Electrodynamics of high-${T}_{c}$ superconductors

Electrodynamics of high- $T_{c}$ superconductors

D. N. Basov and T. Timusk

Rev. Mod. Phys. 77, 721 – Published 26 August 2005

Abstract

Recent studies of the electromagnetic response of high- $T_{c}$ superconductors using terahertz, infrared, and optical spectroscopies are reviewed. In combination these experimental techniques provide a comprehensive picture of the low-energy excitations and charge dynamics in this class of materials. These results are discussed with an emphasis on conceptual issues, including evolution of the electronic spectral weight in doped Mott-Hubbard insulators, the $d$ -wave superconducting energy gap and the normal-state pseudogap, anisotropic superfluid response, electronic phase segregation, emergence of coherent electronic state as a function of both temperature and doping, the vortex state, and the energetics of the superconducting transition. Because the theoretical understanding of these issues is still evolving the review is focused on the analysis of the universal trends that are emerging out of a large body of work carried on by many research teams. Where possible data generated by infrared/optical techniques are compared with the data from other spectroscopic and transport methods.

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DOI:https://doi.org/10.1103/RevModPhys.77.721

Authors & Affiliations

D. N. Basov

Department of Physics, University of California San Diego, La Jolla, California 92093-0319, USA

T. Timusk

Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1

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Vol. 77, Iss. 2 — April - June 2005

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Figure 1
(Color in online edition) Schematic phase diagram displaying characteristic temperatures for electron-doped $A_{2 - x} Ce Cu O_{4}$ and its hole-doped counterpart ${La}_{2 - x} {Sr}_{x} Cu O_{4}$ . The pseudogap temperature $T^{*}$ is from 433. A variety of experimental probes suggest that some form of spin ordering at $T < T_{s}$ can coexist with superconductivity for $x < 0.2$ (488; 219; 334). Electron-doped materials reveal the coexistence of superconductivity and long-range antiferromagnetic (AF) order (289). $T_{G}$ signifies an optical gap observed by 321 and by 503 reminiscent of the pseudogap on the hole-doped side of the phase diagram.Reuse & Permissions
Figure 2
(Color) The phase diagram of Bi-2212 based on transport and infrared measurements. The dashed line labeled $T^{*}$ is the boundary between the high-temperature strange-metal region and the pseudogap region. Its position is based on $c$ -axis transport measurements of 396. The region below it is the pseudogap region. The line denoted $T_{s}$ separates the pseudogap region from the coherent region just above $T_{c}$ , the superconducting transition temperature. Blue dots indicate temperature at which the resonance mode emerges in the data. Adapted from 205.Reuse & Permissions
Figure 3
The effect of the normal-state gaps on dc resistivity. The upper panel shows that the $c$ -axis resistivity begins to increase at around $300 K$ in the sample with the lowest doping level. This is due to the formation of the pseudogap. In the same sample the $a b$ -plane resistivity decreases faster than linear but at a lower temperature. This is due to the development of in-plane coherence. Insets show the doping dependence of the $c$ -axis and $a$ -axis resistivities. Adapted from 418.Reuse & Permissions
Figure 4
(Color in online edition) The superfluid density $(ρ_{s})$ vs the product of the dc conductivity and $T_{c}$ . Inset illustrates a relationship between the superfluid density (shaded area) and the optical conductivity of a superconductor. Adapted from 192.Reuse & Permissions
Figure 5
(Color) Reflectance spectra for La-214 crystals at various dopings. Red lines, $R_{a b} (ω)$ spectra at $300 K$ ; blue lines, $R_{c} (ω)$ spectra at $300 K$ . The interlayer response is nearly insulating even in superconducting phases. This behavior is modified at $T < T_{c}$ : black lines ( $10 − K$ data) show a sharp plasma edge, discussed in Sec. 7A. Data from 133, 134, 124, and 329.Reuse & Permissions
Figure 6
(Color in online edition) Characteristic energy scales in high- $T_{c}$ superconductors. Infrared and optical spectroscopies allow one to probe electronic excitations in the frequency range from $≃ 20 {cm}^{- 1}$ to at least $50 000 {cm}^{- 1}$ . Several other techniques described in this section are suitable for extending the low-energy boundary down to microwave frequencies.Reuse & Permissions
Figure 7
Model calculations of the optical conductivity, scattering rate, and mass enhancement $λ$ . Two bosonic spectral functions are used: a “square” spectrum (left panels) and a form capturing key features of the spin fluctuations spectrum (right panels). The coupling constant is equal to 1. From 357.Reuse & Permissions
Figure 8
Frequency dependence of the (a) real and (b) imaginary parts of the conductivity at 4.5 and $7 K$ compared to the predictions of the BCS weak-coupling limit using the formulas of Mattis and Bardeen with a finite scattering time $τ = 3 \times 10^{7} s$ . Also shown is the normal-state behavior at $T = 9 K$ fitted by the Drude model. The far-infrared data (stars) show no temperature dependence below $10 K$ . In this spectral range, the imaginary part of the conductivity $σ_{2} (ω)$ has large error bars. The inset shows the temperature dependence of the penetration depth $λ (T)$ compared with the predicted behavior. From 354.Reuse & Permissions
Figure 9
Doping dependence of $Y {Ba}_{2} {Cu}_{3} O_{x}$ at $4 K$ ( $100 K$ for the superconducting samples with $x = 6.4$ and 6.92). From 168.Reuse & Permissions
Figure 10
In-plane $(E ⊥ c)$ optical conductivity $σ (ω)$ obtained from a Kramers-Kronig analysis of the reflectivity data for various compositions of $Y {Ba}_{2} {Cu}_{3} O_{6 + x}$ . Adapted from 106.Reuse & Permissions
Figure 11
Evolution of the $c$ -axis optical conductivity spectra below $6 eV$ for ${La}_{2 - x} {Sr}_{x} Cu O_{4}$ . The inset displays infrared region in more detail. From 448.Reuse & Permissions
Figure 12
(Color in online edition) Infrared response of ${Bi}_{2} {Sr}_{2} {Ca}_{2} {CuO}_{8}$ single crystal: Top panel, the in-plane conductivity of optimally doped Bi-2212 untwinned single crystals probed with the polarization of incident light along the $a$ axis. The dashed lines show separately each of the contributions within the multicomponent picture of the electromagnetic response [Eq. (2)]. Bottom panel, the results of the analysis of the data in the top panel within the one-component model, implying a frequency-dependent scattering rate $1 ∕ τ (ω)$ (components are given by the dashed lines). From 361.Reuse & Permissions
Figure 13
Anisotropic conductivity of ${La}_{1.97} {Sr}_{0.03} Cu O_{4}$ (left panels) and ${La}_{1.96} {Sr}_{0.04} Cu O_{4}$ (right panels). Panels (a), (b), (e), and (f) show temperature and frequency dependence of the conductivity. The $a$ axis corresponds to the direction of spin stripes; the $b$ axis is across the direction of spin stripes. The lattice contribution has been subtracted from $σ_{1} (ω)$ in (e) and (f) by fitting the phonon peaks to Lorentzian oscillators. Panels (c) and (d) display the anisotropy of both ac and dc conductivities. Dashed line, $σ_{dc, a} (ω) ∕ σ_{dc, b} (ω)$ ; dotted line, $σ_{1, a} (ω) ∕ σ_{1, b} (ω)$ at $13 K$ . From 134.Reuse & Permissions
Figure 14
$Cu O_{2}$ plane conductivity $σ_{a} (ω)$ for various temperatures in both the normal and superconducting states of Y-123. Panel (a): nearly optimally doped crystal with $T_{c} = 93 K$ from top to bottom, $T = 120, 100, 90$ (dashed), 70, and $20 K$ ; panel (b): underdoped crystal with $T_{c} = 82 K$ at $T = 150, 120, 90, 80$ (dashed), 70, and $20 K$ ; panel (c): underdoped crystal with $T_{c} = 56 K$ at $T = 200, 150, 120, 100, 80, 60$ (dashed), 50, and $20 K$ . Note the persistence of the $500 − {cm}^{- 1}$ threshold above $T_{c}$ (above dashed curves). From 375.Reuse & Permissions
Figure 15
The optical conductivity for the Y-123 system for radiation polarized along the $c$ axis for four different oxygen dopings. In the highly doped material below $T_{c} = 93.5 K$ a gaplike depression opens up below $500 {cm}^{- 1}$ but at $10 K$ the conductivity is nonzero down to at least $100 {cm}^{- 1}$ . In the $x = 0.85$ material, the conductivity is clearly nonmetallic in nature, a trend that resolves itself into a pseudogap for $x < 0.70$ . Adapted from 195.Reuse & Permissions
Figure 16
Interlayer conductivity $σ_{1} (ω)$ of single-layer (left panels) and double-layer cuprates (right panels) at $T_{c}$ and $T ⪡ T_{c}$ . Experimental data: $Hg {Ba}_{2} Cu O_{4}$ (Hg-1201; 400); ${Tl}_{2} {Ba}_{2} Cu O_{6 + x}$ (Tl-2201; 46; 229), ${La}_{2 - x} {Sr}_{x} Cu O_{4}$ (126), ${Nd}_{2 - x} {Ce}_{x} Cu O_{4}$ (NCCO; 400), $Y {Ba}_{2} {Cu}_{3} O_{6.6}$ (YBCO; 195, 196), $\Pr_{y} Y_{1 - y} {Ba}_{2} {Cu}_{3} O_{7 - δ}$ (127), ${Pb}_{2} {Sr}_{2} (Y ∕ Ca) {Cu}_{3} O_{8}$ (368), ${Bi}_{2} {Sr}_{2} Ca {Cu}_{2} O_{z}$ (Bi-2212; 127). In single-layered systems the $σ_{1} (ω, T ⪡ T_{c})$ spectra are suppressed compared to $σ_{1} (ω, T_{c})$ data, in accord with type-II coherence factors relevant to the response of superconductors in the dirty limit. In contrast crossing between normal and superconducting data (shaded areas) is apparent in the conductivity spectra for several different classes of double-layered materials. Adapted from 127.Reuse & Permissions
Figure 17
The $c$ -axis reflectivity $R$ , real part of the conductivity $σ_{1} (ω)$ , and real part of the complex impedance $ρ_{1} (ω)$ above (dashed lines) and below $T_{c}$ (solid lines): left panels, $Y {Ba}_{2} {Cu}_{3} O_{6.93}$ ; right panels, $Y {Ba}_{2} {Cu}_{3} O_{7}$ . The thick gray lines depict fits of reflectance and of the conductivity using the multilayer model. The shaded areas show the electronic contribution due to superconducting carriers. In order to demonstrate that the total $Re ρ (ω)$ is a linear superposition of the subcell contributions, also displayed are the electronic contributions $Re ρ_{A}$ and $Re ρ_{B}$ along with the total dynamical resistivity, including the phonons, as obtained from the fit in the upper panel. The “superconducting” contribution to the conductivity is not confined to the $δ$ function at $ω = 0$ but also appears at finite energies in these bilayered materials (shaded regions in the middle panels). From 170.Reuse & Permissions
Figure 18
An account of the local fields associated with the transverse resonance in multilayered crystals, allowing one to accurately describe anomalies in the phonon line shapes and oscillator strengths: (a) Experimental spectra of the $c$ -axis conductivity, $σ = σ_{1} + i σ_{2}$ , of $Y {Ba}_{2} {Cu}_{3} O_{6.55}$ with $T_{c} = 53 K$ ; top panels, $σ_{1} (ω)$ spectra; bottom panels, $σ_{2} (ω)$ spectra, 엯, experimental data; solid lines, the fits obtained by using the multilayered model described in the text for (b) $T = 75 K$ and (c) $T = 4 K$ ; dotted lines in (b) and (c), the electronic contributions. From 304.Reuse & Permissions
Figure 19
The frequency-dependent scattering rate for various high-temperature superconductors plotted over a wide frequency range. The quantity plotted is $1 ∕ τ^{*}$ , the renormalized scattering rate. Adapted from 137.Reuse & Permissions
Figure 20
(Color in online edition) Electronic self-energy-effects probed by means of IR and ARPES spectroscopy: Left panels, real part, and middle panels, imaginary part of the frequency-dependent scattering rate at a series of doping levels for Bi-2212. UD, underdoped regime; OPT, optimally doped regime; OD, overdoped regime. There is a gradual decrease of the steplike onset in scattering as the doping level increases. The middle panels show that the scattering can be separated into two channels: a sharp peak and a broad background. Black lines in the middle panel show the differential spectra between data at $T \approx T_{c}$ and the lowest temperature. From 204. Right panel shows the self-energy spectra from ARPES measurements of 217.Reuse & Permissions
Figure 21
(Color in online edition) The frequency of the resonance as calculated from the optical scattering rate plotted against maximum $T_{c}$ for a series of high-temperature superconductors from 430. The family of superconductors with a $T_{c}$ that is over $90 K$ form a separate group from the low- $T_{c}$ group. Inset, small squares show the intensity of the $41 − meV$ magnetic resonance, while large squares show the peak $c$ -axis conductivity. Both features appear below $150 K$ and reach half of their peak values at the superconducting transition temperature. Medium squares show the pseudogap amplitude, which develops already at $300 K$ . From 430.Reuse & Permissions
Figure 22
(Color in online edition) Electron boson spectral function $W (ω)$ for an Y-123 system near optimal doping: Bottom panel, inverted data from 81 of the second derivative of the frequency-dependent scattering rate for YBCO compared with a model calculation in the lower panel showing that the higher-frequency fine structure can be used to extract the superconducting gap from the optical data. From 2.Reuse & Permissions
Figure 23
The extended Drude model: left panels, spectra of the $a$ -axis conductivity $σ_{1} (ω)$ , right panels, the scattering $1 ∕ τ (ω)$ extracted from the extended Drude model as described in the text for a series of Y-123 single crystals. Thick solid lines show the data at lowest temperature in the normal state: $10 K$ for a nonsuperconducting crystal with $y = 6.35$ and $T \sim T_{c}$ for superconducting compounds $y = 6.44$ , 6.55, and 6.65. The solid triangles in the left panels mark the positions of the mid-IR absorption bands. In the right panel for $y = 6.65$ the dashed line represents the linear $ω$ dependence. Top panels display the two-component analysis of the conductivity for a $y = 6.35$ crystal [Eq. (2)]. Dashed lines in the top left panel show separately the Drude and Lorentzian contributions. Top-right panel displays the model $1 ∕ τ (ω)$ spectra calculated from the oscillator fit. Clearly, the simple two-component description captures the key features of the experimental data. Adapted from 262.Reuse & Permissions
Figure 24
(Color in online edition) The $T$ dependence of in-plane resistivity (a) and the optical conductivity spectra for oxygen-reduced crystals of Nd-214. Adapted from 321.Reuse & Permissions
Figure 25
${La}_{1.9} {Sr}_{0.1} Cu O_{4}$ at specified temperatures. The inset shows the in-plane resistivity data for ${La}_{1.9} {Sr}_{0.1} Cu O_{4}$ up to $1000 K$ . From 203.Reuse & Permissions
Figure 26
Self-energy effects in IR and ARPES spectra. Triangles and circles show the full width at half maximum of the spectral peaks vs the binding energy of the spectral peak in ARPES, and the carrier scattering rate vs energy for Bi-2212 $(90 K)$ obtained from infrared reflectivity measurements (solid and dashed lines) after 357. Adapted from 224.Reuse & Permissions
Figure 27
Ellipsometric data for the $a$ -axis component of the real part of the (i) conductivity and (ii) dielectric function of $Y {Ba}_{2} {Cu}_{3} O_{6.95}$ $(T_{c} = 91.5 K)$ at $100 K$ (thin solid lines) and at $10 K$ (thick solid lines). The inset in the top panel shows the conductivity data over an extended frequency region. The inset in the bottom panel displays the reflectance spectra calculated from ellipsometry data (52). Ellipsometric results for the optical constants of $Y {Ba}_{2} {Cu}_{3} O_{6.95}$ are in excellent agreement with the data inferred from the Kramers-Kronig analysis of the reflectance spectra and reveal the same characteristic features, including a threshold near $500 {cm}^{- 1}$ as well as residual absorption at lower energies.Reuse & Permissions
Figure 28
Microwave response of Y-123 single crystal near optimal doping: Top panel, the temperature-dependent part of the quasiparticle scattering rate obtained from the microwave data. The nearly straight line on a semilogarithmic scale indicates a scattering rate that varies as $\exp (T ∕ T_{0})$ , a temperature dependence that is also seen in NMR and nuclear quadrupole resonance measurements. From 63. Bottom panel, the $T$ dependence of the real part of the conductivity at several different frequencies in the microwave range (199). From 62.Reuse & Permissions
Figure 29
(Color in online edition) The low- $T$ evolution of the quasiparticle conductivity spectra of an ortho-II $Y {Ba}_{2} {Cu}_{3} O_{6.50}$ crystal: black triangles, $T = 1.3 K$ ; lighter triangles, $2.7 K$ ; squares, $4.3 K$ ; dots, $6.7 K$ . The solid curve is a fit to the $6.7 − K$ data with the Born scattering model described in the text. The dashed curves show that the model fails to capture the observed $T$ dependence. The inset shows that the data obey an unusual frequency-temperature scaling, $σ_{1} (ω, T) = σ_{1} (ω ∕ [T + T_{0}])$ , with $T_{0} = 2.0 K$ . The Drude fit in the inset illustrates the inadequacy of the Lorentzian line shape. From 446.Reuse & Permissions
Figure 30
The role of umklapp processes in the temperature dependence of the scattering rate: Left panel, schematic representation of the Fermi surface associated with a single $Cu O_{2}$ plane of Y-123. The $d$ -wave superconducting gap varies with momentum along the Fermi surface, going to zero at the nodes indicated by solid circles. The wave vectors $Q_{i}$ on the Fermi surface are associated with quasiparticles involved in an umklapp processes. $G$ is a reciprocal-lattice vector. Right panel, a comparison of the theoretical curve for the temperature dependence of the quasiparticle scattering rate $t_{el}^{- 1} (T)$ calculated by taking account of peculiarities of the umklapp processes in a $d$ -wave superconductor with the data of 199. Adapted from 469.Reuse & Permissions
Figure 31
Infrared response of untwinned YBCO single crys-tals: Left panels, frequency dependence of $λ (ω) = c ∕ \sqrt{σ_{2} (ω) ω}$ for Y-124 sample (upper panel), and two different Y-123 samples (middle and bottom panels). Right panels, the real part of the complex conductivity. All curves shown are at $10 K$ . Left panels reveal clear anisotropy of the penetration depth, which is stronger for the double-chained Y-124 material. Adapted from 40.Reuse & Permissions
Figure 32
(Color in online edition) The $c$ -axis penetration depth $λ_{c} (T ⪡ T_{c})$ plotted as a function of the $c$ -axis dc conductivity $σ_{dc}$ for a variety of layered superconductors. We find two distinct patterns of $λ_{c} - σ_{dc}$ scaling. Cuprate superconductors exhibit much shorter penetration depths than noncuprate materials with the same $σ_{dc} (T)$ . This result may be interpreted in terms of a dramatic enhancement of the energy scale from which the condensate is collected as described in the text. See 126 for a discussion of the raw data points and analysis. Inset: In a conventional dirty-limit superconductor the spectral weight of the superconducting condensate (given by $1 ∕ λ_{c}^{2}$ ) is collected primarily from the energy-gap region. The total normal weight is preset by the magnitude of $σ_{dc}$ whereas the product of $λ_{c} σ_{dc}$ quantifies the fraction of the weight that condenses.Reuse & Permissions
Figure 33
The temperature dependence of the superfluid fraction probed separately for $a$ , $b$ , and $c$ directions. From 200.Reuse & Permissions
Figure 34
Temperature dependence of the spectral weight associated with different components of the conductivity in THz region. The spectral weight of the total conductivity from $0.2 to 0.8 THz$ , $\sum (T)$ , the quasiparticle Drude conductivity, $\sum_{qp} (T)$ , and the difference between the two, $\sum (T) - \sum_{qp} (T)$ . The solid line displays the superfluid density $ρ_{s}$ multiplied by 0.18, showing the proportionality between this and the portion of the spectral weight not due to the quasiparticles. From 109.Reuse & Permissions
Figure 35
(Color in online edition) Spin ordering and inhomogeneous superfluid response in an La-214 system: Dots, the doping dependence of the asymmetry of the loss function $W_{JPR} ∕ W_{JPR, x = 0.125 B}$ . This parameter characterizes the degree of inhomogeneity of the superfluid density within the $Cu O_{2}$ planes, as discussed in the text. The correlation length $ξ_{s}$ of both static (squares) and dynamic (rhombs) spin structures is extracted from neutron-scattering experiments (488; 235). Adapted from 123.Reuse & Permissions
Figure 36
An infrared probe of the kinetic-energy change associated with the superconducting transition: Top panels, difference between the effective spectral weight in the interplane conductivity of several cuprates at $T ≃ T_{c}$ and at $T ⪡ T_{c}$ normalized by the magnitude of the superfluid density $ρ_{s}$ extracted from the imaginary part of the conductivity. In these three materials $[N_{n} (ω) - N_{s} (ω)] ∕ ρ_{s}$ does not exceed 0.5 at $ω ≃ 0.1 eV$ . In the case of ${La}_{2 - x} {Sr}_{2} Cu O_{4}$ and ${Tl}_{2} {Ba}_{2} Cu O_{6 + x}$ $[N_{n} (ω) - N_{s} (ω)] ∕ ρ_{s}$ spectra are shown without phonon subtraction (thin lines) and with phonons removed (thick lines). Approximate error bars are displayed at selected energies. Left bottom panel, calculations for a conventional dirty-limit superconductor with the scattering rate $Γ = 20 Δ$ , showing that about 90% of the superfluid density is collected from $ω < 4 Δ$ . This energy scale corresponds to about $0.1 eV$ for cuprates where the photoemission result for $2 Δ$ is used. Right-bottom panel, the spectra of $σ_{1} (ω)$ for a conventional dirty superconductor with $Γ = 20 Δ$ . Adapted from 46.Reuse & Permissions
Figure 37
Temperature dependence of the low-frequency spectral weight $A_{1 + D} (T)$ and the high-frequency spectral weight $A_{h} (T)$ , for optimally doped (top) and underdoped (bottom) ${Bi}_{2} {Sr}_{2} Ca {Cu}_{2} O_{8 - d}$ . Insets: derivatives $- T^{- 1} d A_{1 + D} ∕ d T$ and $T^{- 1} d A_{h} ∕ d T$ . Adapted from 297.Reuse & Permissions
Figure 38
The magnetoconductivity of ${YBa}_{2} {Cu}_{3} O_{7}$ thin film $Re [σ (ω, H)]$ obtained from a Kramers-Kronig analysis. The change induced by the magnetic field can be described as the sum (dotted line) of a low-frequency oscillator of width $Γ_{1} = 10 {cm}^{- 1}$ , $ω_{1} = + 3.15 {cm}^{- 1}$ (single dashed line). Adapted from 268.Reuse & Permissions
Figure 39
Evolution of the real and imaginary parts of $Δ σ = σ (H) - σ (0)$ at $150 GHz$ for a Bi-2212 thin film. Adapted from 282.Reuse & Permissions
Figure 40
A series of surface resistance traces vs field for a ${Bi}_{2} {Sr}_{2} Ca {Cu}_{2} O_{8 + δ}$ crystal for incident microwave frequency between 30 and $50 GHz$ at $4.2 K$ . The inset shows the frequency position of the resonance. Adapted from 443.Reuse & Permissions
Figure 41
Real part of the complex conductivity $σ_{1}$ and loss function $Im (1 ∕ ϵ^{*})$ $Sm {La}_{0.8} {Sr}_{0.2} Cu O_{4 - δ}$ along the $c$ axis for different magnetic fields at $2.3 K$ . The transverse plasmon $ν_{T}$ shows up as a peak in $σ_{1}$ and longitudinal plasmons $ν_{1, K}$ show up as peaks in $Im (1 ∕ ϵ^{*})$ . Adapted from 344.Reuse & Permissions
Figure 42
(Color in online edition) Interlayer response of an underdoped ${YBa}_{2} {Cu}_{3} O_{6.6}$ crystal in a magnetic field: Panel (a): $c$ -axis optical conductivity under a parallel field $H_{a b}$ . The rectangles indicate the weight of low-energy condensate as derived from fitting by the function $Re (ϵ) ≃ ϵ_{\infty} - ρ_{s} ∕ ω^{2}$ shown in the inset for 0 and $7 T$ (field-cooled) at $5.5 K$ . The solid circles in the inset indicate the data points used for the fitting. Panel (b): optical conductivity on a wider scale. Adapted from 242.Reuse & Permissions
Figure 43
Hall coefficient $R_{H}$ vs frequency for thin-film $Y {Ba}_{2} {Cu}_{3} O_{7}$ on a Si substrate sample probed in the Faraday geometry at $95 K$ : ●, the real part of $R_{H}$ derived from a Kramers-Kronig analysis of the transmission data in an $8.9 − T$ magnetic field; ∎, the imaginary part of the same $R_{H}$ . The solid and dashed curves show the predicted values from fitting an expression derived by 202 to the same data. Inset: Real (circles) and imaginary (squares) parts of $\cot Θ_{H}$ , along with the predictions of the theory by 201 (solid and dashed lines). The real part is $1 ∕ ω_{c} τ_{H}$ , and the slope of the imaginary part is $- 1 ∕ ω_{c}$ . Adapted from 226.Reuse & Permissions

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