Hall, Seebeck, and Nernst Coefficients of Underdoped ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\delta }$: Fermi-Surface Reconstruction in an Archetypal Cuprate Superconductor

Charge-density-wave order has been observed in cuprate superconductors whose crystal structure breaks the square symmetry of the ${\mathrm{CuO}}_2$ planes, such as orthorhombic ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_y$ (YBCO), but not so far in cuprates that preserve that symmetry, such as tetragonal ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\delta }$ (Hg1201). We have measured the Hall ($R_H$), Seebeck ($S$), and Nernst ($\nu$) coefficients of underdoped Hg1201 in magnetic fields large enough to suppress superconductivity. The high-field $R_H(T)$ and $S(T)$ are found to drop with decreasing temperature and become negative, as also observed in YBCO at comparable doping. In YBCO, the negative $R_H$ and $S$ are signatures of a small electron pocket caused by Fermi-surface reconstruction, attributed to charge-density-wave modulations observed in the same range of doping and temperature. We deduce that a similar Fermi-surface reconstruction takes place in Hg1201, evidence that density-wave order exists in this material. A striking similarity is also found in the normal-state Nernst coefficient $\nu (T)$, further supporting this interpretation. Given the model nature of Hg1201, Fermi-surface reconstruction appears to be common to all hole-doped cuprates, suggesting that density-wave order is a fundamental property of these materials.

Popular Summary

Copper-oxide materials called cuprates superconduct at record high temperatures. The origin of this remarkable phenomenon, discovered nearly three decades ago, remains an outstanding puzzle in condensed-matter physics, largely because a number of electronic phases appear to coexist and compete in these materials. The identification of those phases is key to understanding high-temperature superconductivity. In some cuprates with orthorhombically distorted $\mathrm{Cu}{\mathrm{O}}_2$ planes, the competing phases are characterized by particular forms of density-wave order, involving charge and/or spin modulations. But, is density-wave order a universal property of cuprates? This question is examined here in $\mathrm{HgB}{\mathrm{a}}_2\mathrm{Cu}{\mathrm{O}}_{4+\delta }$ (Hg1201), a clean model cuprate that has a high superconducting critical temperature and that preserves the lattice symmetry of the square $\mathrm{Cu}{\mathrm{O}}_2$ planes. Our transport measurements provide evidence of the existence of density-wave order in Hg1201, indicating that it is indeed a generic property of cuprate superconductors.

Our transport experiments measure both the Hall and Seebeck coefficients of Hg1201. These measurements have already been established previously as an effective probe of density-wave order through the studies of ${\mathrm{YB}{\mathrm{a}}_2\mathrm{C}{\mathrm{u}}_3\mathrm{O}}_y$ (YBCO), an important cuprate. In YBCO, quantum oscillations combined with negative Hall and Seebeck coefficients in the normal state at low temperature revealed a reconstruction of the Fermi surface and suggested charge-density-wave modulations as the most likely cause. Indeed, charge-density-wave modulations were directly observed via NMR and x-ray-scattering measurements. The dependence of the charge order on both doping and temperature was seen to correlate well with the Hall and Seebeck data, showing the effectiveness of these transport coefficients as probes of density-wave order. The Hall and Seebeck coefficients in Hg1201 we have measured exhibit a sign change from positive to negative as a function of decreasing temperature, revealing a Fermi-surface reconstruction very similar to that of YBCO. This is evidence that a similar density-wave order is present in Hg1201.

Full confirmation that charge-density-wave order is indeed universal in cuprates will go a long way toward resolving the puzzle of the nature of superconductivity in these materials.

Figure 1

(a) The Hall coefficient $R_H$ of Hg1201 as a function of magnetic field $H$, in sample $A$ at various temperatures, as indicated. Also shown is an isotherm at $T=4.2\mathrm{K}$, measured on sample $B$ (black curve, labeled $B$). (b) Normal-state Hall coefficient $R_H$ as a function of temperature, at $H=53\mathrm{T}$ (sample $A$; closed red circles, left axis) and $H=68\mathrm{T}$ (sample $B$; open red circles, left axis). Corresponding data are shown for YBCO at $p=0.10$ and $H=55\mathrm{T}$ (blue circles, left axis; from Ref. 10). Note how in both materials, $R_H(T)$ drops at low temperature to become negative, a signature of Fermi-surface reconstruction [12]. We reproduce the splitting of NMR lines in YBCO at $p=0.10$ and $H=28.5\mathrm{T}$, which reveals the onset of charge order below $T_{\mathrm{CO}}\simeq 50\mathrm{K}$ (green squares, right axis; from Ref. 13).

Figure 2

(a) Seebeck coefficient $S$ of Hg1201 plotted as $S/T$ versus $H$, in sample $A$ at various temperatures, as indicated. (b) Normal-state Seebeck coefficient $S/T$ of Hg1201 as a function of temperature, at $H=28\mathrm{T}$ (red circles) and $H=45\mathrm{T}$ (red triangles). Corresponding data are shown for YBCO at $p=0.10$ in $H=0$ (blue circles) and 28 T (blue triangles), from Ref. 8.

Figure 3

(a) Temperature-doping phase diagram of YBCO, showing the zero-field superconducting phase below $T_c$ (grey dome, from Ref. 36) and the onset of $q=0$ magnetic order below $T_{\mathrm{mag}}$ detected by spin-polarized neutron scattering (down triangles, from Ref. 42). Several characteristic temperatures of the transport properties are displayed: The thick green and grey lines schematically represent the pseudogap temperature $T^{\star }$, from resistivity and Nernst data [23], and $T_H$, from $R_H(T)$ [19], respectively. $T_{\max }$ (full circles) and $T_0$ (empty circles) are determined from $R_H(T)$ [19] (see text). We also show the onset of charge order at $T_{\mathrm{CO}}$ via NMR (full diamonds, from Ref. 13) and the approximate onset of charge modulations via x-ray scattering (open diamonds, from Refs. 14, 15, 16). (b) Corresponding phase diagram for Hg1201, showing $T_c$ (grey dome, from Ref. 22), $T_{\mathrm{mag}}$ (down triangles, from Refs. 39, 40), and $T^{\star }$ from resistivity (full squares, from Refs. 21, 39, 40; open squares, from this work). The characteristic temperatures $T_{\max }$ (full circles) and $T_0$ (empty circles) of $R_H(T)$ are also shown. All dashed lines in both panels are a guide to the eye.

Figure 4

Nernst coefficient $\nu$ of Hg1201 ($T_c=65\mathrm{K}$; red circles, sample $A$) and YBCO ($T_c=66\mathrm{K}$, $p=0.12$; blue circles, from Ref. 23), plotted as $\nu /T$ versus $T/T^{\star }$, where $T^{\star }$ is the pseudogap temperature as defined in the main text. Here, $T^{\star }=340\mathrm{K}$ for Hg1201 and $T^{\star }=250\mathrm{K}$ for YBCO. The magnetic field $H=10\mathrm{T}$ is along the $c$ axis, and the heat gradient is along the $a$ axis. Note the different vertical scales for Hg1201 (red arrow, left axis) and YBCO (blue arrow, right axis). Below $T/T^{\star }=1.0$, the quasiparticle signal falls gradually to reach large negative values, in identical fashion in the two materials.