Connected Network Model for the Mechanical Loss of Amorphous Materials

Abstract

Dissipation in amorphous solids at low frequencies is commonly attributed to activated transitions of isolated two-level systems (TLS) that come in resonance with elastic or electric fields. Materials with low mechanical or dielectric loss are urgently needed for applications in gravitational wave detection, high precision sensors, and quantum computing. Using atomistic modeling, we explore the energy landscape of amorphous silicon and titanium dioxide, and find that the pairs of energy minima that constitute single TLS form a sparsely connected network with complex topologies. Motivated by this observation, we develop an analytically tractable theory for mechanical loss of the full network from a nonequilibrium thermodynamic perspective. We demonstrate that the connectivity of the network introduces new mechanisms that can both reduce low frequency dissipation through additional low energy relaxation pathways, and increase dissipation through a broad distribution of energy minima. As a result, the connected network model predicts mechanical loss with distinct frequency profiles compared to the isolated TLS model. This not only calls into question the validity of the TLS model, but also gives us many new avenues and properties to analyze for the targeted design of low mechanical loss materials.

Figures (14)

• Figure 1: Left: First $100$ nodes of the connected network of inherent structures observed in samples of amorphous silicon (a-Si) and amorphous titanium dioxide (a-TiO$_{2}$). The color of the dots and lines correspond to the energy of the minima $E_{i}$ and the average barrier height $V_{ij} - (E_i+E_j)/2$ respectively. Middle: Distributions of energy minima $E_{i}$ (black solid curve), magnitude of energy asymmetries $\Delta_{ij} = E_j-E_i$ (red dotted), and barrier heights $V_{ij}$ (blue dashed) between inherent structures ($i$,$j$) for a-Si (top) and a-TiO$_{2}$ (bottom). The energetics for a pair of connected inherent structures is shown schematically in the inset. Right: Fraction of the number of basis cycles and probability distribution of the degree (number of connections) of nodes in the network with the dashed line a linear least squares fit to the a-Si data with a slope (scaling exponent) of $-2.84\pm 0.04$. Error bars in the degree distribution are the standard error of the mean from logarithmically binned averages. Basis cycles are the minimal set of closed loops in the network, with examples of a three-state cycle in a-TiO$_{2}$ and a four-state cycle in a-Si shown in the inset.
• Figure 2: Scaled mechanical loss $Q^{-1}\mathcal{V} C/(\beta\gamma_{0}^2)$ as a function of frequency for a) a four-state chain and b) a four-state cycle. Two different curves are shown: the TLS approximation \ref{['eq: TLS Quality Factor']} (black solid line), and the connected network \ref{['eq: CN quality factor']} (red dashed). In all cases the large energy barrier is $0.8$eV with the remaining barriers $0.6$eV.
• Figure 3: Average mechanical loss $Q^{-1}\mathcal{V} C/(\beta\gamma_{0}^2)$ from $1000$ node BA networks (curves) and an equivalent TLS model (black squares). Energies are drawn from a uniform energy distribution with barriers between states constrained to be larger than the energy minima. Results are averaged over 100 random networks. The connectivity of the network is increased by changing the number of connections between additional nodes when forming the network $m= 2,4,8,16$ for red solid, green dashed, blue dotted, and purple dash-dotted curves, respectively.
• Figure 4: Mechanical loss $Q^{-1}\mathcal{V} C/(\beta\gamma_{0}^2)$ as a function of frequency. Two different colors (shades) of curves are shown: the TLS approximation \ref{['eq: TLS Quality Factor']} (black), and the connected network linear-response approximation \ref{['eq: CN quality factor']} (red). The energy of the center two minima is $0,0.15,0.3$ eV for solid, dashed, and dotted curves respectively. In all cases the energy barriers are $0.4$eV above the lower energy level connecting two states.
• Figure 5: Average mechanical loss $Q^{-1}\mathcal{V} C/(\beta\gamma_{0}^2)$ from $1000$ node BA networks (red) and TLS (black). Here energies are drawn from a uniform distribution between $0$ and $E_{\rm max}$, with $E_{\rm max} = 0.1,0.5,0.9$eV for solid, dashed, and dotted curves respectively. Barriers between states are constrained to be larger than the energy minima with a maximum of $0.2$eV. Results are averaged over 100 random networks.