Field-induced valence fluctuations in ${\mathrm{YbB}}_{12}$

We performed high-magnetic-field ultrasonic experiments on ${\mathrm{YbB}}_{12}$ up to 59 T to investigate the valence fluctuations in Yb ions. In zero field, the longitudinal elastic constant $C_{11}$, the transverse elastic constants $C_{44}$ and $\left(C_{11}-C_{12}\right)/2$, and the bulk modulus $C_{\mathrm{B}}$ show a hardening with a change of curvature at around 35 K indicating a small contribution of valence fluctuations to the elastic constants. When high magnetic fields are applied at low temperatures, $C_{\mathrm{B}}$ exhibits a softening above a field-induced insulator-metal transition signaling field-induced valence fluctuations. Furthermore, at elevated temperatures, the field-induced softening of $C_{\mathrm{B}}$ takes place at even lower fields and $C_{\mathrm{B}}$ decreases continuously with field. Our analysis using the multipole susceptibility based on a two-band model reveals that the softening of $C_{\mathrm{B}}$ originates from the enhancement of multipole-strain interaction in addition to the decrease of the insulator energy gap. This analysis indicates that field-induced valence fluctuations of Yb cause the instability of the bulk modulus $C_{\mathrm{B}}$.

Figure 1

Schematic view of the volume strain and the hexadecapole in ${\mathrm{YbB}}_{12}$. (a) Crystal lattice around the Yb ion [4] and volume strain ${\varepsilon }_{\mathrm{B}}$ with the irrep ${\mathrm{\Gamma }}_1$ of $O_h$. Orange arrows indicate the isotropic deformation of the lattice. (b) Hexadecapole $H_0$ with ${\mathrm{\Gamma }}_1$ symmetry obtained as a result of an isotropic change of the Yb ionic radius.

Figure 2

Temperature dependence of (a) the longitudinal elastic constant $C_{11}$, the transverse elastic constants (b) $C_{44}$ and (c) $C_{\mathrm{T}}=\left(C_{11}-C_{12}\right)/2$, and (d) the bulk modulus $C_{\mathrm{B}}$ calculated from $C_{11}$ and $C_{\mathrm{T}}$. The dotted lines indicate the fit of $C_{44}, C_{\mathrm{T}}$, and $C_{\mathrm{B}}$ in the framework of the phenomenological two-band model discussed in Sec. 4. The solid lines indicate the temperature dependence of the elastic constants without multipole contribution. The strains ${\varepsilon }_{ij}$ for $C_{11}, C_{44}, C_{\mathrm{T}}$, and $C_{\mathrm{B}}$ are schematically shown in the inset in each panel. The vertical arrows in each panel indicate the characteristic temperature $T^{★}$ discussed in Sec. 4.

Figure 3

Magnetic-field dependence of the relative variation of the elastic constants $\mathrm{\Delta }{C}_{ij}/C_{ij}=\left[C_{ij}\left(B\right)-C_{ij}(B=0)\right]/C_{ij}(B=0)$ at several temperatures for $B\parallel \left[001\right]$. Field dependence of (a) the longitudinal elastic constant $C_{11}$, the transverse elastic constants (b) $C_{44}$ and (c) $C_{\mathrm{T}}$, and (d) the bulk modulus $C_{\mathrm{B}}$. The data sets are shifted consecutively along the $\mathrm{\Delta }C/C$ axes for clarity. The vertical arrows indicate the insulator-metal transition field $B_{\mathrm{IM}}$. The horizontal arrows show the field-sweep directions.

Figure 4

Temperature-field phase diagram of ${\mathrm{YbB}}_{12}$ for $B\parallel \left[001\right]$. The filled red circles indicate the insulator-metal transition field $B_{\mathrm{IM}}$. The filled red diamond indicates the characteristic temperature $T^{★}$ (see text for details). The color code shows the value of the bulk modulus $C_{\mathrm{B}}$ (field up-sweep).

Figure 5

Temperature dependence of the bulk modulus $C_{\mathrm{B}}$ of ${\mathrm{YbB}}_{12}$ at various magnetic fields for $B\parallel \left[001\right]$ extracted from the up-sweep data shown in Fig. 3, except for the zero-field data. The dotted line indicates the fit describing $C_{\mathrm{B}}$ in the framework of the phenomenological two-band model. The inset shows $C_{\mathrm{B}}$ below 10 K.

Figure 6

Schematic view of the two-band model assuming a constant DOS over each band (red rectangular). (a) DOS at zero field with an energy gap $2\mathrm{\Delta }=140$ K and bandwidth $W=55$ K. (b) DOS at 58 T. The energy gap $2\mathrm{\Delta }=74$ K and the bandwidth $W=55$ K is determined using Eq. (9). The blue dotted curves indicate the schematic view of a more realistic DOS of ${\mathrm{YbB}}_{12}$.

Figure 7

Temperature dependence of the longitudinal elastic constant $C_{11}$.

Figure 8

Temperature dependence of the electric multipole susceptibility of ${\mathrm{YbB}}_{12}$. (a) Hexadecapole susceptibility ${\chi }_{\mathrm{B}}$ of $H_0=O_4^0+5O_4^4$ with ${\mathrm{\Gamma }}_1$. This susceptibility provides the temperature dependence of the bulk modulus $C_{\mathrm{B}}$. (b) Quadrupole susceptibility ${\chi }_{{\mathrm{\Gamma }}_3}$ of $O_u$ and $O_v$ with ${\mathrm{\Gamma }}_3$ related to $(C_{11}-C_{12})/2$. (c) Quadrupole susceptibility ${\chi }_{{\mathrm{\Gamma }}_5}$ of $O_{yz}, O_{zx}$, and $O_{xy}$ with ${\mathrm{\Gamma }}_5$ related to $C_{44}$. As indicated in Eq. (B14), $-{\chi }_{{\mathrm{\Gamma }}_{\gamma }}$ contributes to these elastic constants. (The curves for $O_u$ and $O_v$ as well as for $O_{yz}, O_{zx}$, and $O_{xy}$ virtually lie on top of each other.)

Figure 9

Magnetic-field dependence of the eigenenergy in $H_{\mathrm{total}}$ of Eq. (C3) for $B\parallel \left[001\right]$. The color code shows the Zeeman mixing ratio ${\beta }_i^2$ in the wave functions of Eqs. (C16, C17, C18, C19, C20, C21, C22, C23).

Figure 10

Electric hexadecapole susceptibility ${\chi }_{\mathrm{B}}$ of ${\mathrm{YbB}}_{12}$ for $B\parallel \left[001\right]$. (a) Magnetic-field dependence of ${\chi }_{\mathrm{B}}$ at several temperatures. (b) Temperature dependence of ${\chi }_{\mathrm{B}}$ below 100 K in several magnetic fields. In the inset in (b), ${\chi }_{\mathrm{B}}$ is shown up to 300 K.

Figure 11

Fit of the bulk modulus $C_{\mathrm{B}}$ of ${\mathrm{YbB}}_{12}$ by the hexadecapole susceptibility with energy gaps $\mathrm{\Delta }=70$ and 30 K.