Unusual Interplay between Superconductivity and Field-Induced Charge Order in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_y$

We present a detailed study of the temperature ($T$) and magnetic field ($H$) dependence of the electronic density of states (DOS) at the Fermi level, as deduced from specific heat and Knight shift measurements in underdoped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_y$. We find that the DOS becomes field independent above a characteristic field $H_{\mathrm{DOS}}$, and that the $H_{\mathrm{DOS}}(T)$ line displays an unusual inflection near the onset of the long-range 3D charge-density wave order. The unusual $S$ shape of $H_{\mathrm{DOS}}(T)$ is suggestive of two mutually exclusive orders that eventually establish a form of cooperation in order to coexist at low $T$. On theoretical grounds, such a collaboration could result from the stabilization of a pair-density wave state, which calls for further investigation in this region of the phase diagram.

Figure 1

(a),(b) Magnetic field dependence of the electronic specific heat, $C_{\mathrm{es}}(T,H)/T$, normalized to its value at the field $H_{\mathrm{DOS}}$ above which data saturate (see arrows), $C_{\mathrm{es}}^N/T$. [The curves in panel (b) have been shifted for clarity, by $+0.5$ for $p=0.119$ and by $-0.5$ for $p=0.098$.] At 2.5 K, the data markedly deviate from a linear dependence [shaded area in panel (a); see the text for details]. (c) Field dependence of the ${}^{17}\mathrm{O}$ Knight shift for $p=0.109$. Solid lines are fits to the data, as explained in the main text (see Ref. [34] for the determination of the error bars).

Figure 2

(a) Magnetic field dependence of the specific heat at low temperature ($p=0.109$; a smooth polynomial background has been removed from the data, and each temperature has been shifted by 0.05 for clarity; arrows indicate the $H_{\mathrm{DOS}}$ values). The clear quantum oscillations can be very well described by the standard Lifshitz-Kosevich formula with $F\sim 530\mathrm{T}$ and a warping term of $t_w\sim 15\mathrm{T}$ (red lines), or with $F\sim 520\mathrm{T}$ and $t_w\sim 22\mathrm{T}$ (green line). (b) $C_{\mathrm{es}}^{\mathrm{N}}/T$ (closed squares) and residual $C_p/T$ values (${\gamma }_R$, open squares) as a function of the doping rate, together with the effective mass deduced from quantum oscillations in transport and skin depth measurements (gray circles) [12, 13].

Figure 3

$H_{\mathrm{DOS}}$ versus temperature for the indicated doping contents. The solid circles and squares have been derived from $T$ (see Refs. [20, 34]) and $H$ sweeps [see Figs. 1 and 1] of the specific heat, respectively, and the crossed squares have been deduced from $K_{\mathrm{spin}}(H)$ [see Fig. 1]. As shown, a clear “plateau” is observed in $H_{\mathrm{DOS}}(T)$ for $p=0.109$ [panel (b)] and $p=0.119$ [panel (c)] in the vicinity of the onset of the long-range 3D-CDW [41, 42] (open crosses and shaded areas), highlighting the interplay between those two competing orders. For $p=0.098$ [panel (a)], a change of slope is observed when $H_{\mathrm{DOS}}$ crosses $H_{\mathrm{CDW}}$. The field $H_{\mathrm{scat}}$ marking the onset of scattering by vortices, as deduced from thermal conductivity measurements (diamonds [18, 42]), has also been reported for $p=0.11$. Open diamonds in panel (c) correspond to the fields below which the intensities of the CDW diffraction peaks decrease [4]. Lines are guides to the eyes.

Figure 4

Sketch of the $H-T$ diagram of underdoped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_y$, emphasizing the interplay between the superconducting and CDW orders. Specific heat and Knight shift measurements show that the density of states at the Fermi level reaches its normal-state value above $H_{\mathrm{DOS}}$ (see Fig. 1). Different shades of yellow tentatively depict the intensity of local superconducting fluctuations, with emphasis on the field scale $H_{\mathrm{scat}}$ deduced from thermal conductivity measurements [18, 42]. The greenish region corresponds to the $H_{\mathrm{CDW}}\le H\le H_{\mathrm{DOS}}$ field range, in which superconductivity might be impacted by the presence of 3D CDW order.