Doping dependence of coupling between charge carriers and bosonic modes in the normal state of high-$T_c$ superconductors

Recently, the doping dependence of the optical-conductivity scattering rate has been used by Hwang, Timusk, and Gu to gain some insight in the way the coupling between the charge carriers and the bosonic modes in high-$T_c$ superconductors depends on doping. These authors used the extended Drude analysis, which does not take into account the normal-state pseudogap explicitly. In this work, we calculated the optical conductivity within the rotating antiferromagnetism theory, which models explicitly the pseudogap. Then we analyzed the resistivity as a function of temperature $T$ and doping $p$. We extracted the scattering rate $1∕\tau$ by fitting the ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ resistivity data. We found that for $p$ smaller than a critical value $p_c$, $1∕\tau$ shows a marginal-Fermi-liquid dependence for $T$ greater than a $p$-dependent temperature $T_{\rho }^{*}$. But for $T<T_{\rho }^{*},1∕\tau$ deviates downward from this law. We attribute this depression to a stronger coupling between the charge carriers and the normal-state bosonic modes. Both this coupling and $T_{\rho }^{*}$ vanish at $p_c$ while superconductivity continues to be significant well above it. $p_c$ is interpreted as a quantum critical point, and $T_{\rho }^{*}$ as the pseudogap temperature $T^{*}$ because it is found to agree with the experimental data on $T^{*}$. We propose that a possible candidate for the bosonic modes may be the spin-wave excitations in the rotating frame of the rotating antiferromagnetic order.

Figure 1

The doping dependence of $Q$ and $D_0$ is shown for $T=0.033t$.

Figure 2

(a) The temperature dependence of $Q$ is shown for three values of doping. (b) Both $Q$ and $D_0$ are displayed vs $T$ for two values of doping. Set II of the Hamiltonian parameters is used.

Figure 3

The energy spectra $E_{\pm }\left(\mathbf{k}\right)$ are displayed along the symmetry lines. These spectra are calculated for $T=0.1t$ and two doping densities $p=0.04$ and 0.13 in the underdoped regime.

Figure 4

The energy spectra $E_{\pm }\left(\mathbf{k}\right)$ are displayed along the symmetry lines for doping $p=0.22$ and temperature $T=0.1t$.

Figure 5

The motion of the electrons between sites $A$ and $B$ is illustrated. The electrons hop to the NN and NNN lattice sites with energies $t$ and $t^{\prime }$, respectively. The lattice sites are labeled using even and odd indices because of the bipartite character of the lattice. Solid squares (oval) indicate the sites belonging to the sublattice $A\left(B\right)$.

Figure 6

We display the fit of the ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ resistivity data (Ref. 32) made using an MFL law and set I of the Hamiltonian parameters. (a) $1∕\tau =3.9T+0.4$ for $x\equiv p=0.15$. (b) $1∕\tau =3T+0.11$ for $x=0.2$.

Figure 7

We display the ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ resistivity data (Ref. 32) that were fitted by adjusting the relaxation rate. Set II of the Hamiltonian parameters was used here, but identical curves are obtained for set I. Obviously, what differs for the two sets is $1∕\tau$. The symbols indicate the experimental points for which we carried the calculation. (a) Underdoped to optimal regime. (b) Overdoped regime.

Figure 8

The $T$ dependence of $1∕\tau$ used to fit $\rho \left(T\right)$ for ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ in the underdoped regime is shown. Here, set I of the Hamiltonian parameters was used. The dashed lines are linear fits for $T>T_{\rho }^{*}$.

Figure 9

The $T$ dependence of $1∕\tau$ used to fit $\rho \left(T\right)$ for ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ in the underdoped regime is shown. Set II of the Hamiltonian parameters was used. The dashed lines are linear fits for $T>T_{\rho }^{*}$.

Figure 10

The $T$ dependence of $1∕\tau$ used to fit $\rho \left(T\right)$ in the overdoped regime of the material ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ is shown. Set I of the Hamiltonian parameters was used. The dashed lines are linear fits for $T>T_m$.

Figure 11

The $T$ dependence of $1∕\tau$ used to fit $\rho \left(T\right)$ in the overdoped regime of the material ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ is shown. Set II of the Hamiltonian parameters was used. The dashed lines are linear fits for $T>T_m$.

Figure 12

Comparison of RAFT’s values for $T^{*}\equiv T_{\rho }^{*}$ and $T_m$ with experiment. (a) $T^{*}$ vs doping. The experimental data are from Ref. 36 for the resistivity $\rho$ and susceptibility $\chi$, and from Ref. 37 for the Hall constant $R_H$. (b) We display $T_m$ vs doping shifted by 0.055. The data are from Ref. 38 for ${\mathrm{Y}}_{1-x}{\mathrm{Ca}}_x{\mathrm{Ba}}_2{\left({\mathrm{Cu}}_{1-y}{\mathrm{Zn}}_y\right)}_3{\mathrm{O}}_{7-y}$.

Figure 13

The doping dependence of $T^{*}$ and $T_Q^{*}$, and of ${\lambda }_{cb}\sim \Delta (1∕\tau )$ is shown. (a) $T_Q^{*}$ and $T^{*}$ vs doping. (b) $\Delta (1∕\tau )$ vs doping. Set II of the Hamiltonian parameters was used. The continuous and dashed lines are a guide to the eye only.

Figure 14

The diagonal component of the spectral function for set II of the Hamiltonian parameters is displayed vs $\omega$ for $\mathbf{k}=(\pi ,0)$ in (a), and $\mathbf{k}=(\pi ∕2,\pi ∕2)$ in (b). The diagonal DOS is plotted vs $\omega$ for set II in (c) and set I in (d). In (a)–(c), $T∕t=0.3$, $1∕t\tau =3.5$ (thick line), $T∕t=0.1$, $1∕t\tau =1$ (thin line), $T∕t=0.05$, $1∕t\tau =0.5$ (dashed-dotted line). In (d), $1∕t\tau =2$, 0.5, and 0.25 for $T∕t=0.4$, 0.1, and 0.05, respectively.