Electrodynamics of correlated electron materials

Studies of the electromagnetic response of various classes of correlated electron materials including transition-metal oxides, organic and molecular conductors, intermetallic compounds with $d$ and $f$ electrons, as well as magnetic semiconductors are reviewed. Optical inquiry into correlations in all these diverse systems is enabled by experimental access to the fundamental characteristics of an ensemble of electrons including their self-energy and kinetic energy. Steady-state spectroscopy carried out over a broad range of frequencies from microwaves to UV light and fast optics time-resolved techniques provides complimentary prospectives on correlations. Because the theoretical understanding of strong correlations is still evolving, the review is focused on the analysis of the universal trends that are emerging out of a large body of experimental data augmented where possible with insights from numerical studies.

Figure 1

The ratio of the experimental kinetic energy and the kinetic energy from band theory $K_{\exp }/K_{\mathrm{band}}$ for various classes of correlated metals and also for conventional metals. The data points are offset in the vertical direction for clarity. From 740.

Figure 2

Schematic diagram revealing complimentary approaches to probing electronic correlations using IR and optical methods. Top panels show the momentum-resolved spectral function in a noninteracting metal (left), weakly interacting system (middle), and strongly correlated system (right). Characteristic forms of the real part of the conductivity ${\sigma }_1(\omega )$, the frequency-dependent scattering rate $1/\tau (\omega )$, and effective mass $m^{*}(\omega )$ are displayed. The Drude intraband contribution to the conductivity (the low energy shaded area in the second row) develops a “sideband” in a system with strong electron-boson coupling. The corresponding enhancement of $m^{*}(\omega )$ at energies below a characteristic bosonic mode ${\Omega }_0$ can be registered through the extended Drude analysis (see Sec. 2c). The magnitude of $m^{*}(\omega \to 0)$ is related to the quasiparticle renormalization amplitude $Z$ introduced in Sec. 2c. In a strongly correlated system (right panels) the oscillator strength of the entire intraband contribution is suppressed with the spectral-weight transfer to the energy scale of the order of $U$. The strength of this effect can be quantified through the ratio of $K_{\exp }/K_b$ as in Fig. 1 or equivalently through the ratio of optical and band mass $m_b/m_{\mathrm{opt}}$. Quite commonly this renormalization effect and strong electron-boson interaction act in concert yielding further enhancement of $m^{*}$ over the $m_{\mathrm{opt}}$ at $\omega <{\Omega }_0$.

Figure 3

Schematic representation of characteristic energy scales in correlated electron systems. These different processes give additive contributions to the dissipative parts of optical constants. TMO: transition-metal oxides.

Figure 4

Schematic description of dynamics in condensed matter probed with femtosecond spectroscopy. Prior to photoexcitation (1) the electrons, lattice, and spins are in thermal equilibrium. Photoexcitation creates (2) a nonthermal electron distribution. The initial relaxation proceeds primarily through electron-electron thermalization. Following thermalization, the electrons have excess energy which is transferred to other degrees of freedom on characteristic time scales ranging from $\sim 1\mathrm{ps}$ for electron-phonon relaxation to tens of picoseconds for processes such as pair recovery across a gap. From 41.

Figure 5

Quasiparticle formation in GaAs at room temperature following excitation with 10 fs, 1.55 eV pulses. The dynamic loss function is plotted as a function of frequency at various delays following photoexcitation. The response evolves to a coherent Drude response on a time scale of 175 fs as dressed quasiparticles are formed from an initially uncorrelated state at zero delay. From 385.

Figure 6

Diagrammatic representation of the current-current correlation function and the Bethe-Salpeter equation for the vertex correction to the optical conductivity.

Figure 7

Optical conductivity spectrum (real and imaginary parts) of ${\mathrm{UPd}}_2{\mathrm{Al}}_3$ at temperature $T=2.75\mathrm{K}$. The fit by Eq. (1), with ${\sigma }_{\mathrm{dc}}=1.05\times {10}^5(\Omega \mathrm{cm}{)}^{-1}$ and $\tau =4.8\times {10}^{-11}\mathrm{s}$, documents the excellent agreement of experimental data and the Drude prediction. The characteristic relaxation rate $1/(2\pi \tau )$ is marked by the decrease in ${\sigma }_1$ and the maximum in ${\sigma }_2$ around 3 GHz. From 798.

Figure 8

Absorption spectrum of ${\mathrm{CuGeO}}_3$ measured at 300 K for two different polarizations of the light. The band-gap energy is 3 eV. The peak at 1.75 eV is a phonon-assisted copper $d\mathrm{\text{−}}d$ exciton. From 65.

Figure 9

Optical conductivity of ${\mathrm{SrTi}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_3$ for $x=0.01$, 0.002, 0.009, and 0.02 at 300 K (top panel) and 7 K (bottom panel) (923). The broad, temperature-dependent, midinfrared band between 100 and 750 meV corresponds to (multi-)phonon sidebands of the Drude peak. The narrow Drude peak contains approximately the same amount of spectral weight as the sidebands, implying that $m^{*}/m_b\sim 2$.

Figure 10

Evolution of the optical conductivity from weakly coupled chains via two-leg ladders to 2D layers at $T=4\mathrm{K}$. (Top panel) ${\sigma }_1(\omega )$ of ${\mathrm{CaCu}}_2{\mathrm{O}}_3$ for $E\parallel b$ (solid line), DMRG result (circles) for $J_{\parallel }/J_{\perp }=5$ and $J_{\perp }=1300{\mathrm{cm}}^{-1}$. (Middle panel) $\sigma (\omega )$ of ${\mathrm{La}}_{5.2}{\mathrm{Ca}}_{8.8}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ for $E\parallel c$ (solid line), DMRG calculation (closed symbols), for $J_{\parallel }/J_{\perp }=1.3$, $J_{\mathrm{cyc}}/J_{\perp }=0.2$, and $J_{\parallel }=1000{\mathrm{cm}}^{-1}$ (662). (Bottom panel) ${\sigma }_1(\omega )$ of the 2D bilayer ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_6$ for $E\parallel a$ (solid line). In a bilayer, the two-magnon contribution from spin-wave theory (dashed line) contains an in-plane part (dotted line) and an interplane part (dash-dotted line). Here the in-plane exchange is $J=780{\mathrm{cm}}^{-1}$ and the interplane exchange amounts to $J_{12}/J=0.1$ (326). The two-magnon peak corresponds to $2.88J$ for $J_{12}/J=0.1$, and to $2.73J$ for $J_{12}=0$ (327).

Figure 11

(a) Temperature-dependent ${\sigma }_1(\omega )$ of ${\mathrm{CaRuO}}_3$ shows the power-law scaling at three representative temperatures. The symbols in the inset mark the energy scale where power law ${\omega }_1^{-1/2}$ ceases to hold. From 535. (b) Logarithmic plot of ${\sigma }_1(\omega )$ for ${\mathrm{SrRuO}}_3$. The curves from the top correspond to temperatures $T=8\mathrm{K}$, 40, 60, and 80 K, respectively. Dotted lines are fits to Eq. (27). From 199.

Figure 12

Universal power law of the optical conductivity and the phase angle spectra of optimally doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_{0.92}{\mathrm{Y}}_{0.08}{\mathrm{Cu}}_2{\mathrm{O}}_8$. (a) The phase function of the optical conductivity $\arg \mathbf{(}\sigma (\omega )\mathbf{)}$. The error margins of ${ϵ}_{\infty }=4.5\pm 0.5$ propagate to an uncertainty of $\pm 2°$ of the phase of $\sigma (\omega )$ for $\omega =5000{\mathrm{cm}}^{-1}$. (b) The absolute value of the optical conductivity is plotted on a double logarithmic scale. The open symbols correspond to the power law $|\sigma (\omega )|=C{\omega }^{-0.65}$. From 917.

Figure 13

Comparison at $T\approx T_c$ of the imaginary part [$1/\tau (\omega )$] and the real part [$M(\omega )$] of the optical self-energy, and the electron-boson coupling functions of ${\mathrm{La}}_{1.83}{\mathrm{Sr}}_{0.17}{\mathrm{CuO}}_4$ $(T_c=31\mathrm{K})$ (390) and ${\mathrm{HgBa}}_2{\mathrm{CuO}}_4$ $(T_c=97\mathrm{K})$. From 921.

Figure 14

Far-infrared conductivity of ${\mathrm{MgB}}_2$. (a) ${\sigma }_1/{\sigma }_n$ and (b) ${\sigma }_2/{\sigma }_n$ as a function of frequency for various temperatures: 6 K (dots), 17.5 K (open circles), 24 K (solid squares), 27 K (open squares), 30 K (solid diamonds), and 50 K (open diamonds). The opening of a gap is clearly observed in (a), while (b) shows the characteristic $1/\omega$ inductive response of the condensate. From 432.

Figure 15

Photoinduced change in the superconducting gap in Pb at $T=3.7\mathrm{K}$. Photoexcitation reduces the condensate density which in turn leads to a decrease in the magnitude of the gap and a modified far-infrared transmission. The solid circles are data obtained following photoexcitation with 1.8 nJ pulses, while the open circles are for photoexcitation with 0.4 nJ pulses. The corresponding solid lines are fits to the change in transmission using BCS theory. From 122.

Figure 16

The in-plane $a$-axis optical conductivity of the ${\mathrm{Sr}}_{14-x}{\mathrm{Ca}}_x{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ compound for doping $x=8$ and 11. The insets show the enlargement of the low-frequency conductivity. From 690.

Figure 17

Evolution of the optical conductivity across insulator-to-metal transitions. (a) Bandwidth controlled Mott transition in ${\mathrm{V}}_2{\mathrm{O}}_3$ (48). (b) Filling-controlled Mott transition in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ (904). (c) The Slater transition in ${\mathrm{Cd}}_2{\mathrm{Os}}_2{\mathrm{O}}_7$ (695) [data plotted at the following temperatures (from top to bottom): 300, 250, 220, 210, 200, and 30 K; inset—the temperature dependence of the energy gap determined from the kink in $\sigma (\omega )$]. (d) Kondo system ${\mathrm{Ce}}_3{\mathrm{Bi}}_4{\mathrm{Pt}}_3$ (98).

Figure 18

In-plane optical conductivity of ${\mathrm{La}}_{1.92}{\mathrm{Sr}}_{0.08}{\mathrm{CuO}}_4$ obtained from reflectivity measurements at different temperatures. The arrows on the left axis represent the dc conductivity. The inset shows the temperature dependence of the dc resistivity ${\rho }_{\mathrm{ab}}(T)$ of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ for different doping levels. From 865.

Figure 19

Depiction of a photoinduced phase transition showing the potential energy as a function of a generalized structural coordinate. The basic idea is the so-called “domino effect”: Photoexcitation initiates a structural change which, in turn, drives an electronic phase transition with an order parameter different from that in the ground state. The energy barrier to obtain the new long-range (though likely metastable) ordered phase is greater than $k_{b}T$. The dashed arrows labeled (i) and (ii) depict electronic and vibrational routes toward inducing a phase transition. From 651.

Figure 20

(a) Reflectivity and photocurrent measured on K-TCNQ as a function of frequency taken at room temperature which is well below $T_{\mathrm{SP}}=395\mathrm{K}$. The peak at 1 eV corresponds to an excitonic charge-transfer excitation and the onset of a photocurrent response at $\sim 0.5\mathrm{eV}$ above this is due to the excitation of unbound electron-hole pairs. (b) Spectral dependence of photoinduced absolute reflectivity change $\Delta R$ at various time delays following excitation. The lowest-energy peak arises from polaron formation while the change in the charge-transfer peak suggests “melting” of the spin-Peierls phase. (c) The derivative of reflectivity in (a) with respect to photon energy shows a similar spectral response to the time-resolved data which supports the interpretation of the dynamics. From 668.

Figure 21

Ultrafast dynamics measured on K-TCNQ and $(\mathrm{BEDT}\mathrm{\text{−}}\mathrm{TTF})({\mathrm{F}}_2\mathrm{TCNQ})$. The spin-Pierels compound K-TCNQ (black line) shows a structural bottleneck in that the rise time is 320 fs while the Mott-Hubbard compound K-TCNQ (gray line), with a rise time of 19 fs, does not exhibit a structural bottleneck. In addition, the oscillations in the K-TCNQ data results from the coherent excitation of vibrational modes as shown in the inset. From 939.

Figure 22

(a) The induced change in the near-infrared (800 nm) reflectivity of ${Pr}_{0.7}{\mathrm{Ca}}_{0.3}{\mathrm{MnO}}_3$ as a function of pump photon energy at 30 K. The solid line is the time-integrated reflectivity of the $17\mu \mathrm{m}$ mode. The magnitude of the induced change at 800 nm tracks this mode. (b) Vibrational excitation at $17\mu \mathrm{m}$ initiates a dynamic response (solid black line) that is similar to electronic excitation (i.e., pumping directly at 800 nm), shown as a dotted line. From 755.

Figure 23

Images of the near-field scattering amplitude over the same $4\times 4\mu {\mathrm{m}}^2$ area at the infrared frequency $\omega =930{\mathrm{cm}}^{-1}$. These images are displayed for representative temperatures in the insulator-to-metal transition regime of ${\mathrm{VO}}_2$ to show percolation in progress. The metallic regions give higher scattering near-field amplitude compared with the insulating phase. Adapted from 744.

Figure 24

(a) The optical conductivity of the bandwidth-controlled IMT of the one-band Hubbard model within DMFT approximation. The inset shows the electron spectral function for the same values of $U/W$. From 85. (b) The realistic calculation of the optical conductivity across the bandwidth-controlled IMT of ${\mathrm{V}}_2{\mathrm{O}}_3$. The inset shows the change of the lattice parameters with temperature, which was important to model the transition. From 48. (c) The optical conductivity of the doping controlled IMT in the one-band Hubbard model within DMFT approximation. The inset shows the doping dependence of the Drude weight. From 412. (d) The realistic calculation of the optical conductivity in the material exhibiting the doping driven Mott transition. From 691.

Figure 25

(a) The optical mass $1+\lambda =m^{*}/m_{\mathrm{opt}}$ and the plasma frequency as a function of doping for the $t\mathrm{\text{−}}J$ model within the cluster DMFT approach for a 2D unfrustrated square lattice. (b) The optical conductivity of the same model at two selected doping levels and few temperatures. From 343.

Figure 26

The log-log plot of the spectral weight of the superfluid density ${\rho }_{s,0}$ for a variety of superconductors probed along both the conducting $a\mathrm{\text{−}}b$-plane direction and the interplane $c$-axis direction. From 378 supplemented by iron-pnictide data from 953.

Figure 27

Imaginary part of the far-infrared conductivity in a ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ film as a function of frequency for various times following photoexcitation. An initial reduction in the condensate density yields a reduction in the $1/\omega$ response which recovers in $\sim 2\mathrm{ps}$. From 40.

Figure 28

(a) Peak induced change in $\Delta R/R$ at 1.55 eV and (b) recovery lifetime as a function of hole concentration in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_{1-y}{\mathrm{Dy}}_y{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ at $T=5\mathrm{K}$. An abrupt change in the dynamics occurs at optimal doping ($x\sim 0.15$). Note that the values of hole doping $x$ are obtained through a variation of Dy and the oxygen stoichiometry. From 288.

Figure 29

The real part of the optical conductivity of ${\mathrm{VO}}_2$ film extracted from ellipsometric measurements in mid-IR–UV in combination with reflectance in far-IR. The transformation of these spectra across the IMT reveal spectral-weight transfer over the energy scale beyond 6 eV common to other correlated electron systems. Spectra in the transition region (339–352 K) are representative of electronic phase separation of ${\mathrm{VO}}_2$ in metallic and insulating regions. Applying the effective medium theory to the analysis of these data one can extract the electromagnetic response of metallic puddles. This response is characterized by nearly divergent behavior of the effective mass (inset). From 744.

Figure 30

Optical-pump terahertz probe on ${\mathrm{VO}}_2$ at $T=250\mathrm{K}$. (a) In the 40–85 meV range, the temporal evolution of the optical phonons probes the lattice degree of freedom, while changes above 85 meV are representative of the electronic response. (b), (c) Two-dimensional evolution of the optical conductivity as a function of time and frequency. In (b) the excitation fluence is below the threshold to drive the transition while (c) shows the above threshold dynamics which reveal a prompt increase in the conductivity above 85 meV. From 502.

Figure 31

Summary of the optical response of ${\mathrm{LaMnO}}_3$ upon crossing from the orbitally ordered state to the antiferromagnetic state. (a) The imaginary part of the dielectric function ${ϵ}_{2b}$ along the $b$ axis. As the temperature decreases below $T_N$, there is a transfer of spectral weight from the low-spin (LS) to high-spin (HS) configuration which is consistent with what is expected for $d\mathrm{\text{−}}d$ intersite transition. (b) The imaginary part of the dielectric function ${ϵ}_{2c}$ along the $c$ axis. From 500.

Figure 32

${\sigma }_1$ as a function of photon energy for various temperatures measured on a cleaved single crystal of ${\mathrm{La}}_{0.825}{\mathrm{Sr}}_{0.175}{\mathrm{MnO}}_3$. The polaron peak at $\sim 1\mathrm{eV}$ gradually redshifts with decreasing temperature to a Drude peak. The inset shows the integrated spectral weight as a function of energy for various temperatures. From 864.

Figure 33

(a) Depiction of charge and orbitally ordered state. (b) ${\sigma }_1(\omega )$ and (c) ${ϵ}_1(\omega )$ measured on ${Pr}_{0.7}{\mathrm{C}}_{0.3}{\mathrm{MnO}}_3$ thin film at 4 K. From 454.

Figure 34

Plot of the natural logarithm of the conductivity in the phonon-spin temperature plane [the color bar denotes the magnitude of $\ln (\sigma )$]. The white diagonal line denotes the equilibrium line ($T_{\mathrm{spin}}=T_{\mathrm{phonon}}$) of conventional time-integrated measurements. Photoexcitation provides access to the lower half of the plane. The dynamics can be dominated by changes in the phonon temperature as is the case for arrow 1 or dominated by changes in the spin temperature as indicated by arrow 3. From 39.

Figure 35

Reflectivity of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ for $E\parallel c$ and $E\perp c$ at different temperatures as indicated. From 443.

Figure 36

The far-infrared properties of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ show the development of a plasma edge around $70{\mathrm{cm}}^{-1}$ for $E\parallel c$ related to a Drude contribution of the low-temperature conductivity (735). The total conductivity ${\sigma }_{\mathrm{tot}}(\omega )$ can be separated into a coherent (dashed line) and incoherent (dash-dotted line) contribution.

Figure 37

Frequency dependence of the imaginary parts of the terahertz-dielectric function in (a) ${\mathrm{TbMnO}}_3$ and (b) ${\mathrm{GdMnO}}_3$ with $e\parallel a$ and $b\parallel c$. Open symbols represent experimental data in zero external magnetic field and in the IC-AFMR phase. Solid lines represent model calculations adding an overdamped Lorentzian (dashed lines) to the residual high-frequency contribution. Filled spheres represent the data in the CA-AFMR state obtained by applying $B=2\mathrm{T}$ (${\mathrm{GdMnO}}_3$) and $B=8\mathrm{T}$ (${\mathrm{TbMnO}}_3$) along the $c$ axis. The corresponding zero-field data are shown by filled triangles. From 719.

Figure 38

Optical conductivity of the Ruddlesden-Popper series ${\mathrm{Sr}}_{n+1}{\mathrm{Ir}}_n{\mathrm{O}}_{3n+1}$, where $n=1$, 2, and $\infty$. The insets sketch the $t2g$ density of states in the three materials. From 625.

Figure 39

Optical conductivity and dielectric constant of superlattices consisting of equal thickness of YBCO and LCMO layer. The thickness of each layer in (a), (b), and (c) is 60, 16, and 5 nm, respectively. From 366.

Figure 40

Frequency-dependent absorptivity $A(\omega )=1-R(\omega )$ and optical conductivity of ${\mathrm{UPd}}_2{\mathrm{Al}}_3$ at different temperatures shown over a wide frequency range. The solid symbols on the left axis are from dc measurements; the open symbols in the microwave range are obtained by cavity perturbation technique. In the terahertz range the optical conductivity is determined directly using the transmission and the phase shift measured by the Mach-Zehnder interferometer. The lines are from various optical investigations [transmission through films and reflection of bulk samples (177)] and simultaneously match the directly measured conductivity and dielectric constant (224, 225). Inset: Renormalized band structure calculated from the Anderson lattice Hamiltonian. $E_k$ and $E_f$ denote bands of free carriers and localized $f$ electrons. At low temperatures, a direct gap $\Delta$ opens. The Fermi level $E_F$ is near the bottom of the upper band $E_k^{+}$, resulting in enhanced effective mass of the quasiparticles.

Figure 41

Temperature dependence of the relaxation time on YbAgCu4. Numerical simulations have been performed assuming suppressed scattering of heavy electrons by phonons; the solid line corresponds to simulations by a two-temperature model with a partial reduction of the scattering. An equally good description is obtained by coupled Boltzmann equations when the electron-phonon coupling is suppressed for electronic states within the Abrikosov-Suhl peak in the density of states (184; 11).

Figure 42

(a) Optical conductivity of ${\mathrm{YbB}}_{12}$ at different temperatures. The arrows indicate the shoulder and the conduction onset. (b) Substitution of Yb by Lu decreases correlations; the corresponding energy gap does not shift but disappears (673, 675).

Figure 43

The development of the hybridization gap in the heavy-fermion compound ${\mathrm{CeIrIn}}_5$. (a) The optical conductivity calculated by LDA+DMFT method. From 816. (b) Experimental optical conductivity for the same compound measured by 598. (c) Theoretical momentum-resolved non-$f$ spectral function at low temperature. The two types of hybridization gaps are marked by arrows in (c). (d) Same as (c) but for higher temperature.

Figure 44

The optical conductivity of ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ in the energy region below the band gap of the GaAs host (left panel) and above the gap (right panel). Data reveal systematic enhancement of the intragap conductivity with Mn doping and only weak modification of the van Hove singularities due to interband transitions. Data from 827 and 101. Inset: The peak position of the mid-IR resonance vs the spectral weight below $6450{\mathrm{cm}}^{-1}$, which is proportional to the number of holes added through the doping process as described in detail by 101. A prominent redshift of the resonance is apparent. All infrared data (left panel and the inset) at $T=8\mathrm{K}$. Ellipsometry data in the right panel are at room temperature.

Figure 45

(a) Magneto-optical reflectivity of ${\mathrm{EuB}}_6$ at 16 K; (b) scaling of magnetization and plasma frequency. From 94.

Figure 46

Temperature dependence of the optical scattering rates $\hslash /\tau$ (triangles) of ${\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\mathrm{Si}$ for two different doping concentrations, and dc resistivities scaled as to overlay the scattering rates (solid curves). From 596.

Figure 47

(a) Optical conductivity of ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ from room temperature, and across the antiferromagnetic transition around 140 K, down to 10 K. The partial gap due to SDW is clearly visible. 382, 381 associated the two peaks depicted with the excitations across the SDW gap. (b) Low-frequency conductivity of $\mathrm{Ba}({\mathrm{Fe}}_{0.92}{\mathrm{Co}}_{0.08}{)}_2{\mathrm{As}}_2$ above and below the superconducting transition $T_c=25\mathrm{K}$. The curve given by the shaded area is calculated using the theory of 588 for the lowest temperature with a gap of $2{\Delta }_0^{(1)}=50{\mathrm{cm}}^{-1}$. From 954.

Figure 48

(a) Reflectivity spectra of $(\mathrm{TMTSF}{)}_2{\mathrm{PF}}_6$ measured at different temperatures along the stacking axis $a$ and perpendicular to it. The filled symbols are obtained by a coherent source terahertz spectrometer, the open symbols are calculated from microwave experiments (205; 228). The dashed lines represent a Drude fit. The inset of (b) shows the schematic phase diagram of the deconfinement transition for a system of weakly coupled conducting chains as suggested by 80 and 294. The transition from a Mott insulator to a two- or three-dimensional metallic state occurs at $T=0$ when $t_{\perp }$ reaches a critical value $t_{\perp }^{*}$. At high enough temperature, the increase in $t_{\perp }$ leads to a transition from a Mott-insulating to a one-dimensional Luttinger liquid and further to a dimensional crossover into a metallic state. The development of the SDW gap at $70{\mathrm{cm}}^{-1}$ is seen from the low-temperature conductivity $E\perp a$ plotted in the inset of (c). From 178.

Figure 49

(a) Arrangement of the Ni and Br ions along chains in $[\mathrm{Ni}(\mathrm{chxn}{)}_2\mathrm{Br}]{\mathrm{Br}}_2$ leads to an overlap of the Ni $3d_{x^2}$ and Br $4p_z$ orbitals. From 861. (b) $\mathrm{Im}{\chi }^{(3)}(-\omega ;0,0,\omega )$ spectra of $[\mathrm{Ni}(\mathrm{chxn}{)}_2\mathrm{Br}]{\mathrm{Br}}_2$ at $T=77\mathrm{K}$. The lines are the experimental and calculated results. The energy levels are sketched in the lower panel. (c) Relation between the transition dipole moments of $⟨0|x|1⟩$ and $⟨1|x|2⟩$ for one-dimensional Mott insulators [solid circles: A, $[\mathrm{Ni}(\mathrm{chxn}{)}_2\mathrm{Br}]{\mathrm{Br}}_2$; B, $[\mathrm{Ni}(\mathrm{chxn}{)}_2\mathrm{Cl}]{\mathrm{Cl}}_2$; C, $[\mathrm{Ni}(\mathrm{chxn}{)}_2\mathrm{Cl}]({\mathrm{NO}}_3{)}_2$; D, ${\mathrm{Sr}}_2{\mathrm{CuO}}_3$) compared to other one-dimensional materials]; [empty circles: E, $[\mathrm{Pt}(\mathrm{en}{)}_2][\mathrm{Pt}(\mathrm{en}{)}_2{\mathrm{I}}_2]({\mathrm{ClO}}_4{)}_4$; F, $[\mathrm{Pt}(\mathrm{en}{)}_2][\mathrm{Pt}(\mathrm{en}{)}_2{\mathrm{Br}}_2]({\mathrm{ClO}}_4{)}_4$; G, $[\mathrm{Pt}(\mathrm{en}{)}_2][\mathrm{Pt}(\mathrm{en}{)}_2{\mathrm{Cl}}_2]({\mathrm{ClO}}_4{)}_4$; H, polydihexylsilane (PDHS)]. From 476.

Figure 50

Room-temperature spectra of the imaginary part of the dielectric constant ${ϵ}_2$ of $[\mathrm{Ni}(\mathrm{chxn}{)}_2\mathrm{Br}]{\mathrm{Br}}_2$. (a) Data are taken prior to the photoexcitation (dashed line) and at delay times $t_d$ after the photoexcitation (solid lines) with an excitation density $x_{\mathrm{ph}}$ is 0.5 photon per Ni site. The polarization of the pump and probe is parallel to the chain axis. (b) Dependence of ${ϵ}_2(\omega )$ on the excitation density as indicated ($t_d=01\mathrm{ps}$). The dotted line shows the spectrum before the photoexcitation obtained by direct measurements of the polarized absorption. From 401.

Figure 51

(a) Schematic of the linear chain in a halogen bridged Pt complex. In the ground state a charge density is commensurately modulated; a self-trapped exciton is formed in the excited state (847). (b) Time-resolved differential transmittance of $[\mathrm{Pt}(\mathrm{en}{)}_2][\mathrm{Pt}(\mathrm{en}{)}_2{\mathrm{Br}}_2]·({\mathrm{PF}}_6{)}_4$ following excitation of the intervalence charge-transfer band with 35 fs pulses centered at 800 nm. Measurements were taken at a series of detection wavelengths selected from a broadband femtosecond continuum. (c) The sum of the low-frequency component and the zero-frequency component extracted by linear prediction and singular value decomposition, showing the excited state contributions to the response and the systematic phase shift of the wave packet oscillation with detection wavelength. From 197.

Figure 52

(a) Low-temperature optical conductivity of $\kappa \mathrm{\text{−}}(\mathrm{BEDT}\mathrm{\text{−}}\mathrm{TTF}{)}_2\mathrm{Cu}[\mathrm{N}(\mathrm{CN}{)}_2]{\mathrm{Br}}_x{\mathrm{Cl}}_{1-x}$ for different Br content $x$, which serves as chemical pressure and decreases the effective Coulomb interaction $U/W$. The contributions from intradimer transitions and vibrational modes are subtracted; the ${\sigma }_1(\omega )$ spectra plotted here represent the correlated charge carriers. (b), (c) The frequency dependence of the scattering rate and effective mass extracted from an extended Drude model analysis of the conductivity (a). (d), (e) The corresponding results of DMFT calculations plotted for different $U/W$ and $T=50\mathrm{K}$. Clearly as the Mott-insulating phase is approached, the effective mass and the scattering rate increase significantly. From 599; 241.

Figure 53

Low-temperature optical conductivity of $\kappa \mathrm{\text{−}}(\mathrm{BEDT}\mathrm{\text{−}}\mathrm{TTF}{)}_2\mathrm{Cu}[\mathrm{N}(\mathrm{CN}{)}_2]\mathrm{Cl}$ before and after x-ray irradiation. The dashed curve represents ${\sigma }_1(\omega )$ of the nonirradiated $\kappa \mathrm{\text{−}}(\mathrm{BEDT}\mathrm{\text{−}}\mathrm{TTF}{)}_2\mathrm{Cu}[\mathrm{N}(\mathrm{CN}{)}_2]\mathrm{Br}$. From 788.

Figure 54

(a) The structural arrangement of the $\theta$ phase reveals two stacks of BEDT-TTF molecules in a herringbone fashion. Optical conductivity of (b) $\theta \mathrm{\text{−}}(\mathrm{BEDT}\mathrm{\text{−}}\mathrm{TTF}{)}_2\mathrm{CsZn}(\mathrm{SCN}{)}_4$ and (c)  $\theta \mathrm{\text{−}}(\mathrm{BEDT}\mathrm{\text{−}}\mathrm{TTF}{)}_2\mathrm{RbCo}(\mathrm{SCN}{)}_4$ measured along the stacks ($c$ axis) for different temperatures. The charge-ordered state in the latter is only reached when slowly cooled ($0.1\mathrm{K}/\min $), for rapid cooling ($1\mathrm{K}/\min $) the crystal remains metallic (855). The curves for different temperatures are displaced.

Figure 55

(a) Schematic of graphene-based gated structure. (b) The optical conductivity of graphene extracted from $R(\omega )$ and $T(\omega )$ synchrotron-based microscopy for various gate voltages. The threshold feature is due to interband transitions at $2E_F$ shown schematically in the inset. (c) Magnitude of $v_F$ extracted from the conductivity data in (b). An enhancement of $v_F$ at small biases is indicative of many-body effects in graphene as discussed in the text. From 544.