Theory of Spin Exciton in the Kondo Semiconductor ${\mathrm{YbB}}_{12}$

The Kondo semiconductor ${\mathrm{YbB}}_{12}$ exhibits a spin and charge gap of approximately 15 meV. Close to the gap energy narrow dispersive collective excitations were identified by previous inelastic neutron scattering experiments. We present a theoretical analysis of these excitations. Starting from a periodic Anderson model for crystalline-electric-field- (CEF) split $4f$ states we derive the hybridized quasiparticle bands in slave boson mean-field approximation and calculate the momentum dependent dynamical susceptibility in random phase approximation. We show that a small difference in the hybridization of the two CEF (quasi-) quartets leads to the appearance of two dispersive spin resonance excitations at the continuum threshold. Our theoretical analysis explains the most salient features of previously unexplained experiments on the magnetic excitations of ${\mathrm{YbB}}_{12}$.

Figure 1

Dynamical susceptibility in the $[111]$ direction for $\mathbf{q}=\mathbf{0}$ (direct gap) and $\mathbf{q}=\mathbf{Q}$ (indirect gap) versus energy, for degenerate quasiparticle bands with $V_1=t$ and $\delta V=0$ ($\overline{V}=0.30t$, $J_{\mathbf{Q}}=0$). Inset shows the density of states for two CEF-split quasiparticle bands (${\overline{ϵ}}_{f_1}=0.08t$ and $b=0.41$). The green curve corresponds to the band $V_1=t$ and the red one to the band $V_2=V_1+0.13t$.

Figure 2

Contour plot of the real part of noninteracting dynamical susceptibility for degenerate bands with $V_1=t$ and $\delta V=0$ (${\overline{V}}_1=0.30t$, $J_{\mathbf{Q}}=0$) in the $[111]$ direction.

Figure 3

The imaginary part of the susceptibility for the noninteracting ($J_{\mathbf{Q}}=0$) and interacting case ($J_{{\mathbf{Q}}_1}=J_{{\mathbf{Q}}_2}=0.065t$) for $V_1=t$, $\mathbf{q}=\mathbf{Q}$ and $\delta V=0.13t$. The $J_{{\mathbf{Q}}_{\gamma }}$ are slightly subcritical leading to a finite intrinsic resonance line width.

Figure 4

Contour plot of imaginary part of RPA dynamical susceptibility with Lorentzian interaction $J_{\Gamma }(q)$ for CEF-split quasiparticle bands, ${\overline{ϵ}}_{f_1}=0.08t$ and $b=0.41$. Here $V_1=t$ and $V_2=V_1+0.13t$ ($J_{{\mathbf{Q}}_1}=0.125t$, $J_{{\mathbf{Q}}_2}=0.143t$ and ${\Gamma }_Q=2$ which satisfy the resonance condition in RPA formula: $J_{\Gamma }(\mathbf{Q})=1/\mathrm{Re}\mathbf{[}{\chi }_0^{\Gamma \Gamma }(\mathbf{Q},{\omega }_r)\mathbf{]}$), in the $[111]$ direction. The peaks at the zone boundary ($\mathbf{q}=\mathbf{Q}$) appear at ${\omega }_1=0.047t$ and ${\omega }_2=0.063t$. By choosing $t=0.32\mathrm{eV}$ ($D=1.92\mathrm{eV}$) then ${\omega }_1=15\mathrm{meV}$ and ${\omega }_2=20\mathrm{meV}$ which are comparable with experimental results. The inset shows the contour plot of imaginary part of dynamical susceptibility of noninteracting degenerate bands for comparison ($V_1=t$, $\delta V=0.13t$, ${\overline{V}}_1=0.41t$, and $J_{\mathbf{Q}}=0$) in the $[111]$ direction. The color scale of the inset is the same as in Fig. 2.