Electronic Raman scattering in a multiband model for cuprate superconductors

Charge-charge, current-current, and Raman correlation functions are derived in a consistent way using the unified response theory. The theory is based on the improved description of the conduction electron coupling to the external electromagnetic fields, distinguishing further the direct and indirect (assisted) scattering on the quasistatic disorder. The two scattering channels are distinguished in terms of the energy and momentum conservation laws. The theory is illustrated on the Emery three-band model for the normal state of the underdoped high-$T_c$ cuprates which includes the incoherent electron scattering on the disorder associated with the quasistatic fluctuations around the static antiferromagnetic (AF) ordering. It is shown, for the first time consistently, that the incoherent indirect processes dominate the low-frequency part of the Raman spectra, while the long-range screening which is dynamic removes the long-range forces in the $A_{1g}$ channel. In the mid-infrared frequency range the coherent AF processes are dominant. In contrast to the nonresonant $B_{1g}$ response, which is large by itself, the resonant interband transitions enhance both the $A_{1g}$ and $B_{1g}$ Raman spectra to comparable values, in good agreement with experimental observation. It is further argued that the AF correlations give rise to the mid-infrared peak in the $B_{1g}$ Raman spectrum, accompanied by a similar peak in the optical conductivity. The doping behavior of these peaks is shown to be correlated with the linear doping dependence of the Hall number, as observed in all underdoped high-$T_c$ compounds.

Figure 1

(a) The purely electronic intraband Raman correlation functions in a pure system. (b) The Raman vertex (full rectangle) shown in terms of the bare Raman vertex (full circle) and the interband current vertices (open circles).

Figure 2

(a) The Raman correlation functions in a general case with the long-range forces and the quasielastic scattering processes taken into account. The full rectangle is the Raman vertex of Fig. 1b. The shaded box includes the electron-hole self-energy contributions associated with both the long-range forces and the scattering processes on the disorder. (b) The long-range screening of the Raman correlation functions in the case where the scattering processes on the disorder are absent. The open circles represent the charge vertices and the dashed line is the long-range force $4\pi e^2∕q^2$.

Figure 3

Two typical quadratic (a) and bilinear (b) direct Raman scattering processes in the conduction band proportional to ${\left(H_1^{\prime }\right)}^2$. The self-energy parts on the diagrams treated as constant are encircled (Ref. 41). The crosses represent the quasielastic scattering $H_1^{\prime }$.

Figure 4

The direct $(1\to 2\to 3)$ and indirect (forward, $1\to 2\to 4\to 5$, or backward, $1\to 2\to 4\to 6$) bilinear Raman scattering processes in the conduction band. The solid lines represent the three effective fermionic bands (the indices $c$, $P$, and $N$) for the typical values of the model parameters ${\Delta }_{pd}^{\mathrm{eff}}=0.66\mathrm{eV}$ and $t_{pd}^{\mathrm{eff}}=0.73\mathrm{eV}$ (Refs. 40, 54). The energies are measured with respect to the energy of the $2p_{\sigma }$ oxygen orbitals, so that the dispersionless nonbonding band is placed at $E_p=0$. The dashed lines are the photon dispersions, and the dotted-dashed line is the Fermi energy $\mu =-1.793\mathrm{eV}$ corresponding to the hole doping $\delta =0.1$.

Figure 5

A few direct contributions to the Raman correlation functions in powers of ${\left(H_1^{\prime }\right)}^2$, according to Eqs. (C9, C8, C4). The full rectangle is the effective Raman vertex of Fig. 1b.

Figure 6

The Coulomb screening of the direct (a) and indirect (b) processes in the Raman response functions in presence of the quasielastic scattering. The dotted box includes the electron-hole self-energy contributions associated with the quasielastic scattering processes.

Figure 7

Three typical indirect Raman scattering processes proportional to ${\left(H_1^{\prime }\right)}^2$. The first two include the incoherent scattering of conduction electrons while the third shows the incoherent scattering in the empty band(s) (the notation is the same as in Fig. 4). The effective vertices are encircled (Ref. 41).

Figure 8

(a) The direct and indirect high-frequency contributions [proportional to ${\left(H_1^{\prime }\right)}^2$] to the Raman correlation functions [Fig. 3 and Figs. 7a, 7b]. (b) A few indirect leading terms in powers of ${\left(H_1^{\prime }\right)}^2$.

Figure 9

(a) The effective Raman vertex (open rectangle) in the indirect processes, $\left[{\widehat{\gamma }}_{\nu }^{cc}\right(\mathbf{k},{\omega }_i)-{\widehat{\gamma }}_{\nu }^{cc}({\mathbf{k}}^{\prime },{\omega }_i\left)\right]V_1^{cc}(\mathbf{k}-{\mathbf{k}}^{\prime })∕(\hslash \omega )$. (b) The expansion of the indirect contribution to the Raman correlation functions in powers of ${\left(H_1^{\prime }\right)}^2∕\omega$, with the leading term explicitly shown in (c). The shaded box is the electron-hole propagator which is obtained by the self-consistent solution of the equation shown in (d) (Ref. 41). The diamond is the electron-hole self-energy containing both the single-particle self-energy and vertex corrections.

Figure 10

The dependence of the effective density of states on the Fermi energy $\mu$, for ${\Delta }_{pd}^{\mathrm{eff}}=0.66\mathrm{eV}$, $t_{pd}^{\mathrm{eff}}=0.73\mathrm{eV}$, and $t_{pp}=0$. The label $\nu =1$ denotes the ordinary density of states (divided by a factor of 5), and $\nu =A_{1g}$, $B_{1g}$, and $B_{2g}$ correspond to three Raman polarizations. For clarity the $B_{2g}$ density of states is multiplied by 10. The hole picture is used, i.e., the upper band boundary corresponds to the hole doping $\delta =1$ (measured with respect to half-filling). The doping range $0<\delta <0.3$, relevant to the hole doped high-$T_c$ cuprates, is indicated by two arrows.

Figure 11

Inset: The resonant enhancement of the $A_{1g}$ density of states $n_{A_{1g}}$ for $\delta =0.1$ ($\mu =-1.793\mathrm{eV}$ in Fig. 10), and $\hslash {\Gamma }^{\mathrm{inter}}=0.1\mathrm{eV}$ (solid and dotted line) and $0.15\mathrm{eV}$ (dashed line). The dotted (solid, dashed) line represents the contributions of the real (real and imaginary) part(s) in ${\gamma }_{\nu }^{cc}(\mathbf{k},{\omega }_i)$ to $n_{A_{1g}}$. Main figure: The total bare effective density of states for all three Raman channels for $\hslash {\Gamma }^{\mathrm{inter}}=0.1\mathrm{eV}$. The $B_{2g}$ spectrum is again multiplied by 10.

Figure 12

Main frame: Effective numbers $n_{xx}^{\mathrm{eff}}$ and $\mid n_H\mid$ (representing also the dc conductivity and the inverse Hall coefficient, scaled by $e^2∕\left(m{v}_0{\Gamma }_x^{c,\mathrm{id}}\right)$ and $ec∕v_0$, respectively) as a function of the doping level for ${\Delta }_{\mathrm{AF}}=0$. Inset: The effect of the AF correlations on $n_{xx}^{\mathrm{eff}}$ and $\mid n_H\mid$ for the $d_{x^2-y^2}$ symmetry perturbation $\Delta \left(\mathbf{k}\right)$ with ${\Delta }_{\mathrm{AF}}=50\mathrm{meV}$, in the hole-doped region. The critical doping ${\delta }_0$, where $n_{xy}^{\mathrm{eff}}=0$, is labeled by arrows. $n$ and $p$ denote, respectively, the region of electronlike $(n_H,n_{xy}^{\mathrm{eff}}<0)$ and holelike $(n_H,n_{xy}^{\mathrm{eff}}>0)$ behavior of the charge carriers.

Figure 13

The optical conductivity (45) for the anisotropic-$s$ potential $\Delta \left(\mathbf{k}\right)={\Delta }_{\mathrm{AF}}{[0.5+0.125{(\cos \mathbf{k}\bullet {\mathbf{a}}_1-\cos \mathbf{k}\bullet {\mathbf{a}}_2)}^2]}^{1∕2}$ with ${\Delta }_{\mathrm{AF}}=45\mathrm{meV}$, ${\Delta }_{pd}^{\mathrm{eff}}=0.66\mathrm{eV}$, $t_{pd}^{\mathrm{eff}}=0.73\mathrm{eV}$, $\delta =0.1$, ${\zeta }_1=0.18$, and ${\zeta }_2=0.4$. Main figure: $\hslash {\Gamma }_{\alpha }^{c,\mathrm{id}}=30\mathrm{meV}$ and $\hslash {\Gamma }_{\alpha }^{\mathrm{MIR}}=50\mathrm{meV}$ (suitable to $T=200\mathrm{K}$ spectra in the ${\mathrm{La}}_2{\mathrm{CuO}}_4$ based compounds). Inset: $\hslash {\Gamma }_{\alpha }^{c,\mathrm{id}}=15\mathrm{meV}$ and $\hslash {\Gamma }_{\alpha }^{\mathrm{MIR}}=25\mathrm{meV}$ $(T\approx 100\mathrm{K})$. The data measured in ${\mathrm{La}}_2{\mathrm{CuO}}_{4.12}$ at $T=200\mathrm{K}$ (Ref. 56) connected by the dotted line are given for comparison.

Figure 14

The $B_{1g}$ (a) and $B_{2g}$ (b) electronic Raman spectra obtained by ERVA for the $d_{x^2-y^2}$ symmetry potential $\Delta \left(\mathbf{k}\right)$. The parameters are ${\Delta }_{pd}^{\mathrm{eff}}=0.66\mathrm{eV}$, $t_{pd}^{\mathrm{eff}}=0.73\mathrm{eV}$, $\delta =0.1$, $\hslash {\omega }_i=2\mathrm{eV}$, $\hslash {\Gamma }_{\nu }^{\mathrm{MIR}}=50\mathrm{meV}$, and $\hslash {\Gamma }^{\mathrm{inter}}=0.1\mathrm{eV}$. The curves $A$ $\left(B\right)$: ${\Delta }_{\mathrm{AF}}=0$ (45) meV and $\hslash {\Gamma }_{\nu }^{c,\mathrm{id}}=30\mathrm{meV}$. The curves $C$: ${\Delta }_0=45\mathrm{meV}$ and $\hslash {\Gamma }_{\nu }^{c,\mathrm{id}}=15\mathrm{meV}$ [with the Drude (dotted line) and MIR (dashed line) contributions indicated as well]. The $B_{2g}$ spectrum is multiplied by 10.