Effect of Anisotropic Hybridization in ${\mathrm{YbAlB}}_4$ Probed by Linear Dichroism in Core-Level Hard X-Ray Photoemission Spectroscopy

We have probed the crystalline electric-field ground states of pure $|J=7/2,J_z=\pm 5/2⟩$ as well as the anisotropic $c\text{−}f$ hybridization in both valence fluctuating systems $\alpha$- and $\beta$-${\mathrm{YbAlB}}_4$ by linear polarization dependence of angle-resolved core level photoemission spectroscopy. Interestingly, the small but distinct difference between $\alpha$- and $\beta$-${\mathrm{YbAlB}}_4$ was found in the polar angle dependence of linear dichroism, indicating the difference in the anisotropy of $c\text{−}f$ hybridization, which may be a key to understanding a heavy Fermi liquid state in $\alpha$-${\mathrm{YbAlB}}_4$ and a quantum critical state in $\beta$-${\mathrm{YbAlB}}_4$.

Figure 1

(a) Crystal structure of $\alpha$- and $\beta$-${\mathrm{YbAlB}}_4$ with a view along the $c$ axis. The Yb-Al layer is sandwiched by heptagonal and pentagonal B layers, respectively. The local structures surrounded by the purple quadrangles are shown at the right side as the pictorial views that are oriented to the acute angles of the quadrangles. (b) Schematic top view geometry of incident horizontally (blue arrow) and vertically (red arrow) polarized x rays, sample, and photoelectron analyzer. Polar angle $\theta$ is defined as an angle between the directions toward the photoelectron analyzer and the $c$ axis of the sample, and $\theta =60°$ corresponds that the $c$ axis is oriented toward the incident x ray. Azimuthal angle $\phi$ is defined as the angle of the rotation along the $c$ axis and $\phi =0°$ corresponds to the geometry as drawn in Fig. 1. (c) Polarization-dependent ${\mathrm{Yb}}^{3+}$ $3d_{5/2}$ core-level HAXPES spectra of $\alpha$- and $\beta$-${\mathrm{YbAlB}}_4$ with the polar angle of $\theta =0°$, 45°, and 60°. Red thin and blue thick lines correspond to $s$ and $p$ polarization. The spectra are normalized by the spectral weight of ${\mathrm{Yb}}^{3+}$ $3d_{5/2}$ after the subtraction of Shirley-type backgrounds [18, 19]. (d) Ionic calculation of polarization-dependent Yb $3d_{5/2}$ core-level HAXPES spectra of pure $J_z$ states of a $4f$ hole at each $\theta$.

Figure 2

(a) and (b) The azimuthal angle dependence of ${\mathrm{Yb}}^{3+}$ $3d_{5/2}$ core-level HAXPES spectra in $\alpha$- and $\beta$-${\mathrm{YbAlB}}_4$ at $\theta =60°$ and the ionic calculation of the pure $|\pm 5/2⟩$ state. (c) The azimuthal angle dependence of LDs in $\alpha$- and $\beta$-${\mathrm{YbAlB}}_4$ (closed circle) and the ionic calculation for pure $|\pm 5/2⟩$ state and $|\pm 5/2⟩$ plus a small amount of $|\mp 3/2⟩$ ($\sim 0.9998|\pm 5/2⟩+0.02|\mp 3/2⟩$) or $|\pm 1/2⟩$ ($\sim 0.9975|\pm 5/2⟩+0.07|\pm 1/2⟩$) state (solid line). LD is defined as the difference of normalized intensity between the $s$ and $p$ polarization. The amplitude of LDs of $\alpha$- and $\beta$-${\mathrm{YbAlB}}_4$ is multiplied by three to compare with the calculations.

Figure 3

(a) Polarization-dependent ${\mathrm{Yb}}^{3+}$ $3d_{5/2}$ core-level HAXPES spectra of $\alpha$- and $\beta$-${\mathrm{YbAlB}}_4$ at $\theta =\phi =0°$, and the calculation of the spectra for $|\pm 5/2⟩$ state based on SIAM including isotropic $c\text{−}f$ hybridization. (b) and (c) The experimental LDs (closed circle, square, and triangle) of $\alpha$- and $\beta$-${\mathrm{YbAlB}}_4$ and the calculation for $|\pm 5/2⟩$ state based on the SIAM (solid line) and ionic model (broken line). Red, green, and blue lines on the black lines correspond to the LDs by SIAM at $\theta =0°$, 45°, and 60°. The same parameters are used in the calculations shown in (b) and (c).