Temperature-Dependent Ellipsometry Measurements of Partial Coulomb Energy in Superconducting Cuprates

We performed an experimental study of the temperature and doping dependence of the energy-loss function of the bilayer and trilayer bismuth cuprates family. The primary aim is to obtain information on the energy stored in the Coulomb interaction between the conduction electrons, on the temperature dependence thereof, and on the change of Coulomb interaction when Cooper pairs are formed. We performed temperature-dependent ellipsometry measurements on several ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8-x}$ single crystals: underdoped with $T_c=60$, 70, and 83 K; optimally doped with $T_c=91\mathrm{K}$; overdoped with $T_c=84$, 81, 70, and 58 K; as well as optimally doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{10+x}$ with $T_c=110\mathrm{K}$. Our first observation is that, as the temperature drops through $T_c$, the loss function in the range up to 2 eV displays a change of temperature dependence as compared to the temperature dependence in the normal state. This effect at—or close to—$T_c$ depends strongly on doping, with a sign change for weak overdoping. The size of the observed change in Coulomb energy, using an extrapolation with reasonable assumptions about its $q$ dependence, is about the same size as the condensation energy that has been measured in these compounds. Our results therefore lend support to the notion that the Coulomb energy is an important factor for stabilizing the superconducting phase. Because of the restriction to small momentum, our observations do not exclude a possible significant contribution to the condensation energy of the Coulomb energy associated with the region of $q$ around $(\pi ,\pi )$.

Popular Summary

Ever since the discovery of high-temperature superconductivity in cuprates, the primary question has been why the critical temperature in these materials is much higher than in conventional superconductors. Since the supercurrent in cuprates is certainly carried by pairs of electrons, attempts to answer this question have typically concentrated on various types of mechanisms by which electrons are bound into pairs. However, the lack of consensus about the precise mechanism calls for a different approach. When the temperature of a normal conducting material is lowered below the critical temperature, the pair correlations are expected to change in a characteristic way, along with a change in the Coulomb interaction energy. Here, we use an optical technique to measure the pair-correlation function experimentally in double- and triple-layer bismuth cuprates.

We calculate the Coulomb energy by focusing on underdoped, optimally doped, and overdoped high-purity single crystals of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{10+x}$ and ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8-x}$. We find that the long-range Coulomb energy varies between $-1$ and 1 K, depending on doping, and that the condensation energy ranges from 0 to 2 K per ${\mathrm{CuO}}_2$ unit. Consequently, while the long-range Coulomb energy cannot be solely responsible for the superconductivity, it is nonetheless a major factor in the total energy balance stabilizing the superconducting state. Our experiments demonstrate that it is, in principle, possible to determine the subtle changes of Coulomb correlation energy associated with a superconducting phase transition.

Our findings constitute a promising first step in the experimental exploration of the Coulomb correlation energy as a function of momentum and energy.

Figure 1

S–N difference of the partial Coulomb energy $E_{\mathbf{q}}^C$. Left panel: According to the MIR scenario (schematic). Right panel: Resulting from a BCS model calculation [15] for $d$-wave symmetry, $p=0.16$ hole doping, and the interaction adjusted such as to give $T_c=100\mathrm{K}$. Both panels represent the $q_z=0$ cut in momentum space, corresponding to the electric field polarized along the planes. $E_{\mathbf{q}}^C$ at $\mathbf{q}=0$ corresponds to the integral [Eq. (1)] of the optical in-plane loss function.

Figure 2

Loss-function spectra for Bi2212 with carrier concentrations ranging from underdoped to overdoped for selected temperatures (lines 1 and 3). Temperature dependence of the integrated intensity, $E_C^{\text{iso}}(T)$, of the corresponding samples (lines 2 and 4).

Figure 3

Optimally doped Bi2223. (a) Loss-function spectra, (b) temperature dependence of the integrated intensity $E_C^{\text{iso}}(T)$, and (c) temperature dependence of ${\gamma }_C^{\text{iso}}(T)$.

Figure 4

Temperature dependence of ${\gamma }_C^{\text{iso}}(T)$ for Bi2212 with different carrier concentrations.

Figure 5

The S–N difference of the $\mathbf{q}$-integrated Coulomb energy $\mathrm{\Delta }{E}_C^{\text{mir}}$, together with the total energy difference $-E_{\mathrm{cond}}$ (data reproduced from Ref. [44], with original units converted to the present ones for the sake of comparison) and band-energy difference $\mathrm{\Delta }K$ [33].

Figure 6

Phase diagram summarizing the temperature characterizing the step in ${\gamma }_C^{\text{iso}}(T)$: The temperature of the extremum, coinciding with the superconducting phase transition ($T_{n1}/T_{p1}$), and the upper limit of the step ($T_{n2}/T_{p2}$).

Figure 7

Exponents of the equation $E_C^{\text{iso}}(T)=B+A{T}^{\alpha }$ obtained from fitting the normal state temperature dependence of the intraisosbestic integrated loss-function intensity, shown in Figs. 8, 9, 10, 11, 12, 13, 14, 15. The two values indicated for the 19% doped sample (dark gray) refer to different temperature regions.

Figure 8

Loss function and loss-function integrals of sample Bi2212-60-UD. The value of the fitted exponent $\alpha$ is 2.0 (right middle panel). Panel details are further specified in the main text.

Figure 9

Sample Bi2212-70-UD. Caption details as in Fig. 8. Because the signal for frequencies in zone 3 was not sufficiently stable above 256 K, the temperature range of the panels on the third line is limited below 256 K. The value of the fitted exponent $\alpha$ is 2.0.

Figure 10

Sample Bi2212-83-UD. Caption details as in Fig. 8. The value of the fitted exponent $\alpha$ is 2.3.

Figure 11

Sample Bi2212-91-OpD. Caption details as in Fig. 8. The value of the fitted exponent $\alpha$ is 2.2.

Figure 12

Sample Bi2212-84-OD. Caption details as in Fig. 8. The value of the fitted exponent $\alpha$ is 2 when fitting down to $T_c$ and 1 and 3 when splitting into $T_c-200\mathrm{K}$ and 200–300 K windows.

Figure 13

Sample Bi2212-81-OD. Caption details as in Fig. 8. The value of the fitted exponent $\alpha$ is 1.6.

Figure 14

Sample Bi2212-70-OD. Caption details as in Fig. 8. The value of the fitted exponent $\alpha$ is 1.8.

Figure 15

Sample Bi2212-58-OD. Caption details as in Fig. 8. The value of the fitted exponent $\alpha$ is 1.8.

Figure 16

Sample Bi2223-110-OP. Caption details as in Fig. 8. The value of the fitted exponent $\alpha$ is 2.7. The evolution as a function of temperature above 220 K was not fully reproducible because of a combination of factors having to do with small crystal size and instrument drift. The temperature range for this sample is therefore limited below 220 K.

Figure 17

Example of the $c$-axis correction obtained in 20 iterations: dielectric function (left panel) and corresponding loss function (right panel).

Figure 18

Bi2223 $c$-axis correction using ${\epsilon }_c(\omega ,T)$ from Ref. [77].

Figure 19

Isosbestics points for Bi2212 83K UD.

Figure 20

Bi2223 optimally doped with an extended low-energy region from Ref. [77].

Figure 21

Bi2223 optimally doped with the low-energy region extrapolated.

Figure 22

Relative change of the integral of the loss function between 0 and 2.5 eV with respect to room temperature of sample Bi2212-91-OpD. Black line: The “$q=0$” result, same as in the lower right panel of Fig. 11. Red, blue, green, and orange curves are the finite-momentum extrapolation to finite $q_0=0.31{\AA }^{-1}$, using different extrapolation schemes. Red curve: 2D bilayer expression, Eq. (14), with $n=2$, and $d=3.2\AA$. Blue curve: 2D monolayer expression, Eq. (g11). Green curve: Layered electron gas (LEG) expression, Eq. (g9), with ${\epsilon }_c={\epsilon }_{\mathrm{sc}}=4.5$. Orange: LEG expression, Eq. (g9), with ${\epsilon }_c(T)$ described by Eq. (g1). The overlap of orange and green curves indicates that the $c$-axis loss function has negligible influence on the temperature dependence of the Coulomb energy. The overlap of red, blue, and green curves implies that the extrapolation has negligible dependence on the type of extrapolation scheme chosen. While the black curve is the expression for the “true” Coulomb energy that one would get by simply multiplying the optically measured loss function by the area of the first Brillouin zone, the colored curves are expressions for the “MIR Coulomb energy” as discussed in Sec. 4a. The fact that the colored curves are a substantial fraction of the black one—rather than being related to it approximately by the ratio of the disk $q<0.31{\mathrm{A}}^{-1}$ to that of the first Brillouin zone—is largely attributable to the replacement of $E_C$ by $E_C^{\text{mir}}$; see Eq. (f17) for the 3D case and Eq. (g11) for the 2D case.

Figure 23

Partial integral of the loss function as a function of temperature obtained by using the scattering rate in a Drude model of Norman et al. [65].