Kondo-Ising and tight-binding models for ${\mathrm{TmB}}_4$

In ${\mathrm{TmB}}_4$, localized electrons with a large magnetic moment interact with metallic electrons in boron-derived bands. We examine the nature of ${\mathrm{TmB}}_4$ using full-relativistic ab initio density functional theory calculations, approximate tight-binding Hamiltonian results, and the development of an effective Kondo-Ising model for this system. Features of the Fermi surface relating to the anisotropic conduction of charge are discussed. The observed magnetic moment $\sim 6{\mu }_B$ is argued to require a subtle crystal field effect in metallic systems, involving a flipped sign of the effective charges surrounding a Tm ion. The role of on-site quantum dynamics in the resulting Kondo-Ising type “impurity” model are highlighted. From this model, elimination of the conduction electrons will lead to spin-spin (RKKY-type) interaction of Ising character required to understand the observed fractional magnetization plateaus in ${\mathrm{TmB}}_4$.

Figure 1

(a) Figure 1 shows the structure of a single plane of $\mathrm{Tm}$ atoms within crystalline ${\mathrm{TmB}}_4$. Note the triangle and square tiling. (b) Figure 1 shows the juxtaposition of a plane of $\mathrm{Tm}$ atoms (red) and a plane of B (blue and gray) atoms as viewed from above. The $\mathrm{Tm}$ plane is .20 nm above the boron atoms plane. Coupling of the Tm $f$ orbital to the planar boron $2p_z$ band plays an essential role in the physics and phenomenology of the system. Note the 7 atom boron rings. (c) In Fig. 1 only the dimer boron atoms and the Tm atoms are shown. This figure shows the most simplified structure able to capture the essence of the magnetism and coupling in ${\mathrm{TmB}}_4$. (d) Figure 1 shows the structure of ${\mathrm{TmB}}_4$ as viewed from the side (along the $a$ axis). One sees the stacking of the $\mathrm{Tm}$ layers in red, the blue and gray boron planes, and in addition the chain formed by boron octahedra oriented along the $c$ axis. The apical boron are colored gray.

Figure 2

This shows Tm (red) and its local environment. The blue spheres indicate the dimer boron, while the gray spheres indicate the boron that participate in the formation of octahedra.

Figure 3

(a) Band structure for ${\mathrm{TmB}}_4$ from the DFT calculation, which shows dispersion in the $a\text{−}b$ plane. The inset shows the path taken in the BZ, $\mathrm{\Gamma }\text{−}X\text{−}M\text{−}\mathrm{\Gamma }$. Red indicates Tm $|J=7/2\rangle$ weights, while blue indicates dimer B $|J=3/2,m_J=\pm 1/2\rangle$ weights. We have circled a region that shows strong $4f\text{−}2p_z$ hybridization, which occurs at the $M$ point, with a magenta circle. There are electron pockets around $\mathrm{\Gamma },X,M$, and along the $M\text{−}\mathrm{\Gamma }$ path. (b) Dispersion along the $c$ axis, where the inset shows the path taken in the BZ, $\mathrm{\Gamma }\text{−}Z\text{−}R\text{−}A$. There are electron pockets around the $Z$ point and hole pockets along the $\mathrm{\Gamma }\text{−}Z$ path. $4f\text{−}2p_z$ hybridization can be seen along the $\mathrm{\Gamma }\text{−}Z$ path and around the $Z$ point. There is some $5d\text{−}2p_z$ hybridization around the $Z$ point (not shown).

Figure 4

Fermi surface from the DFT calculation, where the color indicates the magnitude of the Fermi velocity, as indicated by the color map. In (a), we show the full Fermi surface, centered at the $\mathrm{\Gamma }$ point. Surfaces with an exterior blue gradient represent electronlike surfaces, while surfaces with magenta exteriors are holelike surfaces: Thus, the large (magenta) surface along the $\mathrm{\Gamma }\text{−}Z$ path is a quasi one-dimensional hole surface since it is flatter in the $x\text{−}y$ directions compared to $z$, while the others are electron surfaces. We have taken $x\text{−}y$ cross sections of the electronlike surface at the $\mathrm{\Gamma }$ point and of the holelike surface at $2Z/5$, where the cross sections have been given some height in the $z$ direction to show the curvature. In (b) and (c), we focus on a single hybridized $4f\text{−}2p_z$ band, centered at (b) the $M$ point and (c) the $Z$ point to better showcase the symmetry of the hybridized surfaces. The electron surface centered in (c) shows quasi-2D conduction, with a planar effective mass ratio of $|\frac{m_{xx}}{m_{zz}}|=0.067$ near the tip of the surface. For the quasi-1D surface, we estimate an $m_{zz}$ of $0.46m_e$ near $2Z/5$, while for the quasi-2D surface, we estimate an $m_{xx}$ of $0.29m_e$ near the tip. We also estimate carrier concentrations of 0.05, 0.04, 0.01, and 0.006 per cell for the four hole surfaces, and 0.006 and 0.004 per cell for the $Z$ point surfaces. The volume of the unit cell is $1.98·{10}^{-28}{m}^3$.

Figure 5

Band structure of a simplified tight-binding model, where the apical boron has been removed. Tight-binding band dispersion is calculated for $t=-1.0$ eV, $V=-0.04$ eV, and $T=-0.01$ eV with site energies of 1.95 eV for the planar B and 0.1 eV for the Tm $f$ orbital. Red lines reflect contributions from the $\mathrm{Tm}$ $f$ eigenvector, while blue and gray indicate dimer and octahedral boron, respectively. The analog of the two hybrid bands from the DFT calculation can be seen below the Fermi level $\left(E_F=0\right)$ at the $M$ point, circled in magenta. The inset shows the unit cell for this simplified model.

Figure 6

Band structure of the further simplified tight-binding model, where the octahedral boron have been removed. Tight-binding band dispersion is calculated for $t=-1.0$ eV, $V=-0.04$ eV, $t_2=-0.75$ eV, and $T=-0.01$ eV with site energies of $-1.1$ eV for the planar B and 0.1 eV for the Tm $f$ orbital. The analog of the two hybrid bands from the DFT calculation can be seen mixing at the $M$ point near the Fermi level $\left(E_F=0\right)$, circled in magenta. The inset shows the unit cell for this further simplified model.

Figure 7

The full spectrum of $H_{CFZ}$ the crystal field plus Zeeman Hamiltonian with magnetic field along $z$ axis (panels at left), or along $x$ axis (panels at right) from Eq. (8) with $c_0=.01$ meV, and $c_1=0$ meV (top left and top right) or $c_1=0.02$ meV (bottom left and bottom right). The top-left panel illustrates the effective Ising nature of the Tm moment in the $z$ direction with a linear evolution of the lowest energy with field $B\left|\right|z$ over a wide range. We see here the necessity for choosing $c_0>0$; inverting its sign also inverts the spectrum and the ground state then would be non-Ising-like for fields $B_Z\lesssim 5$ Tesla. The top-right panel shows that a field along the $x$ axis $B\left|\right|x$, the lowest level is not linear, but for fields around 6 T the linearity is recovered with a slope (i.e., the magnetic moment) $\sim 6{\mu }_B$. The two lower panels have a nonzero value of the transverse terms, i.e., $c_1\sim 0.02$ meV. This does not create a noticeable gap in the level crossing at zero field in the $z$ direction, while a larger value would do so. It thus appears that we can ignore $c_1$ for many (but not all) purposes, as explained in the text.