Universal relationship between the energy scales of the pseudogap phase, the superconducting state, and the charge-density-wave order in copper oxide superconductors

We report the hole doping dependencies of the pseudogap phase energy scale $2{\mathrm{\Delta }}_{\mathrm{PG}}$, the antinodal (nodal) superconducting energy scales $2{\mathrm{\Delta }}_{\mathrm{SC}}^{\text{AN}}$ ($2{\mathrm{\Delta }}_{\mathrm{SC}}^{\text{N}}$), and the charge-density-wave energy scale $2{\mathrm{\Delta }}_{\mathrm{CDW}}$ extracted from the electronic Raman responses of several copper oxide families. We show for all the cuprates studied that the three energy scales $2{\mathrm{\Delta }}_{\mathrm{PG}}, 2{\mathrm{\Delta }}_{\mathrm{SC}}^{\text{AN}}$, and $2{\mathrm{\Delta }}_{\mathrm{CDW}}$ display the same decreasing monotonic behavior with doping. In particular, $2{\mathrm{\Delta }}_{\mathrm{SC}}^{\text{AN}}$ and $2{\mathrm{\Delta }}_{\mathrm{CDW}}$ have nearly equal values. This suggests a universal scenario in which $2{\mathrm{\Delta }}_{\mathrm{PG}}, 2{\mathrm{\Delta }}_{\mathrm{SC}}^{\text{AN}}$, and $2{\mathrm{\Delta }}_{\mathrm{CDW}}$ are governed by common microscopic interactions that become relevant well above the superconducting transition at $T_c$. This is to be contrasted with the behavior of the nodal superconducting energy scale $2{\mathrm{\Delta }}_{\mathrm{SC}}^{\text{N}}$, which tracks the doping dependence of $T_c$ and, hence, seems to be controlled by different interactions.

Figure 1

For $T\le T_c, B_{1g}$ (antinodal) Raman response functions of (a) an underdoped Hg-1223 (UD 117, $p\approx 0.12$) and (b) Bi-2212 (UD 75, $p\approx 0.11$) single crystal. (c) and (d) For $T\ge T_c$. The SC energy scale $2{\mathrm{\Delta }}_{\mathrm{SC}}$ and the PG energy scale $2{\mathrm{\Delta }}_{\mathrm{PG}}$ are indicated by a red and blue arrow, respectively. In the insets of (a) and (b), the subtracted Raman response (defined in the text) underlines the peak-dip structure. In the insets of (c) and (d), the subtracted Raman response (defined in the text), points out the spectral weight transfer induced by the pseudogap phase in the normal state. The shaded area allows by contrast a better visualization of the peak-dip structure in the SC state and the spectral weight transfer in the normal state.

Figure 2

$B_{2g}$ (nodal) Raman response functions of Hg-1223 (UD 117, $p\approx 0.12$), Hg-1201 (UD 72, $p\approx 0.11$), Y-123 (UD 54, $p\approx 0.08$), and Bi-2212 (UD 75, $p\approx 0.11$) compounds. (a)–(d) Selected temperatures above and below $T_c$. In the insets we plot the subtractions between the Raman responses measured below (red curve) and just above (black curve) $T_c$ and the ones measured at $T_0$. $T_0=290$, 285, 280, and 200 K for Hg-1223, Hg-1201, Y-123, and Bi-2212, respectively. The shaded area allows a better visualization of the CDW signal. (e)–(h) Subtracted Raman responses between selected temperatures above $T_c$ and $T_0$. The peaks labeled by a star in the Y-123 and Bi-2212 are phonon modes induced by oxygen disorder [31, 32]. Note that since the CDW signal is intrinsically weak in the Bi-2212 compound, we applied a slight filtering by using Fourier transform to improve the signal noise ratio of the Raman response in (h).

Figure 3

Universal doping dependencies of the pseudogap, the antinodal superconducting, and the charge-density-wave energy scales [respectively $2{\mathrm{\Delta }}_{\mathrm{PG}}\left(p\right), 2{\mathrm{\Delta }}_{\mathrm{SC}}^{\text{AN}}\left(p\right)$, and $2{\mathrm{\Delta }}_{\mathrm{CDW}}\left(p\right)]$ over four cuprates systems: (a) Hg-1223, (b) Y-123, (c) Hg-1201, and (d) Bi-2212 cuprates. (e)–(h) The doping dependence of the nodal superconducting energy scale $2{\mathrm{\Delta }}_{\mathrm{SC}}^{\text{N}}\left(p\right)$ for Hg-1223, Y-123, Hg-1201, and Bi-2212, respectively. (i)–(l) The doping dependence of the relevant transition temperatures: the pseudogap $T^{*}$, the superconducting $T_c$, and the charge-density-wave $T_{\mathrm{CDW}}$ for Hg-1223, Y-123, Hg-1201, and Bi-2212, respectively. The filled symbols correspond to our Raman data. Our data on Y-123, Hg-1201, and Bi-2212 are supplemented by Raman measurements from other groups (designated by empty symbols), see Ref. [37].

Figure 4

(a) Zero field cooling magnetization curves of Hg-1223 single crystals for several doping levels. The applied magnetic field is perpendicular to the $ab$ plane and its magnitude is of $\approx 10$ Oe. (b) First derivative of the magnetization curves displayed in (a). The location of the peak maximum indicates the value of $T_c$ and its full width at half maximum the transition width.

Figure 5

(a) Doping evolution of the dip depth and the loss of spectral weight generated by the pseudogap phase. (b) Characteristic peak-dip structure extracted from the subtracted Raman response of Bi-2212 (UD 75) single crystal between the SC and the normal state just above $T_c$.

Figure 6

(a) and (c) and (e) and (g): Temperature dependence of the $B_{1g}$ Raman response function of Hg-1223 (with distinct doping levels) up to $T_c$. (b) and (d) and (f) and (h): Temperature dependence of the $B_{1g}$ Raman response function of Hg-1223 (with distinct doping levels) above $T_c$. The pair breaking peak indicated by a red arrow determine the $2{\mathrm{\Delta }}_{\mathrm{SC}}^{\text{AN}}$ scale while the pseudogap $2{\mathrm{\Delta }}_{\mathrm{PG}}$ scale is defined from the energy for which the dip in the continuum ends. The inset in (d) corresponds to a zoom of the $B_{1g}$ Raman response in order to point out the $2{\mathrm{\Delta }}_{\mathrm{SC}}^{\text{AN}}$.

Figure 7

(a) and (b) and (g) and (h): Temperature dependence of the nodal Raman responses ($B_{2g}$) of ${\mathrm{HgBa}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{8+\delta }$ (Hg-1223) for several doping levels. The features related to the CDW and the nodal SC gap are indicated by black arrows. (c) and (d) and (h) and (j) : Nodal Raman responses below $T_c$, after subtracting the one at $T_0$. The $T_0$ values for each doping are listed in the text. In the insets the black curve corresponds to the CDW hump rid of the nodal SC component, the full red curve is a ASG fit of the SC nodal gap subtracted (see text for more details). (e) and (f) and (k) and (l): Nodal Raman responses above $T_c$, after subtracting the one at $T_0$ to highlight the CDW structure (dip and hump).

Figure 8

Left and right panels are, respectively, the subtracted Raman responses of Hg-1223, $\mathrm{\Delta }{\chi }_{B_{2g}}^{\prime \prime }(T\approx 12\text{K},T_0$) and $\mathrm{\Delta }{\chi }_{B_{2g}}^{\prime \prime }(T_c\approx 25\text{K},T_0$) for various doping levels. The $T_0$ values for each doping are listed in the text. The black arrows indicate the location of ${\mathrm{\Delta }}_{\mathrm{CDW}}\left(p\right)$ and ${\mathrm{\Delta }}_{\mathrm{SC}}^{\text{N}}\left(p\right)$. The red dotted line is just a guide for the eyes.