Magnetic breakdown and charge density wave formation: A quantum oscillation study of the rare-earth tritellurides

The rare-earth tritellurides ($R{\mathrm{Te}}_3$, where $R=\mathrm{La}$, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, Y) form a charge density wave state consisting of a single unidirectional charge density wave for lighter $R$, with a second unidirectional charge density wave, perpendicular and in addition to the first, also present at low temperatures for heavier $R$. We present a quantum oscillation study in magnetic fields up to 65 T that compares the single charge density wave state with the double charge density wave state both above and below the magnetic breakdown field of the second charge density wave. In the double charge density wave state it is observed that there remain several small, light pockets, with the largest occupying around 0.5% of the Brillouin zone. By applying magnetic fields above the independently determined magnetic breakown field, the quantum oscillation frequencies of the single charge density wave state are recovered, as expected in a magnetic breakdown scenario. Measurements of the electronic effective mass do not show any divergence or significant increase on the pockets of Fermi surface observed here as the putative quantum phase transition between the single and the double charge density wave states is approached.

Figure 1

(a) Phase diagram of $R{\mathrm{Te}}_3$ shown without the low-temperature magnetic phases (see Appendix) [1, 2] as a function of the in-plane lattice parameter $a$. The top axis marks the specific rare-earth ($R$) ions that yield these lattice parameter values. There are two unidirectional, incommensurate CDWs with $q_1\approx$ 2/$7c^{*}$ and $q_2\approx$ 1/$3a^{*}$. Both are present at low temperatures in the heaviest $R$ (shortest $a$: Tm-Tb), but just $q_1$ for lighter $R$ (longer $a$: Gd, Sm, Nd, Ce, La). The compounds where $R=\mathrm{La}$, Ce, and Nd are known to have a unidirectional CDW ordering with $q_1$ that onsets at temperatures above those measured. (b, c) A 2D tight-binding model that captures the essential structure of the Fermi surface of $R{\mathrm{Te}}_3$ is shown in gray, with the first Brillouin zone shown as the light-green box. Note that this model ignores a subtle $b$-axis warping and bilayer splitting. The full Fermi surface would be reflected across the $k_x=0$ line. In (b)(ii) a portion of the Fermi surface observed by ARPES measurements [3] in the single-CDW state is sketched in green (shown in the unfolded zone). (c) Illustration of the remaining portions of the unfolded Fermi surface resolved by ARPES once gaps ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ have opened [3, 4, 5]: (i) for the double-CDW state (orange) and (ii) for the single-CDW state (green), with the CDW vectors $q_1$ and $q_2$ represented by the green and orange arrows, respectively. (b, c) Adapted from Brouet et al. [3] and Moore et al. [5].

Figure 2

(a) The $b$-axis resistance as a function of the magnetic field, $R\left(B\right)$, normalized by the zero-field value $R_0$ for three representative measurements (${\mathrm{TbTe}}_3$, ${\mathrm{GdTe}}_3$, and ${\mathrm{HoTe}}_3$; as red, black, and dark-blue lines, respectively), one from each DC magnet (see Sec. 2). Smooth, nonoscillating background estimates, $R_{\mathrm{bgrd}}/R_0$ for each measurement are shown as dashed lines. (b) The oscillating component of the data extracted from (a) by subtraction of the smooth background from the data plotted as a function of the inverse field to reveal periodic oscillations indicative of quantum oscillations. (c) Representative mutual inductance data shown as the mixed-down frequency of the tank circuit for ${\mathrm{TmTe}}_3$, with the smooth background estimate shown as the dashed line. (d) The oscillating component of the signal shown by subtraction of the nonoscillating background, $f_{\mathrm{bgrd}}$ as a function of the inverse magnetic field, again revealing periodic oscillations that are consistent with quantum oscillations.

Figure 3

(a) A guide showing the field ranges across which the FFTs shown in (b)–(d) were obtained for each member of the $R{\mathrm{Te}}_3$ series studied here. Each $R$ (top axis) is placed according to its $a$ lattice parameter at 300 K (bottom axis) [22], noting that overlapping ranges have been symmetrically offset in $x$ for clarity. For compounds with two CDWs, the calculated characteristic breakdown field for the lower-temperature (smaller-gap) CDW is shown as a star. The FFTs are thus grouped into three categories; measurements in the double-CDW state with FFTs obtained from a magnetic field range below $B_0$ [orange lines; (b)], measurements in the double-CDW state with FFTs derived from a magnetic field range above $B_0$ [blue lines; (c)], and measurements in the single-CDW state [green lines; (d)]. Note that (b)–(d) are all plotted at the same $x$ scale for comparison. Peaks in the FFTs shown in (c) and (d) are labeled to distinguish primary frequencies from mixing frequencies and harmonics as discussed further in the text. The primary $\alpha$, $\beta$, and $\delta$ frequencies are highlighted in yellow to more clearly show the consistency in their values as $R$ changes. In (d), ${\mathrm{NdTe}}_3$, ${\mathrm{SmTe}}_3$, and ${\mathrm{GdTe}}_3$ No. 2 have an additional FFT of the same data restricted to a higher range of fields to highlight high-frequency components (frequencies below 700 T are omitted from these curves for clarity). Data in these plots were taken at fixed temperatures between 1.5 and 2 K.

Figure 4

An expanded view of the quantum oscillation frequency spectrum for $B<B_0$ in compounds with two CDWs.

Figure 5

Temperature dependence of the quantum oscillation amplitudes for primary orbits in (a) ${\mathrm{ErTe}}_3$, (b) ${\mathrm{DyTe}}_3$, (c) ${\mathrm{TbTe}}_3$, (d) ${\mathrm{GdTe}}_3$, and (e) ${\mathrm{NdTe}}_3$. Fits to the Lifshitz-Kosevitch formula are represented by solid lines and yield the effective mass $m^{*}$. Open symbols and dashed lines correspond to breakdown frequencies. The field range used for the FFT and the frequency of the orbit are shown in the legend.

Figure 6

Summary of effective mass data. (a) Effective masses established for various $R\mathrm{s}$, with dashed lines a guide for the eye to highlight trends upon approach to the transition to a double-CDW state. No trend line is included in the double-CDW state because it is less clear whether like orbits are being compared for different $R\mathrm{s}$. (b) Effective masses plotted as a function of the quantum oscillation frequency, establishing the trend that the effective mass scales with the size of the pocket.

Figure 7

(a) The $b$-axis resistivity data in a fixed magnetic field ($B\parallel b$) in ${\mathrm{GdTe}}_3$ taken on both warming and cooling every 1 T between 0 and 14 T (data offset for clarity). (b) Second derivative in temperature of the data in (a) (offset for clarity). (c) $B-T$ phase diagram derived from the data in (b): filled symbols represent peaks in the second derivative; open symbols are taken from the center of the thermal hysteresis loops in (b), implying a first-order phase transition. The phase boundaries seem to be approaching 0 K just slightly above 14 T. (d) Three FFTs taken in different regions of the $B-T$ phase space in ${\mathrm{GdTe}}_3$: one from within the magnetic phase (2 K, 4.5–12 T; red line), one above the magnetic phase boundary in the field (2 K, 15–24 T; black line), and one from above the magnetic phase in temperature (17 K, 10–14 T; blue line). This illustrates that any alteration of the quantum oscillation spectrum by magnetic ordering must be subtle enough not to affect our conclusions. (e) Magnetic phase transitions determined previously without an applied magnetic field by Ru et al. (black circles; left axis) [31] compared to the magnetic polarization fields observed in applied fields at 2 K for $R=\mathrm{Gd}$, Tb, and Ho (red squares; right axis) as a function of the de Gennes factor. Determination of the polarization fields is discussed in the text. The polarization fields seem to scale with the zero-field transition temperatures.