Prediction of anomalous LA-TA splitting in electrides

Electrides are an emerging class of materials with excess electrons localized in interstices and acting as anionic interstitial quasi-atoms (ISQs). The spatial ion-electron separation means that electrides can be treated physically as ionic crystals, and this unusual behavior leads to extraordinary physical and chemical phenomena. Here, a completely different effect in electrides is predicted. By recognizing the long-range Coulomb interactions between matrix atoms and ISQs that are unique in electrides, a nonanalytic correction to the forces exerted on matrix atoms is proposed. This correction gives rise to an LA-TA splitting in the acoustic branch of lattice phonons near the zone center, similar to the well-known LO-TO splitting in the phonon spectra of ionic compounds. The factors that govern this splitting are investigated, with isotropic fcc-Li and anisotropic hP4-Na as the typical examples. It is found that not all electrides can induce a detectable splitting, and criteria are given for this type of splitting. The present prediction unveils the rich phenomena in electrides and could lead to unprecedented applications.


I. Introduction
The phonon dispersion relation (q) of a crystalline lattice reflects the energy variation in quantum vibration of a solid. This quantity encodes all information about the ionic contributions to the thermodynamics properties and dynamic process at low temperature, which is fundamental * To whom correspondence should be addressed. E-mail: s102genghy@caep.cn -2 -for both equilibrium and non-equilibrium statistics of a solid. Normally, there are 3 acoustic branches and (3N-3) optical branches for a 3-D crystal composed by N (N→∞) atoms. Their frequencies are denoted as A and O, respectively. For non-polar crystals, the long wavelength longitudinal optical (LO) and transverse optical (TO) mode could be degenerate nears the zone center (ZC). However, in ionic crystals, the relative vibrations of positively and negatively charged ions induce macroscopic electric field (E-field), which in turn affects the motion of ions 1,2 .
This long-range Coulomb interaction (LRCI) lifts the aforementioned degenerate, and gives rise to the LO-TO splitting near the Brillouin ZC (LO≠TO, q≈0), which is very important for the accurate description of dielectric properties and phonon transportation in a polar system, and have been widely explored [3][4][5][6][7][8] .
Electrides have promising physical and chemical properties due to the unique electronic structures that originated from the localized ISQs. For example, it open the energy gap and lead to counterintuitive metal-nonmetal transition in Li 25,26 and Na 27 , as well as the accompanied complex structural phase transition in dense Li 26 . This extraordinary behavior of the localized electrons not only modifies the electronic and crystalline structure greatly, but also changes the dielectric and optical properties 28 , which makes a semi-transparent metal become possible 28 .
In physics, electride is analog to ionic compound, and is polar intrinsically. The difference is the negatively charged ISQs take the role of anions in electrides. Therefore, vibrations of positively charged matrix atoms in electride will induce a non-negligible macroscopic E-field, which leads to LRCI and a non-analytical contribution to the dielectric terms near the ZC. In this -3 -work, a theoretical model to describe this non-analytical contribution is proposed. The resultant effect in the phonon dispersion is explored, which gives an anomalous splitting in the long wavelength longitudinal acoustic (LA) and transverse acoustic (TA) branches near the ZC. This novel but counterintuitive LA-TA splitting in electrides is a direct counterpart of the well-known LO-TO splitting in classical polar system, and demonstrates the direct physical consequence of the strong long-range coupling of localized electrons and crystalline matrix atoms. It also acts as a typical example that reveals the direct participation of electrons in sound propagation in condensed matters.
In section II, the methodology and computational details are presented. The results and analysis are given in section III. The criterion and the thumb rule for this extraordinary LA-TA splitting to occur are summarized in section IV, together with relevant discussions. Finally, the conclusion and summary of main findings are given in section V.

A. Basic theory
In quantum mechanics for solid, the electronic and nuclear subsystems can be decoupled by using the Born-Oppenheimer approximation, with the nuclear part described by the lattice dynamics. The motion of ions in a crystalline solid shows a periodic pattern, and vibrates around their equilibrium position. The vibrational frequencies (q) are determined to be the square root of eigenvalues of the dynamical matrix ) ( , q   t s D 29 following the lattice dynamics theory: which is the lattice Fourier transformation of the force constants exerting on the ions. Here where * s Z is the Born effective charge tensor for atom s,  ε is the high frequency static dielectric tensor,  is the unit cell volume. This non-analytic correction lifts the energy degeneracy and induces the LO-TO splitting around the Γ point in ionic crystal.
Electride is unique in that it is intrinsically polar even in the elemental phases such as that of Li and Na [25][26][27][28] . The spatial ion-electron separation and charge transfer in this system leads to dipole moments and the subsequent macroscopic E-field when the matrix atoms move around, very similar to the well-known case of ionic compound. The fact that the highly localized excess electrons in electride form negatively charged ISQs, as well as the observations that ISQs almost remain intact when the matrix atoms vibrate under high-temperature/pressure condations 27,30 and that they could form covalent bonds 31,32 , support us to confidently map an electride onto an ionic -5 -compound, with the matrix atoms and ISQs correspond to the cations and anions, respectively.
Also, this treatment strategy can be regarded as integrating the electronic degree of freedom to the sites and charge of the ISQs, which is a kind of the coarse-graining approach method being widely used in condensed-matter physics. This standard treatment of electride makes it possible to describe the non-analytic contribution of the LRCI in electrides by generalization of Eq.(2) to include ISQ as a new species. The additional components for the non-analytic dynamical matrix and q q Eq.
In Eqs. (3) and (4) the interstitial sites, so that its effective mass m* is very large. As a result, it leads to a relatively small excitation frequency and relatively slow collective motion. Therefore, the vibration of ISQs in electrides can be coupled with matrix atoms. In spite of that, its impact to the phonon dispersions also will be discussed in following sections by using a finite value.

B. Computational details
We explored the LRCI effect on phonon dispersions (Eqs.

III. Results
We noting that the general feature of the splitting is robust, and is independent of the approximations that were used.

FIG. 2. (Color online) Phonon dispersions of isotropic FCC-Li at 40 GPa (a,b) and anisotropic hP4-Na at 300 GPa (c,d) calculated using Bader charge (a,c) and Born effective charge (b,d) (short-dash-dotted lines), which are compared with the bare results that without taking the LRCI contribution into account (black short-dotted lines).
The high-pressure phase of sodium (hP4-Na) is reported as a typical representative of in FCC-Li.
-9 -Structure of hP4-Na is unique because of the lattice anisotropy, which makes the LA-TA splitting also become anisotropic. As shown in Figs. 2(c) and 2(d), Δ(q≈0)|K→Γ is not equal to Δ(q≈0)|A→Γ. It is necessary to point out that in Fig. 2(d) the non-analytical LRCI is almost suppressed when compared to Fig. 2(c). This is because the modern theory of polarization cannot capture the electron localization in the lattice interstice, and the resultant polarization properly. The influence of  ε is explored by artificially changes its value to investigate how the (q) curve is impacted. As shown in Fig. 3(a), when the  ε of hP4-Na are intentionally modified from 1 .
, Δ(q≈0)|K→Γ becomes equal to Δ(q≈0)|A→Γ, and the LA-TA splitting becomes isotropic. On the other hand, both Δ for FCC-Li and hP4-Na show a sharp decrease with increasing dielectric constant, and Δ approaches a small constant when  ε is bigger than 10 (see Fig. 3(b)). This not only reflects the magnitude of LA-TA splitting is inversely proportional to the square root of  ε , as Eq. (2) indicated, but also reveals that  ε is an important source of the anisotropy. Please note that * Z also contains anisotropy information, which however is missed if the Bader charge is used.

FIG. 4. (Color online) Phonon dispersions of (a) FCC-Li at 40 GPa and (b) hP4-Na at 300 GPa calculated with different effective mass MISQ.
In order to get an insight of the effect of the dynamic ISQ response, we lift the constraint of MISQ→∞. Figure 4 That is, it is the energy from the strong electron-phonon coupling that lifts the LA-TA degeneracy.
It also reveals that if ISQs become responsive, the strong electron-phonon coupling will break the current standard theory of lattice dynamics, and the electrides must be treated with both electrons and matrix atoms on the same footing.

IV. Discussion
By a thorough and comprehensive study of dense Li and Na, as well as many other promising -12 -electrides that are not reported here, we bring up with criteria (or rules of a thumb) for the LA-TA splitting in an electride to occur: (i) The Bader charge of the ISQ must be greater than 0.2e; (ii) The  ε must not be bigger than 10, otherwise the LA-TA splitting could be too small to be observed.
We also noticed that in order to have a strong anisotropic LA-TA splitting, the electride should have a very strong anisotropy in the dielectric tensor, or a strong anisotropy in * Z . It is worth noting that the dynamic response of ISQs to lattice vibration has a strong impact to the phonon dispersions. In the case when MISQ→∞ is valid, the LA-TA splitting is governed by Eq.(2), which can be applied to 0-D, 1-D, and 2-D electrides. For the case of a finite value of MISQ, the counterpart to Eqs. (3)(4) for the 1-D and 2-D electrides can be straightforwardly derived following the similar reasoning. It is beyond the scope of this work, and we will not elaborate it here.
The non-analytic correction for the LRCI of Eqs. (2)(3)(4) is applicable as long as there are none zero polarization generated during lattice vibrations. In electride, this term becomes null only when ISQs response to the movement of matrix atoms in such a special manner that all involved atom-ISQ distances scaled linearly to cancel the generated dipoles. This subtle geometry balance is very rare, if not totally impossible. Such scaling correlation maintains the local relative positions of ISQ with respect to the matrix atoms, thus no LRCI occurs. We did not observe this kind of correlation in dense FCC-Li. Nonetheless, if it presents, this subtle balance can be removed by exerting an external magnetic field or electric field.

V. Conclusion
A theory was proposed to model the non-analytic contribution to the lattice phonon of LRCI that should present in electride. An anomalous LA-TA splitting was predicted in dense FCC-Li and -13 -hP4-Na, and the direct effects of participation of localized electrons into lattice vibrational propagation as anionic ISQs were discovered. The strong electron(ISQ)-phonon coupling was revealed. After a thorough and extensive investigation of promising candidates of electrides, the criteria and the rules of a thumb for observable LA-TA splitting are summarized, which provides a guideline for experimentalists to design an appropriate experiment to detect this unconventional LA-TA splitting phenomenon. We also concludes that when electrons and ISQs are highly responsible to the lattice motion, the classical theory of lattice dynamics breaks, and the electrons and matrix atoms must be treated on the same footing with full quantum mechanics for this kind of mobile electrides. In practical applications, crystal defects, especially the potential colossal-charge-state impurities in electrides 46 Figure S4 displays the variation of splitting magnitude Δ=LA-TA with different value of MISQ in FCC-Li at 40 GPa and hP4-Na at 300 GPa. It can be found that the Δ of FCC-Li is always larger than that of hP4-Na. Please note that Δ of FCC-Li and hP4-Na all approach a constant when MISQ is bigger than 50.