Fermi volume as a probe of hidden order

We demonstrate that the volume of the Fermi surface, measured very precisely using de Haas-van Alphen oscillations, can be used to probe changes in the nature and occupancy of localized electronic states. In systems with unconventional ordered states, this allows an underlying electronic order parameter to be followed to very low temperatures. We describe this effect in the field-induced antiferroquadrupolar (AFQ) ordered phase of PrOs${}_4$Sb${}_{12}$, a heavy fermion intermetallic compound. We find that the phase of de Haas-van Alphen oscillations is sensitively coupled, through the Fermi volume, to the configuration of the Pr $f$-electron states that are responsible for AFQ order. In particular, the $\beta$ sheet of the Fermi surface expands or shrinks as the occupancy of two competing localized Pr crystal field states changes. Our results are in good agreement with previous measurements, above 300 mK, of the AFQ order parameter by other methods. In addition, the low-temperature sensitivity of our measurement technique reveals a strong and previously unrecognized influence of hyperfine coupling on the order parameter below 300 mK within the AFQ phase. Such hyperfine couplings could provide insight into the nature of hidden order states in other systems.

Figure 1

Antiferroquadrupolar order in PrOs${}_4$Sb${}_{12}$. (a) $B-T$ phase diagram of PrOs${}_4$Sb${}_{12}$, based on Ref. 13. The blue region is superconducting, with a double superconducting transition indicated by the dark blue lines. The green line denotes the AFQ phase boundary for $B\parallel \left(110\right)$, and the two purple lines are transitions within the AFQ phase, believed to involve changes in the ordered structure.[13] Below 4.7 T, for $B\parallel \left(110\right)$, ${\Gamma }_1$ is the ground state, with a field dependent gap $\Delta$ to the quadrupolar excitations involving ${\Gamma }_4^{\left(2\right),+}$ as indicated in (b). ($\beta =1/k_{B}T$). Above 11 T, this situation is reversed. In the AFQ-ordered region, the ground state involves coherent superpositions of ${\Gamma }_1$ and ${\Gamma }_4^{\left(2\right)}$ on each site. (c) Crystal structure of PrOs${}_4$Sb${}_{12}$, with Pr atoms in pink, Os in green, and Sb in orange. (d) Charge distribution of the Pr 4$f$ electrons in the two lowest-lying crystal field levels. The charge distributions are similar in shape, and mostly buried within the ionic radius of Pr${}^{3+}$, so they are only weakly felt by the conduction electrons on the surrounding Os and Sb ions.

Figure 2

Typical quantum oscillations from PrOs${}_4$Sb${}_{12}$. (a) shows data from 18 T to 4 T. (b) shows the Fourier spectrum of the trace in (a), after the data have been replotted as a function of $1/B$. The $\beta$ frequency corresponds to the broad peak above 1 kT. (c) shows data from 4 to 7 T, between 30 mK and 1 K, to illustrate the temperature dependence of the oscillation amplitudes. (d) shows the results of fitting Eq. (3) to narrow sections of the data at 6.6 T. Note that there is a phase shift between the 200 mK and 800 mK oscillations. The data in (d) have been filtered so that only frequencies between 0.5 and 1.5 kT are present. The solid lines are the fitted curves.

Figure 3

The magnetic field dependence of the dHvA frequency. (a) dHvA frequency $F_f$ vs $B$ at temperatures from 100 mK to 1.2 K. The values of $F_f$ are extracted from fits of Eq. (3) to sets of three periods, as described in the text. The arrows indicate the boundaries of the AFQ phase.[13] The absence of low-field data at high temperature is due to the dHvA thermal damping factor in the Lifshitz-Kosevich amplitude, Eq. (2). The red curve at the bottom is proportional to $-(1-B\frac{\partial }{\partial B})M$, where $M$ is the magnetization derived from the 60 mK $M$ vs $B$ data of Ref. 13. (b) Main figure: simple model curves showing a hypothetical low-temperature field dependence of the Fermi surface area $\mathcal{A}$ (blue line) together with the back-projected frequency $F_f$ that would be seen in quantum oscillation measurements (red line). The dashed green line shows how $F=\hslash \mathcal{A}/2\pi e$ at 4.4 T is back projected to produce the measured frequency $F_f$ (red dot). The inset shows a hypothetical field dependence of $\hslash \mathcal{A}/2\pi e$ and the corresponding measured dHvA frequency $F_f$ at a temperature above the maximum AFQ phase transition temperature. The hypothetical field dependencies of $\mathcal{A}$ were chosen to produce $F_f$ curves that resemble the measured curves in (a) at 100 and 1200 mK.

Figure 4

Temperature dependence at 6.2 T of the dHvA frequency $F_f$ (circles) and the phase $\Delta {\varphi }_f\left(T\right)$ (green triangles). Both show a peak near the antiferroquadrupolar phase transition, $T_{\mathrm{AFQ}}$ (taken from Ref. 13), but the phase data are less noisy, and they reveal an additional downturn below $\sim$250 mK that is not evident in the back-projected frequency $F_f$.

Figure 5

The temperature dependence of $\Delta {\varphi }_f\left(T\right)$ at magnetic fields spanning the AFQ phase. Because of the dHvA thermal damping factor [see Eq. (2)], below 8 T, we only have data up to 950 mK. Error bars are statistical errors from averaging several sweeps with identical conditions. Where errors are not plotted, the error bars are smaller than the point size. The lines are guides to the eye, and the curves are shifted vertically for clarity; without this shift the lowest temperature point would be at 0 radians for every curve. The inset, (b), focuses on data below 300 mK for selected curves from (a). The blue(red) data are outside(inside) the AFQ phase. The lines are guides to the eye.

Figure 6

Energy levels and eigenstates for the two lowest crystal field levels of PrOs${}_4$Sb${}_{12}$ in the presence of a magnetic field, wth quadrupolar and hyperfine couplings. (a) Energy vs field. The red dashed lines are the states of Eq. (B1) with no quadrupolar order and the hyperfine interaction turned off. The $-\delta h$ term mixes ${\Gamma }_4^{\left(2\right),0}$ with ${\Gamma }_1$ so its energy is field dependent, despite ${\Gamma }_1$ being a singlet. Turning on quadrupolar order [see Eq. (B2)] between 4.75 and 11.5 T avoids the level crossing of ${\Gamma }_4^{\left(2\right),+}$ and ${\Gamma }_1$, and makes the ground state an admixture of ${\Gamma }_1$ and ${\Gamma }_4^{\left(2\right)}$ in this field range. The hyperfine Hamiltonian (B3) lifts the degeneracy of the six hyperfine eigenstates. (b) The ground-state manifold of (a). The black line is the lowest hyperfine state, labeled $|0\rangle$, while the yellow line is the highest, labeled $|5\rangle$.

Figure 7

(a) The proportion of ${\Gamma }_4^{\left(2\right)}$ in the six hyperfine states of the ground-state manifold. At $B=0$, all of the hyperfine levels in the electronic ground state are purely ${\Gamma }_1$. At $B>11.75$ T, they are all purely ${\Gamma }_4^{\left(2\right)}$. In the AFQ region, their composition is surprisingly different: at a given field, the ground state, $|0\rangle$ has the most ${\Gamma }_4^{\left(2\right)}$ character, and each successive excited hyperfine state has progressively less. The inset zooms in on the curves between 8 and 9 T. (b) Change in thermal occupation of ${\Gamma }_4^{\left(2\right)}$ from its $T=0$ value, calculated by applying a Boltzmann distribution to the eigenstates in (c). In the AFQ region, occupation of ${\Gamma }_4^{\left(2\right)}$ falls with increasing $T$ because the excited hyperfine states, which have less ${\Gamma }_4^{\left(2\right)}$ character, become thermally occupied.