Ab initio study of the crystal and electronic structure of mono- and bi-alkali antimonides: Stability, Goldschmidt-like tolerance factors, and optical properties

Relaxing within a primitive face-centered cubic (fcc) cell immediately reveals the following five materials (out of the twenty-five under consideration) to be unstable in the $D0_{3}$ structure: K₂LiSb, Rb₂LiSb, Rb₂NaSb, Cs₂LiSb, and Cs₂NaSb. The phonon dispersion relations of the remaining materials (plots in Supplemental Materialref ) reveal ten of those materials to have imaginary phonon frequencies and thus also be mechanically unstable. We further note that all of these materials with imaginary phonon frequencies show instability at the X-point in the fcc Brillouin zone. This then leaves ten materials in our study which are mechanically stable in the $D0_{3}$ structure.

Regarding the unstable phonon displacements, we find that the imaginary branches at the X-point in the fcc Brillouin zone always exhibit a double degeneracy. The complex phonon polarization vectors of these two modes correspond to either the polarization vector shown in Fig. 1b or to a second vector generated via a 90-degree rotation with respect to the axis along the phonon wave-vector direction. The real (gray arrows, Fig. 1b) and imaginary (red arrows, Fig. 1b) parts of these complex polarization vectors correspond to two distinct unstable phonon displacements of the atoms in real space. To understand the significance of these displacements, we note that, as shown in Fig. 1a, the $D0_{3}$ crystal structure for X₂YSb consists of a simple cubic lattice of X atoms, with all cube body centers occupied by alternating Y and Sb atoms. The unstable displacements at the X-point in the fcc Brillouin zone thus consist of either shearing of the X atoms relative to the Y and Sb atoms, or shearing of the Y and Sb atoms relative to the X atoms. These displacements suggest that the relative size of the X atoms to the Y and Sb atoms plays an important role in determining the stability of the material in the $D0_{3}$ structure, with the simple-cubic “cage” formed by the X atoms being destabilized by the Y and Sb atoms at the cage centers if the Y or Sb atoms are either too large or too small compared to the natural size of the cage. With this in mind, we summarize our stability results in Table 1, arranging the X and Y atoms in order of their corresponding atomic radii. Confirming our simple hypothesis, indeed the table demonstrates that combinations with either very small or very large Y atoms and with either very small or very large X atoms can be unstable. The next section below provides a more quantitative analysis of these observations.

To begin the quantitative analysis, Fig. 2 displays the results of Table 1 in the phase space of the atomic radiiSlater (1964) of X ($r_{\mathrm{X}}$) and Y ($r_{\mathrm{Y}}$). As discussed above, we expect there to be upper and lower bounds of stability in terms of the relative size of the X atom to the Y and Sb atoms, corresponding to the ratios $r_{\mathrm{Y}}$/$r_{\mathrm{X}}$ and $r_{\mathrm{Sb}}$/$r_{\mathrm{X}}$, respectively. Therefore, in Fig. 2 we consider four perceptron lines, two corresponding to upper and lower bounds for the ratio $r_{\mathrm{Y}}$/$r_{\mathrm{X}}$ ($1.00<r_{\mathrm{Y}}$/$r_{\mathrm{X}}<1.55$) and two corresponding to upper and lower bounds for the ratio $r_{\mathrm{Sb}}$/$r_{\mathrm{X}}$ ($0.66<r_{\mathrm{Sb}}$/$r_{\mathrm{X}}<1.01$). We see that these four lines indeed separate the stable region from the unstable regions of the phase space, confirming that the relative size of the X atom to the Y and Sb atoms plays a key role in determining the stability of X₂YSb in the $D0_{3}$ structure.

Next, to simplify our results yet further and make them more analogous to the Goldschmidt tolerance factor for perovskites,Goldschmidt (1926) we consider the possibility of constructing one-dimensional tolerance factors. As discussed above, both ratios, $r_{\mathrm{Y}}$/$r_{\mathrm{X}}$ and $r_{\mathrm{Sb}}$/$r_{\mathrm{X}}$, of the stable materials are bounded. We therefore propose the following two simple tolerance factors for determination of stability: the arithmetic mean of the ratios, $t_{1}\equiv(r_{\mathrm{Y}}+r_{\mathrm{Sb}})/(2r_{\mathrm{X}})$, and the geometric mean of the ratios, $t_{2}\equiv\sqrt{r_{\mathrm{Y}}r_{\mathrm{Sb}}}/r_{\mathrm{X}}$. Fig. 3 shows that indeed, both of these Goldschmidt-like tolerance factors work well for all twenty-five materials in our study with the bounds $0.83<t_{1}<1.28$ and $0.81<t_{2}<1.25$, respectively.

Finally, we note that, from the above considerations, we expect the above size-based ratios to be useful as well for the more general class of X₂YM materials, where M can be any Group V metal or semimetal. Moreover, it is at least plausible that approximately the same numerical bounds for stability will hold when $r_{\mathrm{Sb}}$ is replaced with $r_{\mathrm{M}}$.

With the stability of the X₂YSb materials in the $D0_{3}$ structure understood, we next turn to the search for stable structures of the materials we find unstable. In this work, we consider one mono-alkali antimonide and one bi-alkali antimonide, specifically choosing Cs₃Sb and Cs₂KSb, which are technologically interesting for photoemissionCultrera et al. (2011) and photoabsorption,Ettema and de Groot (2002); Kalarasse et al. (2010) respectively. To search for stable structures of these materials, we begin by noting that the most unstable phonon modes occur at the X-point of the face-centered-cubic Brillouin zone (see Supplemental Materialref for phonon dispersion plots), which results in displacements that are commensurate with the conventional cubic unit cell (Fig. 1b).

Beginning with the simpler material, Cs₃Sb, we generate 20 different random perturbations of the atoms from the $D0_{3}$ structure in the conventional cubic cell and then allow the system to relax, as described in Sec. II. These 20 perturbed systems all relaxed into just two distinct structures, only one of which proves to exhibit no imaginary phonon frequencies and thus be mechanically stable. Figure 1c shows the displacements from the $D0_{3}$ positions for the final stable structure. The primary displacements occur for the Cs atoms occupying the Y sites in the X₂YSb $D0_{3}$ structure, which displace along a tetrahedral subset of the eight possible $\langle$111$\rangle$ directions. We also observe notably smaller displacements on the Sb atoms and on the Cs atoms occupying the X sites. Focusing on the largest, Y-site displacements, we note that this displacement pattern can be formed as a linear combination of crystal-symmetry respecting 90-degree rotations of the red phonon displacement pattern in Fig. 1b. Thus, the unstable phonon displacements at the X-point in the Brillouin zone of the $D0_{3}$ structure indeed lead toward the basin of attraction of a stable structure.

Next, we compare the expected peak locations for the powder X-ray diffraction (XRD) pattern from our predicted stable structure with our experimental XRD measurements. (Setup and details described elsewhere.Gaowei et al. (2019); Ding (2017)) We note that our experimental samples posses texture due to the specific growth method and conditions, which favor certain crystal orientations over others. Thus, the relative sizes of the resulting scattering peaks cannot be compared directly to our theoretical calculations, and, moreover, some peaks expected from the powder XRD may be missing from our experiments. Therefore, the most meaningful comparison to our theoretical results is to compare only the peak locations which we actually observe in the experiments. Accordingly, we interpret any measured peaks beyond the calculated $D0_{3}$ peaks to be experimental evidence of breaking of the $D0_{3}$ symmetry, but we do not necessarily expect to detect all peaks calculated from our predicted structures. Finally, to eliminate uncertainty due to subtle differences between density-functional theory and experimental lattice constants, we normalize both our theoretical and experimental peak locations so that the $hkl=111$ Bragg peak appears exactly at a plane separation of $1/\sqrt{3}$ times the cubic lattice constant $a$.

The first two columns of Table 2 show our measured XRD peaks for Cs₃Sb along with the cubic Miller indices $hkl$ corresponding to those peaks. The last two columns show our theoretical powder XRD peaks for Cs₃Sb in the $D0_{3}$ structure and the predicted stable structure, respectively. As expected, the experimental data does not exhibit clearly all of the theoretical peaks. The experiment does show an excellent match to the 222 peak expected for both the $D0_{3}$ and our predicted structure, confirming that the structure in our sample forms a cubic lattice. Furthermore, we note that the 210 peak does not appear in the $D0_{3}$ structure but does appear in our measurements, which clearly indicates symmetry breaking and is consistent with our theoretical findings that this material is not stable in the $D0_{3}$ structure. Finally, the predicted stable structure does show the 210 peak, consistent with our experiments.

Having considered the mono-alkali antimonide Cs₃Sb, we now consider the bi-alkali antimonide Cs₂KSb. Following the same procedure as for Cs₃Sb, we again find a single stable structure with no imaginary phonon frequencies. The resulting structure for Cs₂KSb (Fig. 1d shows nearly the same displacement pattern from the $D0_{3}$ structure as we found for Cs₃Sb, with the Y-site atoms (K) assuming the largest displacements (each along one of four tetrahedral $\langle 111\rangle$ directions, the pattern which can be anticipated from a displacement pattern [Fig. 1b, red arrows] from the most unstable phonon mode in the $D0_{3}$ structure), and both the Sb atoms and the X-site atoms (Cs) showing significantly smaller displacements. Moreover, as can be expected from the above, our theoretical and experimental X-ray diffraction results for this material are indeed very similar to those of Cs₃Sb (Table 2), so that our predicted stable structure is fully consistent with the structure observed in our experiments, which breaks the $D0_{3}$ symmetry and exhibits the 210 peak consistent with a cubic lattice.

Finally, as a control case, we have also performed XRD measurements on K₂CsSb, a promising electron emitterBazarov et al. (2011); Dunham et al. (2013); Schubert et al. (2016) which we predict to be stable in the $D0_{3}$ structure. We find that, indeed, the experimental peaks exhibited by this material are different from those found for Cs₃Sb and Cs₂KSb (Table 2) and that they are now fully consistent with expectations for the $D0_{3}$ structure. We thus find full consistency between our experimental measurements and our predictions for both $D0_{3}$-stable and $D0_{3}$-unstable materials.

Having confirmed experimentally that Cs₃Sb and Cs₂KSb indeed break $D0_{3}$ symmetry, we turn finally to consider the impact of the symmetry breaking on the optical properties of these materials, both of which are of interest for photo-applications.Cultrera et al. (2011); Ettema and de Groot (2002); Kalarasse et al. (2010) We find that, for both materials, the breaking of symmetry changes the nature of the gap from indirect to direct and lowers the direct optical gap significantly, from 1.53 eV to 1.18 eV and from 1.07 eV to 0.57 eV for Cs₃Sb and Cs₂KSb, respectively. (See Supplemental Materialref for more details.) Finally, Fig. 4 shows our ab initio predictions for the optical absorption coefficients for these two materials, where the lowering of the band gaps is evident in the reduction of the location of the absorption edge. Above the absorption edge, the absorption of both materials is relatively unaffected by the breaking of the $D0_{3}$ symmetry due to the fact that the symmetry-breaking displacements are rather small, making the changes in photoexcitation transition rates across the optical gap relatively weak.