Combinatorial exploration of quantum spin liquid candidates in the herbertsmithite material family

Geometric frustration of magnetic ions can lead to a quantum spin liquid ground state where long-range magnetic order is avoided despite strong exchange interactions. The physical realization of quantum spin liquids comprises a major unresolved area of contemporary materials science. One prominent magnetically frustrated structure is the kagome lattice. The naturally occurring minerals herbertsmithite $\left[\mathrm{Zn}{\mathrm{Cu}}_3{\left(\mathrm{O}\mathrm{H}\right)}_6{\mathrm{Cl}}_2\right]$ and Zn-substituted barlowite $\left[\mathrm{Zn}{\mathrm{Cu}}_3{\left(\mathrm{O}\mathrm{H}\right)}_6\mathrm{Br}\mathrm{F}\right]$ both feature perfect kagome layers of spin-$1/2$ copper ions and display experimental signatures consistent with a quantum spin liquid state at low temperatures. To investigate other possible candidates within this material family, we perform a systematic first-principles combinatorial exploration of structurally related compounds [$A{\mathrm{Cu}}_3{\left(\mathrm{O}\mathrm{H}\right)}_6{B}_2$ and $A{\mathrm{Cu}}_3{\left(\mathrm{O}\mathrm{H}\right)}_{6}BC$] by substituting nonmagnetic divalent cations ($A$) and halide anions ($B, C$). After optimizing such structures using density functional theory, we compare various structural and thermodynamic parameters to determine which compounds are most likely to favor a quantum spin liquid state. Convex hull calculations using binary compounds are performed to determine feasibility of synthesis. We also estimate the likelihood of interlayer substitutional disorder and spontaneous distortions of the kagome layers. After considering all of these factors as a whole, we select several promising candidate materials that we believe deserve further attention.

Figure 1

Crystal structures of herbertsmithite and Zn-barlowite. (a) Herbertsmithite viewed along the $c$ axis, showing the kagome arrangement of Cu ions. (b) Zn-barlowite viewed along the $c$ axis. (c) Herbertsmithite viewed along the [110] direction, showing the shifted stacking arrangement of the kagome layers. (d) Zn-barlowite viewed along [110], showing the stacking of the kagome layers and the inequivalence of the Br and F sites.

Figure 2

Structural stability and thermodynamics of candidate compounds. (a) Lowest optical phonon frequency for herbertsmithite-related candidates. (b) Lowest optical phonon frequency for Zn-barlowite-related candidates. (c) Convex hull energies for herbertsmithite-related candidates. (d) Convex hull energies for Zn-barlowite-related candidates. Structurally unstable compounds (identified by $f_0<0$) are denoted with an X. Cations are shown on the vertical axis and anions on the horizontal axis, in order of increasing ionic radius from bottom to top and left to right, respectively. The reference compound (either herbertsmithite or Zn-barlowite) is shown in white and marked with an asterisk. Compounds with parameter values more favorable than the reference compounds are shown with warm colors, and values less favorable are shown with cool colors.

Figure 3

Structural properties of candidate compounds. (a) Cu-O-Cu bond angle for herbertsmithite-related candidates. (b) Cu-O-Cu bond angle for Zn-barlowite-related candidates. (c) Interplane kagome distance for herbertsmithite-related candidates. (d) Interplane kagome distance for Zn-barlowite-related candidates. Structurally unstable compounds are denoted with an X. Cations are shown on the vertical axis and anions on the horizontal axis, in order of increasing ionic radius from bottom to top and left to right, respectively. The reference compound (either herbertsmithite or Zn-barlowite) is shown in white and marked with an asterisk. Compounds with parameter values more favorable than the reference compounds are shown with warm colors, and values less favorable are shown with cool colors.

Figure 4

Dependence of structural properties on ion size. (a) Cu-O-Cu bond angle versus anion radius. For Zn-barlowite, the radius plotted is that of the most electronegative anion. Blue (red) traces correspond to Zn-barlowite (herbertsmithite) relatives. Different cations are plotted as separate traces where darker (lighter) traces correspond to smaller (larger) ion sizes. (b) Interplane kagome distance versus cation radius for herbertsmithite (red) and Zn-barlowite (blue) relatives. Separate traces are plotted for each anion, where small (large) anions are plotted in dark (light) shades. (c) Cu-O-Cu bond angle versus the anion $B$ to anion $C$ ratio for stable compounds. Separate traces are plotted for different cations. (d) $c$-axis length versus cation size (left, dashed line) and $a$-axis length versus anion size (right, solid lines). (e) Frequency of the lowest optical phonon mode versus $c$-axis length for Zn-barlowite (blue) and herbertsmithite (red) relatives. Stable (unstable) compounds are shown with filled (empty) markers. The group IV elements that are less stable in the $2+$ oxidation state (Ge, Sn) are plotted with darker colors because they are almost always unstable, regardless of their $c$-axis length. (f) Cu-O-Cu bond angle versus $a$-axis length for Zn-barlowite (blue) and herbertsmithite (red) relatives. Stable (unstable) compounds are shown with filled (empty) markers. Compounds containing Ge and Sn cations (shown in darker colors) tend to have much smaller bond angles.

Figure 5

Phonon dispersions of final candidates. (a) The phonon dispersion for $\mathrm{Mg}{\mathrm{Cu}}_3{\left(\mathrm{O}\mathrm{H}\right)}_6{\mathrm{Br}}_2$ (blue) overlaid with the reference dispersion for herbertsmithite (gray). (b) The phonon dispersion for $\mathrm{Ca}{\mathrm{Cu}}_3{\left(\mathrm{O}\mathrm{H}\right)}_6\mathrm{I}\mathrm{Cl}$ (blue) overlaid with the reference dispersion for Zn-barlowite (gray). (c) The phonon dispersion for $\mathrm{Mg}{\mathrm{Cu}}_3{\left(\mathrm{O}\mathrm{H}\right)}_6\mathrm{Cl}\mathrm{F}$ (blue) overlaid with the reference dispersion for Zn-barlowite (gray). The absence of imaginary phonon frequencies in all three cases confirms the structural stability of these candidate compounds.