Phase III Cd$_{2}$Re$_{2}$O$_{7}$ Structure: A2B7C2_tI44_98_f_acde_f-001
| Prototype | Cd$_{2}$O$_{7}$Re$_{2}$ |
| AFLOW prototype label | A2B7C2_tI44_98_f_acde_f-001 |
| ICSD | none |
| Pearson symbol | tI44 |
| Space group number | 98 |
| Space group symbol | $I4_122$ |
| AFLOW prototype command |
aflow --proto=A2B7C2_tI44_98_f_acde_f-001
--params=$a, \allowbreak c/a, \allowbreak z_{2}, \allowbreak x_{3}, \allowbreak x_{4}, \allowbreak x_{5}, \allowbreak x_{6}$ |
- Cd$_{2}$Re$_{2}$O$_{7}$ exhibits a number of phases. We will use the notation of (Kapcia, 2020) to describe them:
- Phase I: Above 200K the system takes on the cubic pyrochlore ($E8_{1}$) structure.
- Phase II: in the range 120-200K the system is in the tetragonal $I\overline{4}m2$ #119 structure.
- Phase III: in the range 80-120K the system is in the tetragonal $I4_{1}22$ #98 structure (this structure).
- Phase IV: (Kapcia, 2020) did a first-principles study of this system and found that below 80K Phase III develops a soft phonon mode which transforms the system into an orthorhombic $F222$ #22 structure.
- There are many issues with all of these structures (Norman, 2020):
- Phases II, III, and IV are all close to phase I. If we loosen the tolerance using AFLOW-SYM or FINDSYM the structures are seen to be equivalent to cubic pyrochlore.
- Using the default tolerance, Phase II and Phase IV are equivalent.
\[ \begin{array}{ccc}
\mathbf{a_{1}}&=&- \frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}\\\mathbf{a_{2}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}- \frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{2}c \,\mathbf{\hat{z}}\\\mathbf{a_{3}}&=&\frac{1}{2}a \,\mathbf{\hat{x}}+\frac{1}{2}a \,\mathbf{\hat{y}}- \frac{1}{2}c \,\mathbf{\hat{z}}
\end{array}\]
Basis vectors
| Lattice coordinates | Cartesian coordinates | Wyckoff position | Atom type | |||
|---|---|---|---|---|---|---|
| $\mathbf{B_{1}}$ | = | $0$ | = | $0$ | (4a) | O I |
| $\mathbf{B_{2}}$ | = | $\frac{3}{4} \, \mathbf{a}_{1}+\frac{1}{4} \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (4a) | O I |
| $\mathbf{B_{3}}$ | = | $z_{2} \, \mathbf{a}_{1}+z_{2} \, \mathbf{a}_{2}$ | = | $c z_{2} \,\mathbf{\hat{z}}$ | (8c) | O II |
| $\mathbf{B_{4}}$ | = | $\left(z_{2} + \frac{3}{4}\right) \, \mathbf{a}_{1}+\left(z_{2} + \frac{1}{4}\right) \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{y}}+c \left(z_{2} + \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (8c) | O II |
| $\mathbf{B_{5}}$ | = | $- \left(z_{2} - \frac{3}{4}\right) \, \mathbf{a}_{1}- \left(z_{2} - \frac{1}{4}\right) \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $\frac{1}{2}a \,\mathbf{\hat{y}}- c \left(z_{2} - \frac{1}{4}\right) \,\mathbf{\hat{z}}$ | (8c) | O II |
| $\mathbf{B_{6}}$ | = | $- z_{2} \, \mathbf{a}_{1}- z_{2} \, \mathbf{a}_{2}$ | = | $- c z_{2} \,\mathbf{\hat{z}}$ | (8c) | O II |
| $\mathbf{B_{7}}$ | = | $x_{3} \, \mathbf{a}_{1}+x_{3} \, \mathbf{a}_{2}+2 x_{3} \, \mathbf{a}_{3}$ | = | $a x_{3} \,\mathbf{\hat{x}}+a x_{3} \,\mathbf{\hat{y}}$ | (8d) | O III |
| $\mathbf{B_{8}}$ | = | $- x_{3} \, \mathbf{a}_{1}- x_{3} \, \mathbf{a}_{2}- 2 x_{3} \, \mathbf{a}_{3}$ | = | $- a x_{3} \,\mathbf{\hat{x}}- a x_{3} \,\mathbf{\hat{y}}$ | (8d) | O III |
| $\mathbf{B_{9}}$ | = | $\left(x_{3} + \frac{3}{4}\right) \, \mathbf{a}_{1}- \left(x_{3} - \frac{1}{4}\right) \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $- a x_{3} \,\mathbf{\hat{x}}+a \left(x_{3} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (8d) | O III |
| $\mathbf{B_{10}}$ | = | $- \left(x_{3} - \frac{3}{4}\right) \, \mathbf{a}_{1}+\left(x_{3} + \frac{1}{4}\right) \, \mathbf{a}_{2}+\frac{1}{2} \, \mathbf{a}_{3}$ | = | $a x_{3} \,\mathbf{\hat{x}}- a \left(x_{3} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (8d) | O III |
| $\mathbf{B_{11}}$ | = | $x_{4} \, \mathbf{a}_{1}- x_{4} \, \mathbf{a}_{2}$ | = | $- a x_{4} \,\mathbf{\hat{x}}+a x_{4} \,\mathbf{\hat{y}}$ | (8e) | O IV |
| $\mathbf{B_{12}}$ | = | $- x_{4} \, \mathbf{a}_{1}+x_{4} \, \mathbf{a}_{2}$ | = | $a x_{4} \,\mathbf{\hat{x}}- a x_{4} \,\mathbf{\hat{y}}$ | (8e) | O IV |
| $\mathbf{B_{13}}$ | = | $- \left(x_{4} - \frac{3}{4}\right) \, \mathbf{a}_{1}- \left(x_{4} - \frac{1}{4}\right) \, \mathbf{a}_{2}- \left(2 x_{4} - \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $- a x_{4} \,\mathbf{\hat{x}}- a \left(x_{4} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (8e) | O IV |
| $\mathbf{B_{14}}$ | = | $\left(x_{4} + \frac{3}{4}\right) \, \mathbf{a}_{1}+\left(x_{4} + \frac{1}{4}\right) \, \mathbf{a}_{2}+\left(2 x_{4} + \frac{1}{2}\right) \, \mathbf{a}_{3}$ | = | $a x_{4} \,\mathbf{\hat{x}}+a \left(x_{4} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+\frac{1}{4}c \,\mathbf{\hat{z}}$ | (8e) | O IV |
| $\mathbf{B_{15}}$ | = | $\frac{3}{8} \, \mathbf{a}_{1}+\left(x_{5} + \frac{1}{8}\right) \, \mathbf{a}_{2}+\left(x_{5} + \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $a x_{5} \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (8f) | Cd I |
| $\mathbf{B_{16}}$ | = | $\frac{7}{8} \, \mathbf{a}_{1}- \left(x_{5} - \frac{1}{8}\right) \, \mathbf{a}_{2}- \left(x_{5} - \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $- a x_{5} \,\mathbf{\hat{x}}+\frac{3}{4}a \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (8f) | Cd I |
| $\mathbf{B_{17}}$ | = | $\left(x_{5} + \frac{7}{8}\right) \, \mathbf{a}_{1}+\frac{1}{8} \, \mathbf{a}_{2}+\left(x_{5} + \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{4}a \,\mathbf{\hat{x}}+a \left(x_{5} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (8f) | Cd I |
| $\mathbf{B_{18}}$ | = | $- \left(x_{5} - \frac{7}{8}\right) \, \mathbf{a}_{1}+\frac{5}{8} \, \mathbf{a}_{2}- \left(x_{5} - \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}- a \left(x_{5} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (8f) | Cd I |
| $\mathbf{B_{19}}$ | = | $\frac{3}{8} \, \mathbf{a}_{1}+\left(x_{6} + \frac{1}{8}\right) \, \mathbf{a}_{2}+\left(x_{6} + \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $a x_{6} \,\mathbf{\hat{x}}+\frac{1}{4}a \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (8f) | Re I |
| $\mathbf{B_{20}}$ | = | $\frac{7}{8} \, \mathbf{a}_{1}- \left(x_{6} - \frac{1}{8}\right) \, \mathbf{a}_{2}- \left(x_{6} - \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $- a x_{6} \,\mathbf{\hat{x}}+\frac{3}{4}a \,\mathbf{\hat{y}}+\frac{1}{8}c \,\mathbf{\hat{z}}$ | (8f) | Re I |
| $\mathbf{B_{21}}$ | = | $\left(x_{6} + \frac{7}{8}\right) \, \mathbf{a}_{1}+\frac{1}{8} \, \mathbf{a}_{2}+\left(x_{6} + \frac{1}{4}\right) \, \mathbf{a}_{3}$ | = | $- \frac{1}{4}a \,\mathbf{\hat{x}}+a \left(x_{6} + \frac{1}{2}\right) \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (8f) | Re I |
| $\mathbf{B_{22}}$ | = | $- \left(x_{6} - \frac{7}{8}\right) \, \mathbf{a}_{1}+\frac{5}{8} \, \mathbf{a}_{2}- \left(x_{6} - \frac{3}{4}\right) \, \mathbf{a}_{3}$ | = | $\frac{1}{4}a \,\mathbf{\hat{x}}- a \left(x_{6} - \frac{1}{2}\right) \,\mathbf{\hat{y}}+\frac{3}{8}c \,\mathbf{\hat{z}}$ | (8f) | Re I |
References
- S.-W. Huang, H.-T. Jeng, J.-Y. Lin, W. J. Chang, J. M. Chen, G. H. Lee, H. Berger, H. D. Yang, and K. S. Liang, Electronic structure of pyrochlore Cd$_{2}$Re$_{2}$O$_{7}$, Journal of Physics: Condensed Matter 21, 195602 (2009), doi:10.1088/0953-8984/21/19/195602.
- K. J. Kapcia, M. Reedyk, M. Hajialamdari, A. Ptok, P. Piekarz, A. Schulz, F. S. Razavi, R. K. Kremer, and A. M. Oleś, Discovery of a low-temperature orthorhombic phase of the Cd$_{2}$Re$_{2}$O$_{7}$ superconductor, Phys. Rev. Res. 2, 033108 (2020), doi:10.1103/PhysRevResearch.2.033108.
Found in
- M. R. Norman, Crystal structure of the inversion-breaking metal Cd$_{2}$Re$_{2}$O$_{7}$, Phys. Rev. B 101, 045117 (2020), doi:10.1103/PhysRevB.101.045117.
Prototype Generator
aflow --proto=A2B7C2_tI44_98_f_acde_f --params=$a,c/a,z_{2},x_{3},x_{4},x_{5},x_{6}$