Strong tuning of magnetism and electronic structure by spin orientation

To efficiently manipulate magnetism is a key physical issue for modern condensed matter physics, which is also crucial for magnetic functional applications. Most previous relevant studies rely on the tuning of spin texture, while the spin orientation is often negligible. As an exception, spin-orbit coupled $J_{\mathrm{eff}}$ states of $4d/5d$ electrons provide an ideal platform for emergent quantum effects. However, many expectations have not been realized due to the complexities of real materials. Thus the pursuit for more ideal $J_{\mathrm{eff}}$ states remains ongoing. Here a near-ideal $J_{\mathrm{eff}}=3/2$ Mott-insulating phase is predicted in the family of hexachloro niobates, which avoid some common drawbacks of perovskite oxides. The local magnetic moment is nearly compensated between spin and orbital components, rendering exotic recessive magnetism. More interestingly, the electronic structure and magnetism can be strongly tuned by rotating spin axis, which is rare but crucial for spintronic applications.

Figure 1

(a) Splitting of $d$ orbitals by octahedral crystal field and SOC. The crystal field leads to three low-lying $t_{2\mathrm{g}}$ levels and two higher energy $e_{\mathrm{g}}$ doublets. The SOC further splits the $t_{2\mathrm{g}}$ triplets into two $J_{\mathrm{eff}}$ states: $1/2$ and $3/2$ states. The corresponding orbital shapes are presented. Further considering the spin degeneracy, all these states are duplicated, i.e., two $J_{\mathrm{eff}}=1/2$ and four $J_{\mathrm{eff}}=3/2$ states. (b), (c) Crystal structure of ${\mathrm{K}}_2{\mathrm{NbCl}}_6$ (space group No. 128 $P4/mnc$). (b) Side view. (c) Top view. $\alpha$ denotes the ${\mathrm{NbCl}}_6$ octahedral rotation angle with respect to the [110] axis. Green: K; blue: Nb; red: Cl.

Figure 2

Total density of states (DOS) (gray) and atom-projected DOS (cyan) of ${\mathrm{K}}_2{\mathrm{NbCl}}_6$ near the Fermi level. (a) Calculated using generalized gradient approximation (GGA); (b) $\text{GGA}+U$; (c) $\text{GGA}+\text{SOC}$; (d) $\text{GGA}+U+\text{SOC}$. The Fermi energy is positioned at zero. In (b) and (d), $U_{\mathrm{eff}}=1$ eV is applied on Nb's $4d$ orbitals. Inset: the electron cloud of valence bands near the Fermi level. (e) The local spin (${\mathbf{m}}_s$), orbital (${\mathbf{m}}_l$), and net magnetic moments (${\mathbf{m}}_n={\mathbf{m}}_s+{\mathbf{m}}_l$) of a Nb ion as a function of $U_{\mathrm{eff}}$. The value of ${\mathbf{m}}_n$ is small (ideally to be zero) for the $J_{\mathrm{eff}}=3/2 {\mathrm{\Phi }}_1$ state. When $U_{\mathrm{eff}}$ is small, the Hubbard splitting among ${\mathrm{\Phi }}_i$'s ($i=1–4$) are not sufficient, which reduces the orbital magnetization of occupied state.

Figure 3

(a) The band gap with SOC as a function of $U_{\mathrm{eff}}$ for $\mathrm{spin}//c$ and $\mathrm{spin}//a$, respectively. The metal-insulator transition (MIT) can occur by rotating the spin orientation in the middle $U_{\mathrm{eff}}$ region. (b) The band gap with SOC as a function of polar angle $\theta$ for $U_{\mathrm{eff}}=1$ eV and $U_{\mathrm{eff}}=0.6$ eV, respectively. Here, $\theta =0^{\circ }$ and $\theta ={90}^{\circ }$ stand for $\mathrm{spin}//c$ and $\mathrm{spin}//a$, respectively. Inset: sketch of spin rotation driven by magnetic field. (c), (d) The band structures with SOC for $U_{\mathrm{eff}}=1$ eV: (c) $\mathrm{spin}//c$; (d) $\mathrm{spin}//a$. Insets: the corresponding occupied orbital shapes.

Figure 4

(a) Schematic function of spin-orientation control of magnetism. For pure spin systems (left), when a spin rotates, the amplitude of its magnetic moment ($m_s$) is a constant and the electronic structure will be unchanged. For those SOC $J_{\mathrm{eff}}=3/2$ systems (right), the amplitude of magnetic moment $m_{\text{tot}}$ (contributed by both spin $m_s$ and orbital $m_l$ components) can be tuned accompanying the rotation of spin orientation, which is originated from the change of electronic structure. (b) Magnetocrystalline anisotropy. The energy for spin//$c$ is taken as a reference. Upper axis: rotation of azimuthal angle. Lower axis: rotation of polar angle. (c) The $a/c$ components of local spin and orbital moments (i.e., ${\mathbf{m}}_s$ and ${\mathbf{m}}_l$) as a function of polar angle at $U_{\mathrm{eff}}=1$ eV.