Interpretation of in-plane infrared response of high-$T_c$ cuprate superconductors involving spin fluctuations using quasiparticle spectral functions

The in-plane infrared response of the high-$T_c$ cuprate superconductors was studied using the spin-fermion model, where charged quasiparticles of the copper-oxygen planes are coupled to spin fluctuations. First, we analyzed structures of the superconducting-state conductivity reflecting the coupling of the quasiparticles to the resonance mode observed by neutron scattering. The conductivity $\sigma$ computed with the input spin susceptibility in the simple form of the mode exhibits two prominent features: an onset of the real part of $\sigma$ starting around the frequency ${\omega }_0$ of the mode and a maximum of a related function $W\left(\omega \right)$, roughly proportional to the second derivative of the scattering rate $[1∕\tau ]\left(\omega \right)$, centered approximately at $\omega ={\omega }_0+{\Delta }_0∕\hslash$, where ${\Delta }_0$ is the maximum value of the superconducting gap. The two structures are well-known from earlier studies. Their physical meaning, however, has not been sufficiently elucidated thus far. Our analysis involving quasiparticle spectral functions provides a clear interpretation. Second, we explored the role played by the spin-fluctuation continuum, whose spectral weight is known to be much larger than the one of the mode. We have shown that the experimental spectra of $1∕\tau$ can be approximately reproduced by augmenting the resonant-mode component of the spin susceptibility by a suitable continuum component with a considerably higher spectral weight and with a characteristic width of several hundreds meV. The computed spectra of $1∕\tau$ display a new structure in the midinfrared which is related to the finite width of the occupied part of the conduction band. Third, we investigated the temperature dependence (TD) of $\sigma$ assuming that the normal state spin susceptibility consists of an overdamped low energy mode and the continuum component. The differences between the experimental normal-state spectra and those of the superconducting state, including some interesting effects at higher frequencies, are reasonably well-reproduced. Motivated by recent experimental (ellipsometric) works by Molegraaf and co-workers [H. J. A. Molegraaf et al., Science 295, 2239 (2002)] and Boris and co-workers [A. V. Boris et al., Science 304, 708 (2004)], we further studied the TDs of the effective kinetic energy (KE) and of the intraband spectral weight $I_O$. Calculations for the trivial case of noninteracting quasiparticles in the normal state and a BCS-like superconducting state reveal a strong sensitivity of the TD of $I_O$ to details of the dispersion relation. The TDs of KE and $I_O$ in the interacting case, for the set of the values of the input parameters used throughout this work, are similar to those of the trivial case. The physics beyond the changes occurring when going from the normal to the superconducting state, however, is shown to be more complex, involving, besides the formation of the gap, also a feedback effect of the spin fluctuations on the quasiparticles and a significant shift of the chemical potential.

Figure 1

Fermi surface (thick line) for the dispersion given by Eq. (32) (a) and for the dispersion of Ref. 22 (b). Supplementary contours (thin lines) correspond to excitation energies of $\pm 0.033\mathrm{eV}$ and $\pm 0.050\mathrm{eV}$. A notation of some special points is introduced. The point CS (“cold spot”) is the crossing point of the Fermi surface and the BZ diagonal (dashed-dotted line starting in the $\Gamma$ point). The point HS (“hot spot”) is the crossing point of the Fermi surface and the antiferromagnetic BZ boundary (dashed-dotted line starting in the X point). The point B is located between the CS and the HS and its (dimensionless) coordinates are (1.96, 0.60) and (1.96, 0.78) in (a) and (b), respectively.

Figure 2

Spectral functions of the $\mathbf{k}$-points defined in Fig. 1. Our notation of spectral structures is introduced in (b).

Figure 3

Spectral functions from Fig. 2 (solid lines) compared with those obtained using the values of the input parameters of Ref. 22 (dashed lines).

Figure 4

Contributions of the selected $\mathbf{k}$-points to the real part of the optical conductivity. The onset features are marked by the arrows.

Figure 5

Contributions of the selected $\mathbf{k}$-points to the real part of the optical conductivity from Fig. 4 (solid lines) compared with those obtained using the values of the input parameters of Ref. 22 (dashed lines).

Figure 6

Real and imaginary parts of the optical conductivity (upper panel) and the contributions to ${\sigma }_1\left(\omega \right)$ of the three regions of the irreducible BZ defined in Fig. 7 (bottom panel).

Figure 7

Three regions of the irreducible BZ used in the discussion of the total conductivity.

Figure 8

Real part of the optical conductivity for the three forms of the superconducting gap of Eq. (42). Also shown is the result for ${\Delta }_k=0$.

Figure 9

Real and imaginary parts of the optical conductivity (upper panel) and the three components of ${\sigma }_1\left(\omega \right)$ (bottom panel) from Fig. 6 (solid lines) compared with those obtained using the values of the input parameters from Ref. 22 (dashed lines).

Figure 10

Spectra of the scattering rate and the mass enhancement factor extracted from those of ${\sigma }_1$ and ${\sigma }_2$ shown in Fig. 6.

Figure 11

Spectra of the function $W\left(\omega \right)$ defined by Eq. (6) obtained from the conductivities of Fig. 6 (upper panel). Bottom panel: the contributions of the three regions defined in Fig. 7 $(\Gamma =0)$.

Figure 12

(a) The $\Gamma =0\mathrm{meV}$ spectrum of $W\left(\omega \right)$ from Fig. 11 (solid line) together with the one corresponding to the values of the input parameters from Ref. 22 (dashed line). (b) The same as in Fig. 11b but for the values of the input parameters from Ref. 22.

Figure 13

(a) The ($\mathbf{q}$-integrated) spectra of the imaginary part ${\chi }^{″}$ of the spin susceptibility corresponding to the resonance mode (solid line) and to the continuum term introduced in Eq. (50) (dashed line), respectively. (b) Those used as a starting point in the computations of Sec. 4F. The solid and the dashed line correspond to the superconducting state $(T=20\mathrm{K})$ and to the normal state $(T=100\mathrm{K})$, respectively.

Figure 14

The conductivity and the scattering rate as functions of the parameters $b_{\mathrm{M}}$ and $b_{\mathrm{C}}$ expressing the spectral weights of the mode and the continuum, respectively.

Figure 15

The ($\mathbf{q}$-integrated) imaginary part ${\chi }^{″}$ of the continuum part of the spin susceptibility (a) and the corresponding spectra of the scattering rate (b) as functions of the parameter ${\omega }_C$ introduced in the text.

Figure 16

Calculated temperature dependencies of the conductivity, the scattering rate, and the mass enhancement factor. The dashed-dotted line in part (b) corresponds to an unrealistic case of the normal state, $T=100\mathrm{K}$, and the spin susceptibility containing the sharp mode.

Figure 17

Temperature dependencies of the total intraband spectral weight $I_O$, the effective kinetic energy per unit cell multiplied by $-\pi e^2∕\left(2d{\hslash }^2\right)$ (labeled as $-\mathrm{KE}$), and the chemical potential for noninteracting quasiparticles. Parts (a) and (b) correspond to the dispersion relations of Eq. (32) and of Ref. 22, respectively. For each temperature, the value of the chemical potential was adjusted to yield the total number of electrons per unit cell of 0.74 (a) and 0.84 (b). The symbols correspond to the BCS ansatz with the superconducting gap given by Eq. (33).

Figure 18

The same as in Fig. 17 but for quasiparticles coupled to the spin fluctuations. The lines correspond to the normal state and the closed symbols to the superconducting state. For each value of $T$ the value of $\mu$ was adjusted to yield the total number of electrons per unit cell of 0.76. The open symbols (the symbols × and +) correspond to an unrealistic case of the superconducting state, $\mu =\mu (\text{normal}\text{state},T=20\mathrm{K})$, $\chi (\mathbf{q},\omega )=\chi (\text{normal}\text{state},T=20\mathrm{K},\mathbf{q},\omega )$ $[\chi (\mathbf{q},\omega )=\chi (\text{superconducting}\text{state},T=20\mathrm{K},\mathbf{q},\omega )]$.