Elasticity of disordered binary crystals

Abstract

The properties of crystals consisting of several components can be widely tuned. Often solid solutions are produced, where substitutional or interstitional disorder determines the crystal thermodynamic and mechanical properties. The chemical and structural disorder impedes the study of the elasticity of such solid solutions, since standard procedures like potential expansions cannot be applied. We present a generalization of a density functional–based approach recently developed for one-component crystals to multi-component crystals. It yields expressions for the elastic constants valid in solid solutions with arbitrary amounts of point defects and up to the melting temperature. Further, both acoustic and optical phonon eigenfrequencies can be computed in linear response from the equilibrium particle densities and established classical density functionals. As a proof of principle, dispersion relations are computed for two different binary crystals: A random fcc crystal as an example for a substitutional, and a disordered sodium chloride structure as an example of an interstitial solid solution. In cases where one of the components couples only weakly to the others, the dispersion relations develop characteristic signatures. The acoustic branches become flat in much of the first Brillouin zone, and a crossover between acoustic and optic branches takes place at a wavelength which can far exceed the lattice spacing.

Appendices

Appendix A: Derivation the rotational LMBW-Equation

The generalization of the derivation for one-component systems [51] to the case of several species is straightforward. To begin with, the coupling to external fields can be species-dependent. As a consequence, species-dependent external potentials $\mathcal {V}^{\mathsf {s}}_{\textsf {ext}}$ need to be introduced for which the functional Taylor expansion reads

$$ \delta\mathcal{V}^{\mathsf{s}}_{\textsf{ext}}(\boldsymbol{r})=\sum\limits_{\mathsf{s}^{\prime}}\int\mathsf{d}^{d}r^{\prime} \frac{\delta \mathcal{V}^{\mathsf{s}}_{\textsf{ext}}(\boldsymbol{r})}{{\delta} n^{\mathsf{\mathsf{s}^{\prime}}}(\boldsymbol{r}^{\prime})}{\delta}n^{\mathsf{\mathsf{s}^{\prime}}}(\boldsymbol{r}^{\prime}). $$

(39)

The invariance of the internal state of a crystal under global translations and rotations naturally extends to the case of several species. The spatial variation of a rotation δ𝜃 can be expressed by

$$ \boldsymbol{r}\overset{\boldsymbol{\delta\theta}}{\longrightarrow}\tilde{\boldsymbol{r}}=\boldsymbol{r}+\boldsymbol{\delta\theta}\times\boldsymbol{r}+\mathcal{O}\left( \boldsymbol{\delta\theta}^{2}\right). $$

(40)

We insert the leading order of Eq. (40) into the variation Eq. (39),

$$ \boldsymbol{\delta\theta}\times\boldsymbol{r}\cdot\boldsymbol{\nabla}\delta \mathcal{V}^{\mathsf{s}}_{\textsf{ext}}(\boldsymbol{r}) = \sum\limits_{\mathsf{s}^{\prime}}\int\mathsf{d}^{d}r^{\prime} \frac{\delta \mathcal{V}^{\mathsf{s}}_{\textsf{ext}}(\boldsymbol{r})}{{\delta}n^{\mathsf{\mathsf{s}^{\prime}}}(\boldsymbol{r}^{\prime})}\boldsymbol{\delta\theta}\times\boldsymbol{r}\cdot\boldsymbol{\nabla}n^{\mathsf{\mathsf{s}^{\prime}}}(\boldsymbol{r}^{\prime}). $$

(41)

In combination with the DFT relation

$$ \upbeta\frac{\delta \mathcal{V}^{\mathsf{s}}_{\textsf{ext}}(\boldsymbol{r})}{{\delta}n^{\mathsf{\mathsf{s}^{\prime}}}(\boldsymbol{r}^{\prime})}=c^{\mathsf{ss^{\prime}}}(\boldsymbol{r},\boldsymbol{r}^{\prime})-\delta_{\mathsf{ss^{\prime}}}\frac{\delta({\Delta}\boldsymbol{r})}{n^{\mathsf{s}}(\boldsymbol{r})} $$

(42)

and the cyclicity of the cross product, we obtain the rotational LMBW-Eq. (23b)

$$ \begin{array}{@{}rcl@{}} &&\boldsymbol{r}\times\boldsymbol{\nabla}\left[\ln n^{\mathsf{s}}(\boldsymbol{r})+\upbeta \mathcal{V}^{\mathsf{s}}_{\textsf{ext}}(\boldsymbol{r})\right]\\ &&{}={\sum}_{\mathsf{s^{\prime}}}\int\mathsf{d}^{d}r^{\prime} c^{\mathsf{ss^{\prime}}}(\boldsymbol{r},\boldsymbol{r}^{\prime})\boldsymbol{r}^{\prime}\times\boldsymbol{\nabla}^{\prime}n^{\mathsf{s^{\prime}}}(\boldsymbol{r}^{\prime}). \end{array} $$

(43)

This yields Eq. (23b) in the special case of vanishing external potential.

Appendix B: Derivation of long-wavelength limit

The proof that the dynamical matrix $\tilde {\boldsymbol {\mathsf {\Lambda }}}$ has d branches of eigenvalues which vanish like q² for long wavelength starts from Eq. (26). It reads, expanded up to linear order in wavevector

$$ \begin{array}{@{}rcl@{}} &&\frac{\tilde{\boldsymbol{\mathrm{\Lambda}}}^{\mathsf{11, \textsf{exc}}}(\boldsymbol{q})}{\varrho_{0}}{}\\ &&=\sum\limits_{\mathsf{s,s^{\prime}}}\iint \mathsf{d}^{d}r \mathsf{d}^{d}r^{\prime} [1-i\boldsymbol{q}\cdot(\boldsymbol{r}-\boldsymbol{r}^{\prime})]c^{\mathsf{ss^{\prime}}}(\boldsymbol{r},\boldsymbol{r}^{\prime}){}\\ &&\times \left[(i\boldsymbol{\nabla}+\boldsymbol{q})n^{\mathsf{s}}(\boldsymbol{r})i\boldsymbol{\nabla}^{\prime} -i\boldsymbol{\nabla}n^{\mathsf{s}}(\boldsymbol{r})\boldsymbol{q}\right]n^{\mathsf{s^{\prime}}}(\boldsymbol{r}^{\prime})+\mathcal{O}(\boldsymbol{q}^{2}) \\ &&=\sum\limits_{\mathsf{s}} \int\mathsf{d}^{d}r (i\boldsymbol{\nabla}+\boldsymbol{q})n^{\mathsf{s}}(\boldsymbol{r}) \times{} \\ && \times i\sum\limits_{\mathsf{s^{\prime}}}\int\mathsf{d}^{d}r^{\prime} c^{\mathsf{ss^{\prime}}}(\boldsymbol{r},\boldsymbol{r}^{\prime}) \boldsymbol{\nabla}^{\prime} n^{\mathsf{s^{\prime}}}(\boldsymbol{r}^{\prime}){}\\ && -i\sum\limits_{\mathsf{s^{\prime}}} \int\mathsf{d}^{d}r^{\prime} n^{\mathsf{s^{\prime}}}(\boldsymbol{r}^{\prime})\sum\limits_{\mathsf{s}}\int\mathsf{d}^{d}r c^{\mathsf{s's}}(\boldsymbol{r}^{\prime},\boldsymbol{r})\boldsymbol{\nabla} n^{\mathsf{s}}(\boldsymbol{r})\boldsymbol{q}{}\\ &&+i\boldsymbol{q}\cdot\sum\limits_{\mathsf{s,s^{\prime}}}\iint \mathsf{d}^{d}r \mathsf{d}^{d}r^{\prime} (\boldsymbol{r}-\boldsymbol{r}^{\prime})c^{\mathsf{ss^{\prime}}}(\boldsymbol{r},\boldsymbol{r}^{\prime}){} \\ && \times \boldsymbol{\nabla}n^{\mathsf{s}}(\boldsymbol{r}) \boldsymbol{\nabla}^{\prime}n^{\mathsf{s^{\prime}}}(\boldsymbol{r}^{\prime}) + \mathcal{O}(\boldsymbol{q}^{2}) . \end{array} $$

(44)

In the second term of Eq. (44), we used $c^{\mathsf {ss^{\prime }}}(\boldsymbol {r},\boldsymbol {r}^{\prime })=c^{\mathsf {s's}}(\boldsymbol {r}^{\prime },\boldsymbol {r})$ which follows from the definition Eq. (1). The first two terms now display the right-hand side of Eq. (23a). Before we rewrite these parts, application of the same Eq. (23a) brings the third term to vanish by

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{\mathsf{s,s^{\prime}}}\iint \mathsf{d}^{d}r \mathsf{d}^{d}r^{\prime} \boldsymbol{r}\left[\boldsymbol{\nabla}n^{\mathsf{s}}(\boldsymbol{r})\right]c^{\mathsf{ss^{\prime}}}(\boldsymbol{r},\boldsymbol{r}^{\prime})\boldsymbol{\nabla}^{\prime}n^{\mathsf{s^{\prime}}}(\boldsymbol{r}^{\prime}) {}\\ &={}& \sum\limits_{\mathsf{s}} \int\mathsf{d}^{d}r \boldsymbol{r}\left[\boldsymbol{\nabla}n^{\mathsf{s}}(\boldsymbol{r})\right]\frac{\boldsymbol{\nabla}n^{\mathsf{s}}(\boldsymbol{r})}{n^{\mathsf{s}}(\boldsymbol{r})} \\ &={}&\sum\limits_{\mathsf{s,s^{\prime}}}\iint \mathsf{d}^{d}r \mathsf{d}^{d}r^{\prime} \boldsymbol{r}^{\prime}\left[\boldsymbol{\nabla}^{\prime}n^{\mathsf{s^{\prime}}}(\boldsymbol{r}^{\prime})\right]c^{\mathsf{s's}}(\boldsymbol{r}^{\prime},\boldsymbol{r})\boldsymbol{\nabla}n^{\mathsf{s}}(\boldsymbol{r}) . \end{array} $$

(45)

For the remaining terms, we end up with

$$ \begin{array}{@{}rcl@{}} \frac{\tilde{\boldsymbol{\mathrm{\Lambda}}}^{\mathsf{11, \textsf{exc}}}(\boldsymbol{q})}{\varrho_{0}} &=&i\sum\limits_{\mathsf{s}}\int\mathsf{d}^{d}r \left[ (i\boldsymbol{\nabla}+\boldsymbol{q})n^{\mathsf{s}}(\boldsymbol{r})\right]\frac{\boldsymbol{\nabla}n^{\mathsf{s}}(\boldsymbol{r})}{n^{\mathsf{s}}(\boldsymbol{r})}+{}\\ && -i\sum\limits_{\mathsf{s}}\int\mathsf{d}^{d}r n^{\mathsf{s}}(\boldsymbol{r})\frac{\boldsymbol{\nabla}n^{\mathsf{s}}(\boldsymbol{r})}{n^{\mathsf{s}}(\boldsymbol{r})}\boldsymbol{q} + \mathcal{O}(\boldsymbol{q}^{2}) . \end{array} $$

(46)

Looking at Eq. (19b), this can be easily identified as minus the ideal gas contribution in $\tilde {\boldsymbol {\mathrm {\Lambda }}}^{\mathsf {11}}(\boldsymbol {q})$ up to terms of order $\mathcal {O}(\boldsymbol {q}^{2})$. In the off-diagonal block $\tilde {\boldsymbol {\mathrm {\Lambda }}}^{\mathsf {12}}(\boldsymbol {q}) = \tilde {\boldsymbol {\mathrm {\Lambda }}}^{\mathsf {21}^{\dagger }} (\boldsymbol {q})$, only terms of $\mathcal {O}(\boldsymbol {q}^{0})$ can be ruled out by the same calculations as shown above. This ensures the existence of linear dispersion relations in the long-wavelength limit q →0. All structures considered in this paper are inversion-symmetric such that Λ(q) is even in q. In that special case, the off-diagonal blocks start out like q². Appendix C gives the specification of inversion symmetry that identifies the acoustic modes in the equations of our approach for the total momentum current.

Appendix C: Discussion of inversion symmetry

Within the context of crystals, inversion symmetry is commonly defined with reference to lattice sites as the equilibrium positions of particles. In the present approach, however, the Bravais lattice is not defined from a set of spatially periodic particle positions but only from the spatial periodicity of the equilibrium density n. In that sense, it is hard to identify a privileged point within a given crystal unit cell with respect to which n can be checked for inversion symmetry. The following definition of inversion symmetry also refers only to properties of a given species-independent crystal equilibrium density n where

$$ n(\boldsymbol{r}) = \sum\limits_{\boldsymbol{g}\in\mathbb{G}}n_{\boldsymbol{g}}e^{i\boldsymbol{g}\cdot\boldsymbol{r}}. $$

(47)

We consider n as inversion symmetric in the sense of the present binary crystal approach if and only if two particle species can be distinguished with equilibrium densities n¹ and n² such that n = n¹ + n² with

$$ \begin{array}{@{}rcl@{}} n^{\mathsf{s}}(\boldsymbol{r})&=&\sum\limits_{\boldsymbol{g}\in\mathbb{G}}n^{\mathsf{s}}_{\boldsymbol{g}}e^{i\boldsymbol{g}\cdot\boldsymbol{r}} \end{array} $$

(48)

and

$$ \begin{array}{@{}rcl@{}} n^{\mathsf{s}}_{\boldsymbol{g}} = n^{\mathsf{s}}_{-\boldsymbol{g}} = {n_{\boldsymbol{g}}^{\mathsf{s}}}^{\ast}, \forall \boldsymbol{g}\in\mathbb{G} \quad \mathsf{s}= 1,2 \end{array} $$

(49)

where the realness of the $n^{\mathsf {s}}_{\boldsymbol {g}}$ follows from the definition Eq. (2). This means that two species equilibrium particle densities with a common reciprocal lattice and a common center of inversion can be unambiguously introduced. Note that in non-primitive crystals such as diamond, this requirement can at best be approximately fulfilled.

With the property Eq. (49) at hand, step by step the inversion symmetry of the dynamical matrix in the wave vector, $\tilde {\boldsymbol {\mathsf {\Lambda }}}(\boldsymbol {q}) = \tilde {\boldsymbol {\mathsf {\Lambda }}}(-\boldsymbol {q})$, can be inferred: Starting from the definition Eq. (20), immediately follows the inversion symmetry of $C^{\mathsf {ss^{\prime }}}(\boldsymbol {r},\boldsymbol {r}^{\prime })$ in both its arguments. This can be used in Eq. (22) to obtain the realness of the density fluctuation correlation matrix, $J_{\boldsymbol {g}\boldsymbol {g}^{\prime }}^{\mathsf {ss^{\prime }}} = J_{\boldsymbol {g}\boldsymbol {g}^{\prime }}^{\mathsf {ss^{\prime \ast }}}$. The inversion symmetry in q of the dynamical matrix Λ then follows from its definition in Eq. (14) by sign changes in the summation ${\sum }_{\boldsymbol {g}\boldsymbol {g}^{\prime }}\rightarrow {\sum }_{-\boldsymbol {g}-\boldsymbol {g}^{\prime }}$ and subsequent substitution. As a consequence $\tilde {\boldsymbol {\mathsf {\Lambda }}}$ is even in q and, as required for a long-wavelength decoupling, the off-diagonal blocks are of leading order q²,

$$ \tilde{\boldsymbol{\mathrm{\Lambda}}}^{\mathsf{21}} = \tilde{\boldsymbol{\mathrm{\Lambda}}}^{\mathsf{12}^{\dagger}} =\mathcal{O}(\boldsymbol{q}^{2}). $$

(50)

Appendix D: Derivation of Voigt symmetries

As mentioned in the main text, in order to show the Voigt symmetries of the dynamical matrix [2], we work closely along the calculations in the single-component case[50]: Concerning μ_αβ, we combine both Eq. (2323) as “r × Eq. (23a)–Eq. (23b)” to obtain

$$ \boldsymbol{0} = \sum\limits_{\mathsf{s^{\prime}}}\int\mathsf{d}^{d}r^{\prime} c^{\mathsf{ss^{\prime}}}(\boldsymbol{r},\boldsymbol{r}^{\prime})\boldsymbol{r}^{\prime}\times\boldsymbol{\nabla}^{\prime}n^{\mathsf{s^{\prime}}}(\boldsymbol{r}^{\prime}) . $$

(51)

Summation of this equation over s yields the symmetry Eq. (30a). In order to show Eq. (30b), some tedious recombinations [50] of Eq. (29b) lead to

$$ \begin{array}{@{}rcl@{}} &&\frac{4V}{k_{\textsf{B}}T} \sum\limits_{\mathsf{s,s^{\prime}}} \lambda_{\alpha\upbeta\gamma\delta}^{\mathsf{s,s^{\prime}}} \\ && = \sum\limits_{\mathsf{s,s^{\prime}}}\iint\mathsf{d}^{d}r \mathsf{d}^{d}r^{\prime} c^{\mathsf{ss^{\prime}}}(\boldsymbol{r},\boldsymbol{r}^{\prime})\left( r_{\delta} \nabla_{\gamma}^{\prime}n^{\mathsf{s^{\prime}}}(\boldsymbol{r}^{\prime}) + r_{\gamma} \nabla_{\delta}^{\prime}n^{\mathsf{s^{\prime}}}(\boldsymbol{r}^{\prime})\right)\\ &&\times \left( \underbrace{ r_{\alpha}\nabla_{\upbeta} + r_{\upbeta}\nabla_{\alpha} } \underline{- r^{\prime}_{\upbeta}\nabla_{\alpha}- r^{\prime}_{\alpha}\nabla_{\upbeta}} \right)n^{\mathsf{s}}(\boldsymbol{r}) \end{array} $$

(52a)

and

$$ \begin{array}{@{}rcl@{}} &&\frac{4V}{k_{\textsf{B}}T} \sum\limits_{\mathsf{s,s^{\prime}}} \lambda_{\gamma\delta\alpha\upbeta}^{\mathsf{s,s^{\prime}}} \\ &&= \sum\limits_{\mathsf{s,s^{\prime}}}\iint\mathsf{d}^{d}r \mathsf{d}^{d}r^{\prime} c^{\mathsf{ss^{\prime}}}(\boldsymbol{r},\boldsymbol{r}^{\prime})\left( r_{\upbeta} \nabla_{\alpha}^{\prime}n^{\mathsf{s^{\prime}}}(\boldsymbol{r}^{\prime}) + r_{\alpha} \nabla_{\upbeta}^{\prime}n^{\mathsf{s^{\prime}}}(\boldsymbol{r}^{\prime})\right)\\ &&\times \left( \underbrace{ r_{\gamma}\nabla_{\delta} + r_{\delta}\nabla_{\gamma} } \underline{- r^{\prime}_{\delta}\nabla_{\gamma}- r^{\prime}_{\gamma}\nabla_{\delta}} \right)n^{\mathsf{s}}(\boldsymbol{r}) . \end{array} $$

(52b)

The underlined parts in both Eqs. (52) can be identified by an interchange of both the integration and the summation variables, viz. $\boldsymbol {r}\leftrightarrow \boldsymbol {r}^{\prime }$ and $\mathsf {s}\leftrightarrow \mathsf {s}^{\prime }$. For the terms with the underbraces in Eq. (52), we employ the translational LMBW equation Eq. (23a) in reverse direction to the primed integral and summation:

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{\mathsf{s}^{\prime}}\int\mathsf{d}^{d}\boldsymbol{r}^{\prime}c^{\mathsf{ss^{\prime}}}(\boldsymbol{r},\boldsymbol{r}^{\prime})\left( r_{\upbeta} \nabla_{\alpha}^{\prime} + r_{\alpha} \nabla_{\upbeta}^{\prime}\right)n^{\mathsf{s^{\prime}}}(\boldsymbol{r}^{\prime})\\ &&=r_{\upbeta}\left( \frac{\nabla_{\alpha} n^{\mathsf{s}}(\boldsymbol{r})}{n^{\mathsf{s}}(\boldsymbol{r})} + \nabla_{\alpha}\mathcal{V}^{\mathsf{s}}_{\textsf{ext}}(\boldsymbol{r}) \right) + r_{\alpha}\left( \frac{\!\nabla_{\upbeta} n^{\mathsf{s}}(\boldsymbol{r})}{n^{\mathsf{s}}(\boldsymbol{r})} \!+ \!\nabla_{\upbeta}\mathcal{V}^{\mathsf{s}}_{\textsf{ext}}(\boldsymbol{r})\! \right) .\\ \end{array} $$

(53)

The terms with underbraces in Eq. (52) can also be identified which shows λ_αβγδ = λ_γδαβ and thus the symmetry of Eq. (30b).