Inelastic light scattering from correlated electrons

Inelastic light scattering is an intensively used tool in the study of electronic properties of solids. Triggered by the discovery of high-temperature superconductivity in the cuprates and by new developments in instrumentation, light scattering in both the visible (Raman effect) and x-ray part of the electromagnetic spectrum has become a method complementary to optical (infrared) spectroscopy while providing additional and relevant information. The main purpose of the review is to position Raman scattering with regard to single-particle methods like angle-resolved photoemission spectroscopy, and other transport and thermodynamic measurements in correlated materials. Particular focus will be placed on photon polarizations and the role of symmetry to elucidate the dynamics of electrons in different regions of the Brillouin zone. This advantage over conventional transport (usually measuring averaged properties) provides new insights into anisotropic and complex many-body behavior of electrons in various systems. Recent developments in the theory of electronic Raman scattering in correlated systems and experimental results in paradigmatic materials such as the A15 superconductors, magnetic and paramagnetic insulators, compounds with competing orders, as well as the cuprates with high superconducting transition temperatures are reviewed. An overview of the manifestations of complexity in the Raman response due to the impact of correlations and developing competing orders is presented. In a variety of materials, observations which may be understood and a summary of important open questions that pave the way to a detailed understanding of correlated electron systems, are discussed.

Figure 1

(Color) Characteristic Raman-scattering spectrum taken on $\left({\mathrm{Y}}_{0.92}{\mathrm{Ca}}_{0.08}\right){\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.3}$. Light scattering contributions from phonons, magnons, and electrons are plotted in blue, green, and red, respectively. $p$ denotes the number of carriers per copper oxide plaquette. The inset shows the photon polarization in the $\mathrm{Cu}{\mathrm{O}}_2$ plane and the realted form factor in the corresponding Brillouin zone. Courtesy of Matthias Opel.

Figure 2

(Color) Schematic drawing of the light path. The laser light with energy $\hslash {\omega }_i$ is first spatially filtered. A prism monochromator (PMC) is used to suppress the plasma lines of the laser medium, only the coherent one at $\hslash {\omega }_i$ passes the slit. The polarization is prepared with polarizer P1 and the Soleil-Babinet compensator (SB). The $\lambda ∕2$ retarder in front of P1 allows one to adjust the power. Before hitting the sample the light is once again spatially filtered to maintain an approximately Gaussian intensity profile in the spot. If the angle of incidence is not close to zero, phase shift effects at the sample surface must be taken into account since the polarization inside the sample is important. High speed optics collects the scattered light. The polarization state is selected by a $\lambda ∕4$ retarder and polarizer P2. The $\lambda ∕2$ retarder in front of the entrance slit rotates the light polarization for maximal transmission of the spectrometer (here single stage). Except for the compensator most retarders and polarizers work also for light propagating at small angles (up to approximately $\pm 5°$) with respect to the optical axis. The configuration shown here is usually referred to as backscattering geometry since incoming and outgoing photons have essentially opposite momenta in the sample.

Figure 3

(Color online) Cartoon showing light scattering via nonresonant intraband scattering.

Figure 4

(Color online) Cartoon showing light scattering via interband transitions.

Figure 5

Feynman diagrams contributing to the effective light-scattering operator $M$. (a) represents single-photon absorption, (b) two-photon scattering, while photon emission (absorption) followed by absorption (emission) is shown in (c) and (d). The panels on the right are the time-reversed partners of the left diagrams.

Figure 6

Feynman diagrams contained in the cross section. (a) A renormalized photon propagator in the solid, while (b)–(d) vanish due to parity and thus (a)–(d) do not contribute to inelastic light scattering. Of the remaining contributions, (e) refers to intraband (sometimes nonresonant) scattering within a single band, (f) and (g) are referred to as mixed contributions while (h)–(j) describe transitions within a single or between different bands via intermediate band states. If the light energy is equal or close to the energy difference of the states involved, resonance effects with a strong enhancement of the cross section occur. Therefore the contributions themselves are sometimes referred to as resonant.

Figure 7

(Color) Schematic weighting of the light-scattering transition for polarization orientations transforming as $B_{1g}$ and $B_{2g}$ for a $D_{4h}$ crystal. High-symmetry points are indicated. Here a typical Fermi surface for optimally doped cuprates is represented by the solid line, and the orientations of the incident and scattered polarization light vectors are shown with respect to copper-oxygen bond directions.

Figure 8

Feynman diagrams for nonresonant Raman scattering. The wavy and solid lines denote photon and electron propagators, respectively. The cross-hatched rectangle is the reducible charge vertex. The symbol $\gamma$ denotes the stress-tensor vertex of the corresponding electron-photon interaction. From 350.

Figure 9

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. From 350.

Figure 10

Feynman diagrams for the resonant contributions to Raman scattering. From 350.

Figure 11

Feynman diagrams for a typical parquetlike renormalization. This resonant diagram has simultaneous horizontal and vertical renormalizations by the two-particle reducible charge vertex. From 350.

Figure 12

Raman response of an electron gas. From 308.

Figure 13

Raman response from impurity scattering in an otherwise noninteracting system. From 443.

Figure 14

Raman response as a function of temperature obtained by 413 for a system with a nested Fermi surface.

Figure 15

Inverse Raman slope [see Eq. (55)] close to a metal-insulator transition at a value $U=\sqrt{2}$ in the Falicov-Kimball model for $D=\infty$ (124). All energies and temperatures are measured in terms of the hopping $t$.

Figure 16

Polarization-dependent Raman response in a spin-fermion model for a fixed temperature for three different values of the coupling constant. From 96.

Figure 17

(Color online) Cartoon of the two-magnon scattering process in a 2D Heisenberg antiferromagnet. An incident photon causes an electron with spin $\sigma$ to hop leaving a hole and creating a double occupancy in the intermediate state with energy $\propto U$. One particle of the double with spin $-\sigma$ hops back to the hole site liberating a photon with energy $\sim U-zJ\sigma$, leaving behind a locally disturbed antiferromagnet with $z$ exchange bonds broken in the final state as indicated by dotted lines.

Figure 18

The dependence of (a) the two-magnon $B_{1g}$ Raman intensity (shown in the inset for ${\omega }_i=8t$) and of (b) the absorption spectrum on the incoming photon energy ${\omega }_i$ in a 20-site cluster of the Hubbard model with $U=10t(J=0.4t)$. The solid line in (b) is obtained by performing a Lorentzian broadening with a width of $0.4t$ on the delta functions denoted by vertical bars. From 388.

Figure 19

Raman response in the half-filled Falicov-Kimball model $(U=2t)$ as a function of transfered frequency for various temperatures for fixed incident photon frequency ${\omega }_i=2t$. The thickest curve is $T∕t=0.05$, and the temperature increases to 0.2, 0.5, and 1 as the curves are made thinner. From 349.

Figure 20

Raman response for an $s$-wave superconductor for $q\xi \pi ∕2=0.1$, 0.5, 1.0, 2.0, 4.0, and 8.0, with $\xi =\hslash v_F∕\pi \Delta$ the Pippard-BCS coherence length $\xi$ and $q\simeq 1∕\delta$ the momentum transfer in a metal with skin depth $\delta$. From 210.

Figure 21

Raman spectra for unscreened $A_{1g}$ (top), $B_{1g}$ (middle), and $B_{2g}$ (bottom) response as a function of reduced temperature $t=T∕T_c$. Here $2\Delta \sim 37\mathrm{meV}$, and the higher peak in $A_{1g}$ and $B_{1g}$ channels is a van Hove feature. From 48.

Figure 22

Comparison of the screened $A_{1g}$ response obtained for different number of $d_{x^2-y^2}$ gap harmonics $\Delta \left(\mathbf{k}\right)={\Delta }_0\{[\cos \left(k_{x}a\right)-\cos \left(k_{y}a\right)]∕2+{\Delta }_1{[\cos \left(k_{x}a\right)-\cos \left(k_{y}a\right)]}^3∕8+{\Delta }_2{[\cos \left(k_{x}a\right)-\cos \left(k_{y}a\right)]}^5∕32\}$. Here $a–d$ correspond to the set ${\Delta }_{1,2}=(0,0)$, (1,0), (0,1), and (1,1), respectively, and ${\Delta }_0$ has been rescaled to give the same value for the maximum gap. Generally the large peak at $2\Delta$ (as shown in Fig. 21) is suppressed by backflow terms. From 100.

Figure 23

Electronic Raman spectra of $B$ symmetries for three different doping levels including coupling of $d$-CDW and $d$-superconducting amplitudes. The superscript $0$ denotes the spectra in the absence of collective modes. From 446.

Figure 24

Raman spectra of ${\mathrm{Nb}}_3\mathrm{Sn}$. Lower curves in (a) and (b) are at $40\mathrm{K}$ and upper curves are at $1.8\mathrm{K}$. Data in (c) and (d) are at $1.8\mathrm{K}$. Below $T_c=18\mathrm{K}$ the intensity at low energies is strongly suppressed with respect to the normal state. Beyond a threshold of approximately $50{\mathrm{cm}}^{-1}$ a new peak appears. The symmetries are $E_g$ [(a), (d)], $T_{2g}$ (b), $E_g+A_{1g}$ (c top), $E_g$ (c middle), and $A_{1g}$ (c bottom). The $A_{1g}$ data in (c) are obtained by subtracting the middle from the upper curve. Smooth solid lines at low energy are theoretical fits to a broadened Maki-Tsuneto function (see Sec. 2D6). From 101.

Figure 25

Raman spectra in $E_g$ symmetry of ${\mathrm{V}}_3\mathrm{Si}$. From 272.

Figure 26

Raman spectra of ${\mathrm{Nb}}_3\mathrm{Sn}$ at (a) $E_g$ and (b) $A_{1g}$ symmetries. The integrated spectral weight (limits indicated by arrows) stays constant to within 3% in $E_g$ symmetry while increasing by a factor of 3 in $A_{1g}$ symmetry on cooling from $T_c$ to $6\mathrm{K}$. For clarity the data points are omitted and only the results of a smoothing procedure are displayed. The scatter of the data is smaller than $0.1\text{unit}$ around the lines. From 156.

Figure 27

Raman spectra for $xx$ and $zz$ polarization geometries in the superconducting state of $\mathrm{Mg}{\mathrm{B}}_2$ taken by 315. The solid lines are fits to the data using the theory of 88 for disordered $s$-wave superconductors using a single-gap and two-gap model for $zz$ and $xx$ polarizations. The $xx$ spectrum has also been interpreted in terms of a collective bound state by 445.

Figure 28

Raman spectra of $2H\text{−}\mathrm{Nb}{\mathrm{Se}}_2$ above and below $T_c$. The lower curves of each panel are taken in the normal state $\left(9\mathrm{K}\right)$ and the upper ones at $2\mathrm{K}$ well below $T_c$ with the sample immersed in superfluid He. At the $A$ $(\Vert -\perp )$ and $E$ (⊥) symmetries the superconducting spectra are offset by 40 and 20 counts, respectively. According to the strength of the CDW mode (labeled by C) samples B and M have slightly different impurity concentrations. From 365.

Figure 29

Raman spectra in $A$ symmetry of $2H\text{−}\mathrm{Nb}{\mathrm{Se}}_2$ (sample B) in various magnetic fields at $2\mathrm{K}$. From 365.

Figure 30

Temperature dependence of the Raman response measured in FeSi by 288. Here ${\Delta }_c$ denotes the position of energy gap developing in the continuum at low temperatures, transferring spectral intensity into the peak at energy ${\Delta }_0$. The sharp low-energy features are phonons.

Figure 31

Raman response in a charge-ordered system. (a) Temperature dependence of the Raman spectra in the $LL$ scattering geometry in ${\mathrm{Bi}}_{0.19}{\mathrm{Ca}}_{0.81}\mathrm{Mn}{\mathrm{O}}_3$. (b) Fractional change in the integrated quasielastic Raman scattering intensity $(50–350{\mathrm{cm}}^{-1})$ as a function of $T∕T_{\mathrm{co}}$ for samples with charge-ordering temperatures $T_{\mathrm{co}}=165$ (circles) and $210\mathrm{K}$ (triangles). Inset: Fractional change in the quasielastic scattering amplitude $A$ and fluctuation rate $\Gamma$ as a function of $T∕T_{\mathrm{co}}$ obtained via a fit of the data with Eq. (49). Lines are guides to the eye. (c) Example of a closed-loop path for charge motion in the charge-ordered phase $(x=0.5)$ which is not precluded either by the orbital configuration or by the spin environment. Filled and empty circles represent ${\mathrm{Mn}}^{3+}$ and ${\mathrm{Mn}}^{4+}$ sites, respectively. From 439.

Figure 32

(Color) Schematic phase diagram of the cuprates. On the hole-doped $\left(p\right)$ side long-range antiferromagnetic (AF) order disappears rapidly. The maximal superconducting (SC) transition temperature $T_c$ is strongly material dependent but always observed at $p\simeq 0.16$. On the electron-doped $\left(n\right)$ side the AF phase is more extended. $T_c$ does not exceed $30\mathrm{K}$ at $n\simeq 0.14$. $T^{*}$ represents the approximate crossover temperature to the pseudogap regime (387). On the left-hand side and the right-hand side the structures of prototypical ${\mathrm{Nd}}_{2-x}{\mathrm{Ce}}_x\mathrm{Cu}{\mathrm{O}}_4$ and ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$, respectively, are shown. The atoms are Cu: red; O: blue; La,Sr and Nd,Ce: yellow. All cuprates are tetragonal or close to tetragonal with small material-dependent deviations. ${\mathrm{Nd}}_{2-x}{\mathrm{Ce}}_x\mathrm{Cu}{\mathrm{O}}_4$ crystallizes in $T^{\prime }$ structure without O octahedra and a slightly shorter $c$ axis. ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ and all other hole-doped materials have octahedra which are cut into half for materials with more than one $\mathrm{Cu}{\mathrm{O}}_2$ plane.

Figure 33

Doping dependence of the two-magnon $B_{1g}$ spectra in a number of compounds at $300\mathrm{K}$. From 377.

Figure 34

Raman spectra in the (a) normal state and (b) superconducting state for $A_{1g}+B_{2g}$ (left panel) and $B_{1g}(+A_{2g})$ (right panel) orientations in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$. From 434.

Figure 35

Fit to the $B_{1g}$ and $B_{2g}$ data on as-grown ($T_c=86\mathrm{K}$, top panel) and oxygen-annealed ($T_c=79\mathrm{K}$, bottom panel) ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ from 371. Inset: log-log plot of the low-frequency $B_{1g}$ response, showing crossover from linear to cubic behavior at a characteristic frequency ${\omega }^{*}∕{\Delta }_0=0.38$ and 0.45 for the top and bottom panels, respectively. From 90.

Figure 36

Raman and neutron-resonance peak energies as a function of the critical temperature $T_c$ in $\mathrm{Y}{\mathrm{Ba}}_2{\left({\mathrm{Cu}}_{1-x}{\mathrm{Ni}}_x\right)}_3{\mathrm{O}}_{6.95}$ for $x=0$, 0.01, and 0.03. The horizontal line for the $B_{1g}$ peak positions is just a guide to the eye, while the $A_{1g}$ peak and the neutron resonance are fitted by a straight line representing $5k_B{T}_c$. From 134.

Figure 37

Pressure dependences of the superconducting peaks in $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.95}$ for $B_{1g}$ (full circles) and $A_{1g}$ (open circles) symmetries. The position of the $B_{1g}$ phonon is also indicated (triangles). Inset: Comparison of the pressure-dependent results to the Raman data on ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ samples at ambient pressure with different doping levels (196). From 142.

Figure 38

Resonance enhancement of the of the $A_{1g}$ Raman response in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{10+\delta }$ at $10\mathrm{K}$. From 232.

Figure 39

(Color) Doping dependence of the low-energy electronic Raman response of ${\mathrm{Pr}}_{2-x}{\mathrm{Ce}}_x\mathrm{Cu}{\mathrm{O}}_4$ single crystals and thin films for $B_{2g}$, $B_{1g}$, and $A_{1g}$ channels obtained with $647\text{−}\mathrm{nm}$ excitation. The columns are arranged from left to right in order of increasing cerium doping. Abbreviations UND, OPT, and OVD refer to underdoped, optimally doped, and overdoped samples, respectively. The normal-state response (light/red) measured just above the respective $T_c$ is decomposed for the $B_{2g}$ and $B_{1g}$ channels into a Drude-like component with a constant carrier lifetime (green dotted line) and an extended continuum (yellow dotted line). Superconducting spectra (dark/blue) are taken at $T\approx 4\mathrm{K}$. For the OPT crystal a low-frequency ${\omega }^3$ power law is shown at $B_{1g}$ symmetry. From 314.

Figure 40

(Color) Doping dependence of the $B_{1g}$ and $B_{2g}$ Raman spectra in Bi-2212 at 5 and $100\mathrm{K}$. From 377.

Figure 41

(Color) Compilation of the peak position ${\Omega }_{\text{peak}}$ in the Raman response in the superconducting state normalized to $T_{\mathrm{c}}^{\max }$. $B_{1g}$ (open diamonds) and $B_{2g}$ (squares) orientations are shown for a variety of compounds. The respective references are color coded as follows: Bi-2212 [red (408), green (375; 377), cyan (196), yellow (38; 241), pink (166), and gold (258)], LSCO [blue (377)], Bi-2223 [gray (258)], Tl-2201 [purple (285; 191; 137, 138)], Tl-2223 [olive (171; 369)], and Hg-1201 [black (136)]. The dashed black line is the interpolation formula $6T_c∕T_c^{\max }=6[1-82.6{(p-0.16)}^2]$. For comparison, results for twice the maximal leading edge gap of ARPES (circles) (54) and for the peak-to-peak energy in the tunneling density of states (triangles) (442) are included.

Figure 42

Direct comparison of the low-energy $B_{1g}$ and $B_{2g}$ Raman response functions measured at temperatures indicated (all below $T_c$) in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ for different values of Sr doping. The scale is the same for all frames. From 280.

Figure 43

(Color) Direct comparison of the low-energy $B_{1g}$ and $B_{2g}$ Raman response functions measured at temperatures indicated (all above $T_c$) in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ for different doping levels $p=0.15$, 0.20, and 0.22 (from left to right). From 407.

Figure 44

(Color) Raman relaxation rates (resistivities) at $B_{1g}$ and $B_{2g}$ symmetries of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$. $B_{1g}$ and $B_{2g}$ are sensitive to antinodal and nodal regions of the Brillouin zone as indicated. Dashed lines represent dc transport data. Using plasma frequencies from IR spectroscopies, the Drude formula is used for the conversion of resistivities to relaxation rates. From 158.

Figure 45

(Color online) Raman relaxation rates at $B_{1g}$ and $B_{2g}$ symmetries of ${\mathrm{Nd}}_{1.85}{\mathrm{Ce}}_{0.15}{\mathrm{Cu}}_2{\mathrm{O}}_4$ (213). $B_{1g}$ and $B_{2g}$ are sensitive to antinodal and nodal regions, respectively. The dashed line represents recent transport data for ${\mathrm{Pr}}_{1.85}{\mathrm{Ce}}_{0.15}{\mathrm{Cu}}_2{\mathrm{O}}_4$ thin films (78). The Drude model with a plasma energy of $1\mathrm{eV}$ (174) is used for the conversion of the resistivity to relaxation rates.

Figure 46

(Color) Low-energy response of underdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$. A Drude-like peak (443) with a characteristic energy ${\Omega }_c(x,T)$ is revealed after subtraction of the 2D response of the $\mathrm{Cu}{\mathrm{O}}_2$ planes. At $x=0.02$ (a) and 0.10 (b) the additional response is observed in $B_{2g}$ and $B_{1g}$ symmetry, respectively. The styles of the lines (colors) do not correspond to similar temperatures but rather highlight the scaling of the response with temperature: similar spectra are obtained if the temperatures differ by approximately a factor of 2. From 383.

Figure 47

(Color) Raman response ${\chi }_{\mu }^{″}(\omega ,T)$ of $\left({\mathrm{Y}}_{0.97}{\mathrm{Ca}}_{0.03}\right){\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.1}$ in (a) $B_{1g}$ and (b) $B_{2g}$ symmetry. The doping level is close to $p=0.02$. From 158.

Figure 48

(Color) Temperature-dependent Raman response for $cc$ polarization in ladder compounds. Upper inset: Fit of the data with Eq. (49) plus a small background. Lower inset: Two-leg ladder structure. From 146.