single-mode and accepting-mode approximations systematically underestimate non-radiative rates by orders of magnitude. Their approach enables parameter-free predictions of complete optical cycles and ODMR spectra for spin defects in diamond and SiC. Furthermore, the analytical NAC implementation is presented as a step toward ab initio non-adiabatic molecular dynamics in extended systems and the study of polaron and exciton dynamics.