Frustration enhanced by Kitaev exchange in a ${\tilde{j}}_{\text{eff}}=\frac{1}{2}$ triangular antiferromagnet

Triangular Heisenberg antiferromagnets are prototypes of geometric frustration, even if for nearest-neighbor interactions quantum fluctuations are not usually strong enough to destroy magnetic ordering: stronger frustration is required to stabilize a spin-liquid phase. On the basis of static magnetization and electron spin resonance measurements, we demonstrate the emergence of ${\tilde{j}}_{\text{eff}}=\frac{1}{2}$ moments in the triangular-lattice magnet ${\mathrm{Na}}_2\mathrm{Ba}\mathrm{Co}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$. These moments are subject to an extra source of frustration that causes magnetic correlations to set in far above both the magnetic ordering and Weiss temperatures. Corroborating the ${\tilde{j}}_{\text{eff}}=\frac{1}{2}$ ground state, theory identifies ferromagnetic Kitaev exchange anisotropy as an additional frustrating agent, altogether putting forward ${\mathrm{Na}}_2\mathrm{Ba}\mathrm{Co}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$ as a promising Kitaev spin-liquid material.

Figure 1

(a) Energy diagram showing the splitting of the ${}^{4}F$ state of ${\mathrm{Co}}^{2+}$ in a cubic crystal field, the SO coupling, trigonal distortion, and the Zeeman splitting due to the magnetic field $B_0$ in the ${}^{4}T_{1g}$ multiplet. (b) Fragment of a triangular layer in the ${\mathrm{Na}}_2\mathrm{Ba}\mathrm{Co}{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$ crystal structure, with $\mathrm{Co}{\mathrm{O}}_6$ octahedra bridged by $\mathrm{P}{\mathrm{O}}_4$ tetrahedra. For a pair of neighboring Co atoms, the leading contribution to hopping stems from the depicted Wannier functions. Note that the respective Co $d$ orbitals lie in the same plane.

Figure 2

(a) Main panel: Powder $M\left(H\right)$ dependence at $T$ = 2 K (data points). A delayed increase of the $M\left(H\right)$ curve compared to the Brillouin function of noninteracting spins $S=1/2$ (dashed curve) indicates a sizable AFM interaction. Accounting for it, as explained in the text, yields a good agreement with the experiment (red solid curve). Inset: Single-crystalline $M\left(H\right)$ dependence for the $\mathbf{H}\parallel c$ axis and $\mathbf{H}\parallel ab$ plane. (b) Powder ${\chi }^{-1}\left(T\right)$ dependence at ${\mu }_{0}H$ = 20 mT (data points). The solid line depicts the calculated curve according to the population model (see the text).

Figure 3

(a) Single-crystalline ESR signals (field derivative of the microwave absorption) at $\nu =9.6$ GHz for the $\mathbf{H}\parallel c$ axis at selected temperatures. Inset: Angular dependence of the $g$ factor at $\nu =9.6$ GHz and at $T=4$ K (data points). The solid line is a fit to the dependence $g\left(\alpha \right)={[g_{\text{ab}}^2{cos}^2\left(\alpha \right)+g_{\text{c}}^2{sin}^2\left(\alpha \right)]}^{1/2}$. (b) $T$ dependence of the ESR linewidth of the single- and polycrystalline samples at $\nu =9.6$ GHz (data points). Solid lines are the calculated dependences according to the population model (see the text). (c) Left scale: $\nu$ vs field position of the peak of the powder HF-ESR spectrum at $T=3$ K (dots). The solid line is the fit to the dependence $h\nu =g{\mu }_{\text{B}}H$. Right scale: Powder HF-ESR spectra at selected frequencies $\nu$ at $T=3$ K and at selected temperatures at $\nu =708$ GHz.

Figure 4

Left: Full relativistic GGA band structure (thin black curves) and the eigenvalues of the Fourier-transformed Wannier Hamiltonian (thick violet curves). All bands are doubly degenerate due to the time-reversal symmetry. Minute dispersions along $\mathrm{\Gamma }$-Z are indicative of a quasi-two-dimensional electronic structure. Right: Total density of states (DOS) and Co $3d$ contributions. The Fermi level is at zero energy.