Randomness-induced redistribution of vibrational frequencies in amorphous solids

Much of the discussion in the literature of the low-frequency part of the density of states of amorphous solids was dominated for years by comparing measured or simulated density of states to the classical Debye model. Since this model is hardly appropriate for the materials at hand, this created some amount of confusion regarding the existence and universality of the so-called “boson peak” which results from such comparisons. We propose that one should pay attention to the different roles played by different aspects of disorder, the first being disorder in the interaction strengths, the second positional disorder, and the third coordination disorder. These have different effects on the low-frequency part of the density of states. We examine the density of states of a number of tractable models in one and two dimensions and reach a clearer picture of the softening and redistribution of frequencies in such materials. We discuss the effects of disorder on the elastic moduli and the relation of the latter to frequency softening, reaching the final conclusion that the boson peak is not universal at all.

Figure 1

(Color online) Density of vibrational states for a one-dimensional crystal (continuous line) and for exponentially distributed random interaction strength (dashed line).

Figure 2

(Color online) Density of vibrational states for a one-dimensional chain with interactions distributed by the uniform distribution, Eq. (34). Different symbols pertain to different values of the parameter $\Delta$, see inset.

Figure 3

(Color online) Density of vibrational states for a one-dimensional chain with interactions distributed by the Weibull distribution, Eq. (36). Different symbols pertain to different values of the parameter $\alpha$, see inset.

Figure 4

(Color online) Density of vibrational states for a one-dimensional chain with interactions distributed by the inverse distribution, Eq. (40). Different symbols pertain to different values of the parameter $\lambda$, see inset.

Figure 5

(Color online) Comparison of the vibrational density of states for a common value ${\gamma }_{av}=0.6838$. The symbols are explained in the inset.

Figure 6

(Color online) Typical realization of the elastic triangular antiferromagnet, Eq. (43). Solid points represent up spins and empty points represent down spins. Dashed lines connect interacting up spins and solid lines connect interacting down spins. Dotted lines connect pairs of interacting up and down spins.

Figure 7

(Color online) Distribution of angles between nearest neighbors in the elastic triangular antiferromagnet, Eq. (43), for various values of $ϵ$, see inset.

Figure 8

(Color online) Vibrational density of states in the elastic triangular antiferromagnet, Eq. (43), for various values of $ϵ$, see inset.

Figure 9

(Color online) Vibrational density of states normalized by Debye’s prediction in the elastic triangular antiferromagnet, Eq. (43). Symbols are the same as in Fig. 8

Figure 10

(Color online) Density of states for the model with nonlinear elasticity, Eq. (45), with $\kappa =0.25$ and different values of $ϵ$, see inset.

Figure 11

(Color online) Density of states for the model with nonlinear elasticity, Eq. (45), (Fig. 10) normalized by Debye’s model. Symbols are the same as in Fig. 10.

Figure 12

(Color online) Effect of the nonlinear magnetoelastic coupling, Eq. (47), on the density of states for $\nu =0.3$ and different $ϵ$ values, see inset.

Figure 13

(Color online) The density of states with nonlinear magnetoelastic coupling (Fig. 12) divided by the Debye behavior. The symbols are the same as in Fig. 12.

Figure 14

(Color online) Density of states of the diluted elastic triangular antiferromagnet, Eq. (49), with $ϵ=0.1$ and various $p$ values, as indicated in the inset.

Figure 15

(Color online) Density of states of the diluted elastic triangular antiferromagnet, Eq. (49), (Fig. 14) normalized by the Debye prediction. Symbols are the same as in Fig. 14.

Figure 16

(Color online) Shear moduli of the three models Eq. (43) (circles), Eq. (45) (squares), and Eq. (47) (diamonds) for different values of the disorder control parameter $ϵ$.

Figure 17

(Color online) Bulk modulus of the three models Eq. (43) (circles), Eq. (45) (squares), and Eq. (47) (diamonds) for different values of the disorder control parameter $ϵ$.

Figure 18

(Color online) Shear modulus of the diluted model, Eq. (49), as a function of $p$ and $ϵ$. The shear modulus increases when $ϵ$ increase but it decreases when $p$ increases.

Figure 19

(Color online) Bulk modulus of the diluted model, Eq. (49), as a function of $p$ and $ϵ$, symbols as in Fig. 18. The bulk modulus decreases when $ϵ$ increases but it decreases like the shear modulus when $p$ increases.

Figure 20

(Color online) The Debye frequency of the three models, Eq. (43) (circles), Eq. (45) (squares), and Eq. (47) (diamonds) for different values of the disorder control parameter $ϵ$.

Figure 21

(Color online) The Debye frequency of the diluted model, Eq. (49), as a function of $p$ and $ϵ$, symbols as in Fig. 18.

Figure 22

(Color online) $g\left(\omega \right)/\omega$ for the diluted model with different values of $ϵ$. Note that the boson peak amplitude is reduced when $ϵ$ which randomizes the particle positions is increased.