Stability of the 1144 phase in iron pnictides

A series of iron arsenides (e.g., ${\text{CaRbFe}}_4{\text{As}}_4, {\text{SrCsFe}}_4{\text{As}}_4$) have been discovered recently, and have provoked a rise in superconductor searches in a different phase, known as the 1144 phase. For the presence of various chemical substitutions, it is believed that more 1144 compounds remain to be discovered. In this work, we perform general model analysis as well as scenario calculation on a basis of density functional theory to investigate phase stability in a variety of compounds. We predict that the 1144-type phase could be stabilized in ${\text{EuKFe}}_4{\text{As}}_4, {\text{EuRbFe}}_4{\text{As}}_4, {\text{EuCsFe}}_4{\text{As}}_4, {\text{CaCsFe}}_4{\text{P}}_4, {\text{SrCsFe}}_4{\text{P}}_4, {\text{BaCsFe}}_4{\text{P}}_4, {\text{InCaFe}}_4{\text{As}}_4, {\text{InSrFe}}_4{\text{As}}_4$, etc. Remarkably, it involves rare earths, trivalence elements (e.g., indium) and iron phosphides, which greatly expands the range of its existence and suggests a promising prospect for experimental synthesis. In addition, we find that the formation of many random doping compounds (e.g., ${\text{Ba}}_{0.5}{\text{Cs}}_{0.5}{\text{Fe}}_2{\text{As}}_2, {\text{Ba}}_{0.5}{\text{Rb}}_{0.5}{\text{Fe}}_2{\text{As}}_2$) is driven by entropy and could be annealed to a 1144-type phase. Eventually, we plot a phase diagram about two structural factors $\mathrm{\Delta }a$ and $\mathrm{\Delta }c$, giving a bird's-eye view of stability of various 1144 compounds.

Figure 1

(a) Crystal structure of 122/1144. If the $X$ and $Y$ sites are occupied by the same type of atoms, one obtains the 122 phase $X{\text{Fe}}_2{\text{As}}_2$; if $X\text{and}Y$ sites are occupied by different atoms, one obtains the 1144 phase $XY{\text{Fe}}_4{\text{As}}_4$. (b) Top view of a single Fe-As layer. The Fe atoms locate in a plane, forming a square lattice. Two types of arsenic atoms located above and below the Fe plane, as designated by 1 and 2. The arrows indicate spin directions, which are stripelike AFM. (c) Interlayer magnetic structure. Two neighboring Fe-As with AFM coupling.

Figure 2

(a) The 1144-122($s$) phase diagram in Ref. [2]. We also add Eu-contained combinations into the graph (black triangles). Each point represents a particular $X-Y$ combination. The parameter is defined as $\mathrm{\Delta }a=a_{X{\text{Fe}}_2{\text{As}}_2}-a_{Y{\text{Fe}}_2{\text{As}}_2}\mathrm{\Delta }c=c_{X{\text{Fe}}_2{\text{As}}_2}-c_{Y{\text{Fe}}_2{\text{As}}_2}$, where $a$ and $c$ are the lattice constants of the tetragonal 122 unit cell. Note that, to plot this figure, one needs only to know the lattice for two 122 phases, and no need for the lattice of 1144 phase. The combinations that have resulted in the 1144 phase in experiment are colored in red, while the ones of 122(s) phase are in blue. ${\text{BaCsFe}}_4{\text{As}}_4$ remains to be determined (shown as an orange star). It is clear that linear correlation does not hold between $\mathrm{\Delta }a$ and $\mathrm{\Delta }R$. (b) The linear correlation $\mathrm{\Delta }c=\gamma \mathrm{\Delta }R$, where $\gamma$ is about 4.0. The number notation for both (a) and (b) is 1-BaNa, 2-SrNa, 3-CaNa, 4-BaK, 5-SrK, 6-BaRb, 7-CaK, 8-SrRb, 9-SrCs, 10-CaRb, 11-CaCs, 12-BaCs.

Figure 3

(a) Schematics of 1144 structures. The $X$ boxes (red) and (blue) are shorthand representation for building blocks. (b) For 1144 phase, $X$ and $Y$ boxes are in different layers, thus the mismatch only occurs in $x-y$ plane. (c) For 122($s$) phase, mismatch occurs in both $x-y$ plane and $z$ axis. The probability for mismatch to occur is 1/2 due to a random distribution of $X-Y$ boxes.

Figure 4

The temperature dependence of $\mathrm{\Delta }G$. The dashed line is for free energy with configuration entropy only. The solid line includes the effect of vibration entropy and zero-point energy. These curves are calculated with theoretical lattice constants. Two critical temperatures ($\mathrm{\Delta }G=0)$ are denoted for each compound. Positive $\mathrm{\Delta }G$ is defined as favor the 1144 phase.

Figure 5

The AFM and FM spin configurations of ${\text{EuFe}}_2{\text{As}}_2$. The spin in Fe-As layer is same as Fig. 1, but not explicitly shown in the graph. The magnetic moment on each Eu is about $6.5{\mu }_B$ according to our calculations.

Figure 6

(a) A generalized phase diagram with new parameters: $\mathrm{\Delta }a=-|a_X-a_Y|, \mathrm{\Delta }c=|c_X-c_Y|$. Defining $\mathrm{\Delta }a$ as negative values is just to give a similar appearance as the original phase diagram [Fig. 2] for easy comparison. The color indicates the value of $\mathrm{\Delta }H$ (meV/atom). Spots above the line 1 are showing $X-Y$ combinations that have achieved the 1144 phase in experiment. Our calculation shows that the real boundary between 1144 and 122($s$) phases is around line 2, which means that combinations between line 1 and line 2 possibly form the 1144-type phase. (b) The stability of various 1144 structures. The size of hexagon is proportional to $\mathrm{\Delta }H$. Structures with negative $\mathrm{\Delta }H$ not shown. Evidently, the stability increases going from left bottom to right top. In the off-diagonal region, 1144 phase stability is poor, thus it is indicated as 122($s$) region.