An activation-relaxation technique study of two-level system impact on internal dissipation using DFT-based moment tensor potential

This study employs a DFT-trained Moment Tensor Potential coupled with the Activation-Relaxation Technique to reveal that while amorphous silicon models capture similar macroscopic dissipative properties to traditional potentials, they exhibit significantly different atomic-scale two-level system characteristics, specifically a higher prevalence of complex bond-exchange mechanisms and isolated, independent TLS oscillations.

The Gist

The Big Picture: Listening to the Universe's Whispers

Imagine you are trying to hear a tiny pin drop in a hurricane. That is essentially what scientists do with Gravitational Wave Detectors (like LIGO). These massive machines listen for ripples in space-time caused by colliding black holes.

To hear these cosmic whispers, the mirrors inside the detectors must be incredibly still. But there's a problem: even at room temperature, the atoms inside the mirror coatings are jittering around. This jitter creates "thermal noise," which drowns out the faint signals from space.

The scientists in this paper are trying to figure out exactly why these atoms are jittering and how to stop it. They are focusing on a specific type of atomic jitter called a Two-Level System (TLS).

The Analogy: The "Molecular Switch"

Think of the atoms in the mirror coating (amorphous silicon) not as a rigid crystal, but as a messy pile of marbles. In this mess, some atoms get stuck in a weird spot. They have two options:

1. Option A: Stay where they are.
2. Option B: Jump to a spot very close by.

These two spots are almost equal in energy, like a ball sitting in a shallow dip right next to another shallow dip. The ball can easily roll back and forth between them. This back-and-forth motion is the Two-Level System (TLS).

When billions of these "molecular switches" flip back and forth randomly, they create friction. In physics terms, this is internal dissipation (or heat loss), which translates to the "noise" that ruins our ability to hear the universe.

The Problem: Guessing vs. Knowing

For years, scientists tried to simulate these atoms using Empirical Potentials.

• The Analogy: Imagine trying to predict how a complex machine works by looking at a cartoon drawing of it. The drawing gets the general shape right (the gears are round, the levers are long), but it doesn't know the exact physics of how the metal bends or snaps.
• The Reality: These "cartoon" models (specifically one called the modified Stillinger-Weber potential) gave scientists a rough idea of the noise, but they missed the fine details. They were like a weather forecast that gets the temperature right but misses the wind direction.

The Solution: The "Super-Model" (MTP)

In this study, the researchers used a new, high-tech tool called a Moment Tensor Potential (MTP).

• The Analogy: Instead of a cartoon, this is like a hyper-realistic 3D simulation trained on the actual laws of quantum mechanics (Density Functional Theory, or DFT). It's like having a master mechanic who has studied the blueprints of every single atom in existence.
• The Method: They used a technique called ARTn (Activation-Relaxation Technique). Imagine shaking a box of marbles and watching every single time one marble finds a new spot. The computer did this millions of times to find every possible "switch" (TLS) in the silicon.

The Surprising Discoveries

When they compared the "Cartoon" model (mSW) with the "Super-Model" (MTP), they found some big differences:

1. More Switches, But Smaller Ones:
   The Super-Model found twice as many of these atomic switches as the Cartoon model. However, the switches in the Super-Model were more compact.

   • Analogy: The Cartoon model thought the noise came from a few people doing a big, slow dance across the room. The Super-Model revealed that actually, there are twice as many people, but they are doing quick, tight spins in place.

2. Complex Moves vs. Simple Hops:
   The Cartoon model mostly saw atoms doing a simple "hop" (jumping to a neighbor). The Super-Model showed that atoms were actually doing complex "dance moves" involving swapping partners (called Wooten-Winer-Weaire bond exchanges).

   • Analogy: The old model thought everyone was just walking to the next chair. The new model showed that people were actually swapping seats with their neighbors in a complex chain reaction.

3. Isolated Dancers:
   The Super-Model showed that these switches are mostly isolated. They don't talk to each other.

   • Analogy: It's not a synchronized flash mob; it's just a room full of people tapping their feet independently. This is good news because it means the noise is predictable and not a chaotic chain reaction.

Why This Matters

The most important result? When the scientists calculated the "noise" based on the Super-Model (MTP), it matched real-world experiments perfectly. The "Cartoon" model was close, but the Super-Model was spot on.

The Takeaway:
To build the next generation of gravitational wave detectors (which will listen to the universe with even greater precision), we need to understand the atoms at a microscopic level. We can't just use rough approximations. We need the "Super-Model" to tell us exactly how the atoms move.

By understanding that these atoms are doing complex, compact dance moves rather than simple hops, scientists can now design better mirror coatings to stop the jitter, allowing us to hear the faintest whispers from the birth of the universe.

Technical Summary

1. Problem Statement

Gravitational Wave Detectors (GWDs), such as LIGO and Virgo, are limited in sensitivity by background thermal noise, particularly in the mirror coatings. This noise originates from microscopic mechanical fluctuations associated with Two-Level Systems (TLS)—atomic-scale defects where atoms oscillate between two metastable states separated by a low energy barrier.

While amorphous silicon (a-Si) is a promising candidate for next-generation GWD mirrors due to its low internal dissipation and high refractive index, the atomistic mechanisms driving TLSs in a-Si are not fully understood. Previous studies relied on empirical potentials (specifically the modified Stillinger-Weber, or mSW, potential). However, empirical potentials are often fitted to reproduce structural properties but fail to accurately capture dynamical properties and transition-state barriers compared to Density Functional Theory (DFT). The challenge lies in the fact that DFT is computationally too expensive to simulate the large system sizes and statistical sampling required to identify rare TLS events in amorphous materials.

2. Methodology

The authors employed a hybrid approach combining high-fidelity machine learning with advanced transition-state search algorithms:

• Machine-Learned Potential (MLP): They utilized a Moment Tensor Potential (MTP) trained on DFT data. This potential was specifically developed for silicon (and SiO₂) to provide DFT-level accuracy in energy and forces while maintaining computational efficiency that scales linearly with system size.
• Sample Generation: Twenty-eight independent 1000-atom a-Si models were generated using a "melt-and-quench" protocol (heating to 3000 K, cooling to 300 K) using the MTP.
• Event Search: The Activation-Relaxation Technique nouveau (ARTn) was used to extensively explore the energy landscape. ARTn is an open-ended method that identifies saddle points (transition states) and connects them to local minima.
  • 30 event searches were launched per atom, totaling ~30,000 searches per configuration.
  • Events were filtered to identify TLSs based on specific criteria: a barrier height ($V$) between 0.2 and 0.7 eV (relevant for GWD noise frequencies) and low energy asymmetry ($\Delta \leq V/3$).
• Comparison & Validation:
  • Results were compared against previous studies using the mSW potential.
  • Structural properties (density, radial distribution function) were compared to experimental data.
  • Saddle points and minima were re-converged using mSW and other ML potentials (Ind16, Ind20) to validate the stability and accuracy of the MTP-derived structures.

3. Key Contributions

• First MTP-based TLS Study in a-Si: This work represents the first application of a DFT-trained Moment Tensor Potential to systematically identify and classify TLSs in amorphous silicon.
• Quantification of Empirical Potential Limitations: The study provides direct evidence that empirical potentials (mSW) significantly underestimate the density of complex TLSs and misrepresent the nature of the atomic rearrangements involved.
• Mechanistic Classification: The authors categorized TLSs into three distinct mechanisms:
  1. Bond-hopping: An atom reduces coordination to benefit a neighbor.
  2. Wooten-Winer-Weaire (WWW) bond exchange: Two atoms exchange a neighbor while maintaining coordination.
  3. Uncategorized: Complex events not fitting the above.
• Validation of Independence: The study confirmed that TLSs in MTP-based models are largely isolated and oscillate independently, validating the assumption that they can be treated as non-interacting contributors to noise.

4. Key Results

• Structural Fidelity: MTP-generated samples showed excellent agreement with experimental data (density $\approx$ 2.23 g/cm³, energy per atom $\approx$ 0.14 eV above crystal) and were slightly less strained than mSW models.
• TLS Density Discrepancy:
  • MTP: Found 4.18 TLSs per 1000 atoms.
  • mSW: Found 1.95 TLSs per 1000 atoms.
  • The MTP model predicts more than double the density of TLSs compared to the empirical potential.
• Mechanism Shift:
  • While bond-hopping events occurred at similar densities in both models, the MTP model revealed a dominance of complex bond-exchange (WWW) events.
  • In MTP models, bond-exchange events accounted for 40% of TLSs (compared to a lower proportion in mSW), while uncategorized events made up 41%.
  • MTP TLSs involved fewer atoms (average 14) but required larger displacements, suggesting a "smoother" but more rigid energy landscape where local relaxation requires bond breaking rather than simple diffusion.
• Energy Landscape: MTP TLSs showed a bias toward higher energy barriers (>0.4 eV) compared to mSW, attributed to the lower strain energy stored in the MTP structures.
• Loss Angle ($Q^{-1}$) Calculation: When calculating the mechanical loss angle (internal friction) using the derived TLS distribution, the MTP results showed excellent overlap with experimental measurements of a-Si dissipation. In contrast, the mSW-based calculation (even after correcting a scaling factor error from previous literature) deviated from experimental data.

5. Significance

• Accuracy of Modeling: The study demonstrates that while empirical potentials can reproduce average structural properties, they fail to capture the correct atomistic mechanisms and statistical density of TLSs. This is critical because the specific nature of the defect determines how it interacts with strain and contributes to noise.
• Implications for GWD: The finding that complex bond-exchange mechanisms are twice as common in DFT-based models suggests that strategies to reduce thermal noise in gravitational wave detectors must account for these more complex, localized atomic rearrangements, not just simple bond hops.
• Methodological Advancement: The work validates the use of Moment Tensor Potentials as a viable bridge between the accuracy of DFT and the statistical requirements of amorphous material modeling. It establishes that machine-learned potentials can provide reliable, high-fidelity insights into activated processes in disordered systems where traditional empirical potentials fall short.

In conclusion, this paper argues that to truly understand and mitigate thermal noise in future gravitational wave detectors, researchers must move beyond empirical potentials and adopt DFT-trained machine learning potentials to accurately model the complex, atomistic origins of internal dissipation.