Layered Kondo Lattice Model for Quantum Critical $\beta \mathrm{\text{−}}{\mathrm{YbAlB}}_4$

We analyze the magnetic and electronic properties of the quantum critical heavy fermion superconductor $\beta \mathrm{\text{−}}{\mathrm{YbAlB}}_4$, calculating the Fermi surface and the angular dependence of the extremal orbits relevant to the de Haas–van Alphen measurements. Using a combination of the realistic materials modeling and single-ion crystal field analysis, we are led to propose a layered Kondo lattice model for this system, in which two-dimensional boron layers are Kondo coupled via interlayer Yb moments in a $J_z=\pm 5/2$ state. This model fits the measured single-ion magnetic susceptibility and predicts a substantial change in the electronic anisotropy as the system is pressure tuned through the quantum critical point.

Figure 1

Local coordination of Yb atom (large spheres, blue), suspended between two planes of boron heptagons (small green spheres) and surrounded in-plane by a distorted rectangular of Al atoms (red spheres). The proposed $|m_J=\pm 5/2⟩$ ground-state wave function is also shown, see Eq. (4). Boron conduction electrons $c^{†}({\mathbf{r}}_{i\pm })|0⟩$ interact with Yb spins ${\mathbf{S}}_j$ via layered Kondo lattice model Eq. (2).

Figure 2

Projected density of states (DOS), showing relative contribution of different atoms. The DOS at the Fermi level corresponds to the specific heat coefficient $\gamma \equiv C_v/T=6.7\mathrm{mJ}/(\mathrm{mol}\mathrm{Yb}\times {\mathrm{K}}^2)$ per Yb atom.

Figure 3

LDA-solution calculated (a) Fermi surfaces and (b) angle-dependent de Haas–van Alphen frequencies of extremal orbits showing only the lightest bands with frequencies $F<10\mathrm{kT}$ (higher frequencies up to 50 kT are also present but are unlikely to be observed in dHvA measurement). Two $\mathrm{Yb}\mathrm{\text{−}}f$ bands contribute most to the Fermi surface: band 89 (blue circles) and 91 (red crosses). In addition, LDA predicts small hole pockets due to band 93 (green rhombi). Each band is doubly degenerate in the spin-orbit mixed basis; hence, only odd-numbered bands are shown.

Figure 4

(a) Theoretical fit to experimental susceptibility data [7] ${\chi }_c(T)$ (circles) and ${\chi }_{ab}(T)$ (rhombi) in the range 50–300 K using the single-ion model shown in the inset and the fitted Curie-Weiss temperature ${\theta }_{\mathrm{CW}}$. The best fit yields the lowest crystal field splitting $\Delta =80\mathrm{K}$, the coefficient of admixture $\gamma \approx 0.28$ and ${\theta }_{\mathrm{CW}}^{\mathrm{th}}=-230\mathrm{K}$, close to the experimental value ${\theta }_{\mathrm{CW}}^{\exp }\sim -210\mathrm{K}$.