Manipulation of $J_{\mathrm{eff}}=\frac{3}{2}$ states by tuning the tetragonal distortion

The spin-orbit entangled quantum states in $4d/5d$ compounds, e.g., the $J_{\mathrm{eff}}=\frac{1}{2}$ and $J_{\mathrm{eff}}=\frac{3}{2}$ states, have attracted great interests for their unique physical roles in unconventional superconductivity and topological states. Here, the key role of tetragonal distortion is clarified, which determines the ground states of $4d^1/5d^1$ systems to be the $J_{\mathrm{eff}}=\frac{3}{2}$ one (e.g., ${\mathrm{K}}_2{\mathrm{NbCl}}_6$) or the $S=\frac{1}{2}$ one (e.g., ${\mathrm{Rb}}_2{\mathrm{NbCl}}_6$). By tuning the tetragonal distortion via epitaxial strain, the occupation weights of $d_{xy}/d_{yz}/d_{xz}$ orbitals can be subtly modulated, competing with the spin-orbit coupling. Consequently, quantum phase transitions between the $S=\frac{1}{2}$ state and the $J_{\mathrm{eff}}=\frac{3}{2}$ state, as well as between different $J_{\mathrm{eff}}=\frac{3}{2}$ states, can be achieved, resulting in significant changes of local magnetic moments. Our prediction points out a reliable route to engineer new functionality of $J_{\mathrm{eff}}$ states in these quantum materials.

Figure 1

Schematic of the collaborative effect of SOC and $Q_3$ mode distortion for the $4d^1/5d^1$ electron states in the octahedral crystal field. Here, $Q_3$ is defined as $(2l_z-l_x-l_y)/\sqrt{6}$, where $l$ denotes the bond length along a particular axis [34]. For an undistorted octahedron (i.e., $Q_3=0$), the SOC leads to four degenerate low-lying $J_{\mathrm{eff}}=\frac{3}{2}$ levels (${\mathrm{\Phi }}_1$, ${\mathrm{\Phi }}_2$, ${\mathrm{\Phi }}_3$, and ${\mathrm{\Phi }}_4$) and high-lying $J_{\mathrm{eff}}=\frac{1}{2}$ doublets (not shown here). For the $d^1$ configuration, the $J_{\mathrm{eff}}=\frac{3}{2}$ levels are quarter-filled. For nonzero $Q_3$, the $J_{\mathrm{eff}}=\frac{3}{2}$ levels are split into two groups: ${\mathrm{\Phi }}_1/{\mathrm{\Phi }}_2$ and ${\mathrm{\Phi }}_3^{\prime }/{\mathrm{\Phi }}_4^{\prime }$, the latter of which are mixed with the $J_{\mathrm{eff}}=\frac{1}{2}$ states, more or less. Once this mixture is heavy, ${\mathrm{\Phi }}_3^{\prime }/{\mathrm{\Phi }}_4^{\prime }$ are close to the $S=\frac{1}{2}$ states. The sign of $Q_3$ determines whether ${\mathrm{\Phi }}_1$ (${\mathrm{\Phi }}_2$) or ${\mathrm{\Phi }}_3^{\prime }$ (${\mathrm{\Phi }}_4^{\prime }$) is the lowest-energy one.

Figure 2

Numerical results of the atomic orbital model to demonstrate the scenario proposed in Fig. 1. (a) Eigenenergies, (b) $t_{2\mathrm{g}}$ orbital weights, and (c) spin ($m_s$) and orbital ($m_l$) magnetic moments (along the $z$ axis), as a function of $Q_3$ intensity, characterized by $\omega$. The $\omega$ and eigenenergy are in units of $\lambda$, the SOC coefficient. In (b), the red and blue symbols are for ${\mathrm{\Phi }}_3^{\prime }$ (${\mathrm{\Phi }}_4^{\prime }$) and ${\mathrm{\Phi }}_5^{\prime }$ (${\mathrm{\Phi }}_6^{\prime }$), respectively. $m_n$: net magnetic moment ($=m_l+m_s$).

Figure 3

(a)–(c) Schematic crystal structures of $A_{2}M{\mathrm{Cl}}_6$. (b) Top view of $A$ = Rb, space group $I4/mmm$ (No. 139). (c) Top view of $A$ = K, space group $P4/mnc$ (No. 128). The smaller K ion leads to a smaller in-plane lattice constant, which results in the elongation (i.e., $Q_3>0$) and rotation of octahedra. Since these octahedra are almost isolated, the effect of octahedral rotation to the electronic states is negligible. (d)–(g) Comparison of density of states (DOS) between the $\mathrm{GGA}+U$ and $\mathrm{GGA}+U+\mathrm{SOC}$ calculations. Here, $U_{\mathrm{eff}}=1$ and 0.7 eV are applied on Nb's $4d$ and Ta's $5d$ orbitals, respectively. Insets: the electron clouds of valence bands near the Fermi level, i.e., the $d^1$ electrons.

Figure 4

Biaxial strain tuned electronic state transitions between $S=\frac{1}{2}$ and $J_{\mathrm{eff}}=\frac{3}{2}$ (${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_3$) in (a), (b) ${\mathrm{Rb}}_2{\mathrm{NbCl}}_6$ and (c), (d) ${\mathrm{Rb}}_2{\mathrm{TaCl}}_6$. (a) and (c) The band gaps (with/without SOC). (b) and (d) The magnetic moments of a Nb/Ta ion (with SOC). The boundary $Q_3=0$ (vertical broken line) locates at $\varepsilon \sim -3.8\%$ and $\varepsilon \sim -3.0\%$, respectively. Insets: the electron clouds of valence bands near the Fermi levels.