Electronic Raman scattering in correlated materials: A treatment of nonresonant, mixed, and resonant scattering using dynamical mean-field theory

We solve for the electronic Raman scattering response functions on an infinite-dimensional hypercubic lattice employing dynamical mean-field theory. This contribution extends previous work on the nonresonant response to include the mixed and resonant contributions. We focus our attention on the infinite-dimensional spinless Falicov-Kimball model, where the problem can be solved exactly, and the system can be tuned to go through a Mott-Hubbard-like metal-insulator transition. Resonant effects vary in different scattering geometries, corresponding to the symmetries of the charge excitations scattered by the light. We do find that the Raman response is large near the double resonance, where the transfered frequency is close to the incident photon frequency. We also find a joint resonance of both the charge-transfer peak and the low-energy peak when the incident photon frequency is on the order of the interaction strength. In general, the resonance effects can create order of magnitude (or more) enhancements of features in the nonresonant response, especially when the incident photon frequency is somewhat larger than the frequency of the nonresonant feature. Finally, we find that the resonant effects also exhibit isosbestic behavior, even in the $A_{1\mathrm{g}}$ and $B_{2\mathrm{g}}$ sectors, and it is most prominent when the incident photon frequency is on the order of the interaction energy.

Figure 1

Feynman diagrams for nonresonant Raman scattering. The wavy lines denote photon propagators and the solid lines denote electron propagators. The cross-hatched rectangle is the reducible charge vertex. In the $B_{1g}$ channel, only the bare (first) diagram enters, while in the $A_{1g}$ channel both diagrams enter. The symbol $\gamma$ denotes the stress-tensor vertex of the corresponding electron-photon interaction.

Figure 2

Feynman diagrams for the mixed contributions to Raman scattering. The symbols $j_f$ and $j_i$ remind us to include the relevant vertex factors from the current operator in the electron-photon interaction. The mixed contribution vanishes in the $B_{2g}$ channel, it consists of only the bare diagrams on the top line in the $B_{1g}$ channel (and turns out to be a $1∕D$ correction), and all diagrams enter for the $A_{1g}$ channel.

Figure 3

Feynman diagrams for the resonant contributions to Raman scattering. Only the first two diagrams in the first four lines, and the first diagram in the last two lines contribute in the $B_{1\mathrm{g}}$ and $B_{2\mathrm{g}}$ sectors. The $A_{1\mathrm{g}}$ response includes all diagrams.

Figure 4

Interacting single-particle density of states for $U=0.5$, 1.0, 1.5, 2.0, and 3.0 ($U$ increases as the pseudogap becomes stronger). Note how the DOS first develops a depression near the chemical potential and then develops a pseudogap as the metal-insulator transition occurs (the DOS vanishes only at $\omega =0$ in the insulator).

Figure 5

Stokes Raman response for the three symmetry channels in a dirty metal with $U=0.5$. The Raman scattering response function is plotted as a function of the transfered frequency for incident photon frequencies ranging from 0.25 to 4.5 in steps of 0.25 (the thickness of the lines aids in distinguishing the different curves). This data is at low temperature $(T=0.05)$ where the results are the “sharpest.”

Figure 6

Stokes Raman response for the three symmetry channels in a strongly scattering “metal” with $U=1.0$. The Raman scattering response function is plotted as a function of the transfered frequency for incident photon frequencies ranging from 0.25 to 4.5 in steps of 0.25. This data is at the temperature $(T=0.5)$ where the nonresonant response has enhanced low-energy spectral weight in the $B_{1g}$ channel.

Figure 7

Stokes Raman response for the three symmetry channels in a correlated insulator with $U=3.0$. The Raman scattering response function is plotted as a function of the transfered frequency for incident photon frequencies ranging from 0.25 to 5.0 in steps of 0.25. This data is at a high temperature $(T=1.0)$ where the nonresonant response has enhanced low-energy spectral weight in $B_{1g}$ and $A_{1g}$ channels.

Figure 8

Separation of different contributions to the Stokes response for $U=3$, $T=1.0$, and ${\omega }_i=4.0$ in the (a) $B_{1g}$ and (b) $A_{1g}$ channels. The solid line is the total response, the dotted line is the resonant piece, the dashed line is the nonresonant piece, and the chain-dotted line is the mixed contribution.

Figure 9

Raman response at low energy for $U=3$ and $T=0.2$. The incident photon frequency changes from 0.1 to 1.0 in steps of 0.1.

Figure 10

Raman response at $\Omega =3$ for $U=3$ and various temperatures. The horizontal axis is the incident photon frequency. The thickest curve is $T=0.05$, and the temperature increases to 0.2, 0.5, and 1 as the curves are made thinner. Note that the curves for the lowest two temperatures are the largest, and are hard to separate.

Figure 11

Raman response at $\Omega =0.5$ for $U=3$ and various temperatures. The horizontal axis is the incident photon frequency. The thickest curve is $T=0.05$, and the temperature increases to 0.2, 0.5, and 1 as the curves are made thinner. Note that the $T=0.05$ curves are more than three orders of magnitude smaller than the $T=0.2$ curves, and cannot be distinguished from the horizontal axis in the figure.

Figure 12

Low-energy isosbestic behavior of the resonant Raman response for $U=3$ and ${\omega }_i=3.5$. The different thicknesses correspond to different temperatures, with thicker curves corresponding to lower temperatures (four curves are plotted for $T=1$, 0.5, 0.2, and 0.05). Note how the curves all cross at an isosbestic point, just slightly smaller than $U∕2$.

Figure 13

Feynman diagrams for a typical parquetlike renormalization. This resonant diagram has a simultaneous horizontal and vertical renormalization by the two-particle reducible charge vertex. Note that such a renormalization is only possible in the $A_{1\mathrm{g}}$ sector.