Ioffe-Regel criterion and diffusion of vibrations in random lattices

Using a stable random matrix approach, we consider the diffusion of vibrations in $3d$ harmonic lattices with strong force-constant disorder. Above some frequency ${\omega }_{\mathrm{IR}}$, corresponding to the Ioffe-Regel crossover, the notion of phonons becomes ill defined. They cannot propagate through the system and transfer energy. Nevertheless, most of the vibrations in this range are not localized. We show that they are similar to diffusons introduced by Allen et al. [P. B. Allen, J. L. Feldman, J. Fabian, and F. Wooten, Phil. Mag. B 79, 1715 (1999)] to describe heat transport in glasses. The crossover frequency ${\omega }_{\mathrm{IR}}$ is close to the position of the boson peak. By changing the strength of disorder we can vary ${\omega }_{\mathrm{IR}}$ from zero value (when the rigidity is zero and there are no phonons in the lattice) up to a typical frequency in the system. Above ${\omega }_{\mathrm{IR}}$, the energy in the lattice is transferred by means of diffusion of vibrational excitations. We calculated the diffusivity of the modes $D\left(\omega \right)$ using both the direct numerical solution of the Newton equations and the formula of Edwards and Thouless. It is nearly a constant above ${\omega }_{\mathrm{IR}}$ and goes to zero at the localization threshold. We show that apart from the diffusion of energy, a diffusion of particle displacements in the lattice takes place as well. Above ${\omega }_{\mathrm{IR}}$, a displacement structure factor $S(\mathbf{q},\omega )$ coincides well with a structure factor of random walk on the lattice. As a result, the vibrational line width is $\Gamma \left(q\right)=D_u{q}^2$, where $D_u$ is a diffusion coefficient of particle displacements. These findings may be important for the interpretation of experimental data on inelastic x-ray scattering and mechanisms of heat transfer in glasses. We also show that the scaling with model parameters in our model maps directly onto the scaling observed in the jamming transition model.

Figure 1

Structure of the dynamical matrix $M$ in the $2d$ case. Particles 1–12 interact with the central black particle.

Figure 2

Distributions of random elastic spring constants in $3d$ simple cubic lattice.

Figure 3

The normalized DOS $g\left(\omega \right)$ for a random matrix $M=A{A}^T$ built on a simple cubic lattice with $N=20\times 20\times 20$ particles and averaged over 1000 realizations. (Inset) The participation ratio $P\left(\omega \right)$ for $N={10}^3$ (a) and ${27}^3$ (b) for one realization.

Figure 4

The spacial eigenmode structure of random matrix $M=A{A}^T$ for the lowest frequency ${\omega }_{\min }$ in a two-dimensional square lattice $400\times 400$.

Figure 5

Young modulus $E$ as a function of $\mu$ for dynamical matrix $M=A{A}^T+\mu M_0$ built on a simple cubic lattice with $N=100\times 100\times 100$ particles (one realization). Black dots are calculated values, the line is the best least-square fit.

Figure 6

The normalized DOS $g\left(\omega \right)$ for the dynamical matrix $M=A{A}^T+\mu M_0$ and five different $\mu$ (0, 0.001, 0.01, 0.1, 1) calculated with the precise numerical KPM solution for a simple cubic lattice with $N={200}^3$ (full lines). Straight lines correspond to Eq. (18) with sound velocity $v=\sqrt{E}$. Filled and open diamonds correspond to the phonon contribution to the DOS below and above the Ioffe-Regel crossover frequency ${\omega }_{\mathrm{IR}}$, respectively (see further text for details). (Inset) Dependence ${\omega }_{\max }\left(\mu \right)\propto \sqrt{\mu }$.

Figure 7

Participation ratio for different $\mu$ as a function of $\omega$ for $N={27}^3$ (one realization). The arrows indicate the positions of ${\omega }_{\max }$ in $g\left(\omega \right)$ for the corresponding values of $\mu$ (see Fig. 6).

Figure 8

The normalized DOS $g\left(\omega \right)$ for the dynamical matrix $M=A{A}^T+\mu M_0$ and different $\mu$ (0, 0.001, 0.01, 0.1, 1) calculated with the precise numerical KPM solution for a simple cubic lattice with $N={200}^3$ (full lines). The condition (2) is violated. (Inset) Dependence ${\omega }_{\max }\left(\mu \right)\propto \sqrt{\mu }$.

Figure 9

The Lorentz dispersion curves for different wave vectors $\mathbf{q}\parallel \langle 100\rangle$ and $\mu =0.1$. Closed diamonds correspond to the calculated values of $S(\mathbf{q},\omega )$ and lines are fitting curves according to Eq. (23). The number of particles $N={50}^3$ (one realization). (Insets) Lorentz dispersion curves for $q=0.5$ and $0.75$.

Figure 10

The dependence ${\omega }_{\mathbf{q}}$ on $q$ for $\mathbf{q}\parallel \langle 100\rangle$ for various $\mu$ (1, 0.1, 0.01) in a cubic sample with $N={50}^3$ (one realization). Filled and open diamonds are the maxima of $S(\mathbf{q},\omega )$ as a function of $\omega$ for each discrete value of $q_n$ for frequencies below and above the Ioffe-Regel crossover, respectively (see text below for details). Solid lines correspond to halves of the maxima. Dashed lines show the $\omega =vq$ linear dependence with sound velocity $v=\sqrt{E}$. Horizontal dotted lines correspond to the maximum frequency ${\omega }_{\max }$ in $g\left(\omega \right)$ (taken from Fig. 6). Insets show the group velocity $v_g=d\omega /dq$ as a function of $\omega$.

Figure 11

The phonon linewidth $\Delta \omega$ as a function of $\omega$ for different $\mu$ in a cubic sample with $N={50}^3$ (one realization). Different symbols correspond to different $\mathbf{q}$ directions. $\diamond$ for $\mathbf{q}\parallel \langle 100\rangle$, $▵$ for $\mathbf{q}\parallel \langle 110\rangle$, $\square$ for $\mathbf{q}\parallel \langle 111\rangle$. Filled and open symbols refer to excitations below and above the Ioffe-Regel crossover frequency ${\omega }_{\mathrm{IR}}$, respectively (see text for details).

Figure 12

The ratio $l\left(\omega \right)/\lambda$ as a function of $\omega$ for different $\mu$. Different symbols correspond to different $\mathbf{q}$ directions as explained in Fig. 11. The full horizontal line (separating filled and open symbols) corresponds to Ioffe-Regel crossover $l\left(\omega \right)=\lambda /2$.

Figure 13

The displacement structure factor $S(\mathbf{q},\omega )$, Eq. (20) (symbols) for $\mu =0$ and for three different frequencies. The sample size is $N={50}^3$. The averaging is performed over 300 realizations. Different symbols correspond to different $\mathbf{q}$ directions: $\diamond$ for $\mathbf{q}\parallel \langle 100\rangle$, $▵$ for $\mathbf{q}\parallel \langle 110\rangle$, and $\square$ for $\mathbf{q}\parallel \langle 111\rangle$. Full lines correspond to the structure factor $S_{\mathrm{rw}}(\mathbf{q},\omega )$ of the random walk on the lattice given by Eq. (28) with $D_{\mathrm{rw}}=0.7$. Dashed line corresponds to the limit $q\ll 1$ [see Eq. (31)].

Figure 14

The correlation function $C(\mathbf{r},\omega )$ for $\mu =0$ and six frequencies $\omega$ (0.14, 0.31, 0.49, 0.66, 0.84, 1.01) for a sample with $N={50}^3$ particles averaged over 300 realizations. The full lines are our numerical results obtained from Eq. (20). Each line starts from $r=r_{\min }$, which is about 2.5 interatomic distances (marked by arrows). The dashed line corresponds to Eq. (35) with $D_{\mathrm{rw}}=0.7$.

Figure 15

The normalized structure factor $S_n(\mathbf{q},\omega )$ as a function of $q$ for some direction in $\mathbf{q}$ space and for each frequency $\omega$ for various values of $\mu$ (0, 0.01, 0.1, 1). The sample size is $N={50}^3$. The averaging is performed over 100 realizations. Left sides of all plots are for $\mathbf{q}\parallel \langle 111\rangle$, right sides are for $\mathbf{q}\parallel \langle 100\rangle$. White horizontal dashes show the Ioffe-Regel crossover frequency ${\omega }_{\mathrm{IR}}$. For $\mu =1$, the frequency ${\omega }_{\mathrm{IR}}$ is slightly different for different $\mathbf{q}$ directions. The black full line corresponds to Eq. (32) for the random walk on a simple cubic lattice with diffusion constant $D_{\mathrm{rw}}=0.7$.

Figure 16

The same normalized structure factor $S_n(\mathbf{q},\omega )$ as in Fig. 15 but in $\mathbf{q}$ space in plane $q_x{q}_y$ ($q_z=0$) for $\omega =0.5$. The left picture corresponds to $\mu =0$ (a) and the right to $\mu =0.1$ (b).

Figure 17

The dependence of $R^2\left(t\right)$ in the case of $\mu =0$ for one sample with $N=100\times 100\times 100$ particles and 14 different frequencies $\omega =0.5,1,1.5,...,7$ (from top to bottom). The numbers indicate integer frequencies. The slope of each line corresponds to each black dot in Fig. 18. Two points at $\omega =2$ and 6 correspond to two distributions of energy $E(x,t)$ over the sample for delocalized and localized modes, respectively. They are shown in Fig. 19 (see below).

Figure 18

The dependence of diffusivity $D\left(\omega \right)$ on $\omega$ for $\mu =0$. Black dots are calculated by the direct solution of Newton's equations from Eqs. (40) and (42) and Fig. 17 for $N={100}^3$ particles (one realization). Full lines for $N={10}^3,{14}^3,{20}^3$ are calculated using the formula of Edwards and Thouless (47) with $c=1$ (see below). The averaging for lines is performed over frequencies in the small interval $(\omega -\delta \omega ,\omega +\delta \omega )$ with $\delta \omega =0.25$ and over several thousand realizations.

Figure 19

Black points (diamonds and triangles) show the distribution of energy $E(x,t)$ contained in the layer $x$ as a function of $x$ for two different frequencies $\omega =2$ and 6 at times $t=234$ and 900, respectively, calculated numerically with the Newton method. Full lines are theoretical predictions for delocalized (diffusive) and localized modes given by Eqs. (43) and (44) with $R^2\approx 166$ and $\approx 22$, respectively.

Figure 20

The diffusivity $D\left(\omega \right)$ for various $\mu$ (0, 0.01, 0.1, 1) for a sample with $N={14}^3$ (crosses). The diffusivity was calculated using the formula of Edwards and Thouless (47) with $c=1$ and averaged over two thousand realizations. The arrows indicate frequencies ${\omega }_{\max }$ in the DOS $g\left(\omega \right)$ for corresponding values of $\mu$. Open symbols correspond to the phonon diffusivity (50) below the Ioffe-Regel crossover frequency ${\omega }_{\mathrm{IR}}$.

Figure 21

The normalized DOS $g\left(\omega \right)$ for dynamical matrix $M=A{A}^T+\mu M_0$ with $\mu =1$ and different percentage $100\%-p$ of cut out springs calculated with the precise numerical KPM solution for a simple cubic lattice with $N={200}^3$ (full lines). Straight lines are calculated according to Eq. (18) with sound velocity $v=\sqrt{E}$. The Young modulus $E$ is calculated in the same way as in Sec. 3.

Figure 22

The normalized DOS $g\left(\omega \right)$ for the dynamical matrix $M=A{A}^T+\beta B{B}^T$ with different $\beta$ calculated with the precise numerical KPM solution for a simple cubic lattice with $N={100}^3$ (full lines). Straight lines are calculated according to Eq. (18) with the sound velocity $v=\sqrt{E}$. The Young modulus $E$ is calculated in the same way as in Sec. 3.