Abstract
Landau suggested that the lowtemperature properties of metals can be understood in terms of longlived quasiparticles with all complex interactions included in Fermiliquid parameters, such as the effective mass m^{⋆}. Despite its wide applicability, electronic transport in bad or strange metals and unconventional superconductors is controversially discussed towards a possible collapse of the quasiparticle concept. Here we explore the electrodynamic response of correlated metals at half filling for varying correlation strength upon approaching a Mott insulator. We reveal persistent Fermiliquid behavior with pronounced quadratic dependences of the optical scattering rate on temperature and frequency, along with a puzzling elastic contribution to relaxation. The strong increase of the resistivity beyond the Ioffe–Regel–Mott limit is accompanied by a ‘displaced Drude peak’ in the optical conductivity. Our results, supported by a theoretical model for the optical response, demonstrate the emergence of a bad metal from resilient quasiparticles that are subject to dynamical localization and dissolve near the Mott transition.
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Introduction
Conduction electrons in solids behave differently compared to free charges in vacuum. Since it is not possible to exhaustively model the interactions with all constituents of the crystal (nuclei and other electrons), Landau postulated quasiparticles (QP) with charge e and spin \(\frac{1}{2}\), which can be treated as nearly free electrons but carry a renormalized mass m^{⋆} that incorporates all interaction effects^{1}. In his Fermiliquid picture, the conductivity of metals scales with the QP lifetime τ, which increases asymptotically at low energy as the scattering phase space shrinks to zero^{1}. Electron–electron interaction involves a quadratic energy dependence of the scattering rate γ = τ^{−1} on both temperature T and frequency ω^{2,3,4,5}, expressed as:
Here γ_{0} stems from residual scattering processes at zero energy (e.g., impurities), and p is a numerical constant; the coefficient B controls the overall rate of variation with energy and increases with the effective mass m^{⋆}. In most metals with large electronic bandwidth W the behavior described by Eq. (1) is not seen because of the small m^{⋆}, so that the intrinsic contribution to scattering is negligible compared to other sources of dissipation (Fig. 1). Electronic correlations can strongly enhance the effective mass, m^{⋆}/m_{b} ≫ 1 (in a local Fermi liquid the QP weight scales as \(Z\propto {({m}^{* }/{m}_{b})}^{1}\), where m_{b} is the band mass), making the energydependent terms of Eq. (1) the dominant contributions in the dc resistivity ρ(T) and optical conductivity σ_{1}(ω).
While the quasiparticle concept has proven extremely powerful in describing good conductors, QPs become poorly defined in case of excessive scattering. In metals, the scattering rate is expected to saturate when the mean free path approaches the lattice spacing, known as Ioffe–Regel–Mott (IRM) limit^{6,7}. However, this bound is often exceeded (ρ ≫ ρ_{IRM}) in correlated, ‘Mott’ systems^{8}. In view of the apparent breakdown of Boltzmann transport theory, it is controversially discussed whether charge transport in such bad metals^{6,7} is in any way related to QPs^{9,10,11} or whether entirely different excitations come into play^{12}. By investigating the lowenergy electrodynamics of a strongly correlated metal through comprehensive optical measurements, here we uncover the prominent role of QPs persisting into this anomalous transport regime, providing evidence for the former scenario. Our results also demonstrate the emergence of a ‘displaced Drude peak’ (DDP, see inset of Fig. 1) indicating incipient localization of QPs in a regime where their lifetime is already heavily reduced by strong electronic interactions.
Results
We have chosen the molecular chargetransfer salts κ[(BEDTSTF)_{x}(BEDTTTF)_{1−x}]_{2}Cu_{2}(CN)_{3} (abbreviated κSTF_{x}), which constitute an ideal realization of the singleband Hubbard model on a halffilled triangular lattice. In the parent compound of the series (x = 0), strong onsite Coulomb interaction U = 2000 cm^{−1} (broad maximum of σ_{1}(ω) in Fig. 2e, f) gives rise to a genuine Mottinsulating state^{13,14} with no magnetic order^{15} down to T = 0. Partial substitution of the organic donors by Secontaining BEDTSTF molecules with more extended orbitals^{16} increases the transfer integrals t ∝ W (Fig. 2a–c). As a result, the correlation strength U/W is progressively reduced with x, allowing us to tune the system through the “bandwidthcontrolled” Mott metalinsulator transition (MIT), covering a wide range in k_{B}T/W and ℏω/W within the parameter ranges accessible in our transport and optical experiments (see Supplementary Notes 2 and 3).
ρ(T) of κSTF_{x} (Fig. 2d) reveals a textbook Mott MIT resembling the pressure evolution of κ(BEDTTTF)_{2}Cu_{2}(CN)_{3}^{17,18,19}, which turns metallic around 1.3 kbar. As x rises further, Fermiliquid behavior ρ(T) = ρ_{0} + AT^{2} stabilizes below a characteristic T_{FL} that increases progressively with x, while \(A\propto {({m}^{\star })}^{2}\) is reduced^{20} as correlations diminish (Fig. 2h–k). As common for halffilled Mott systems^{10,21}, ρ(T) rises faster than T^{2} above T_{FL}, seen by the effective temperature exponent β ≫ 2 in Fig. 2d, and exceeds ρ_{IRM} = hd/e^{2} = 4 mΩ cm (h = 2πℏ is Planck’s constant, e the elementary charge, and d = 16 Å the interlayer spacing). We note that the bad metal formed here does not exhibit a linearinT resistivity that occurs in many other materials^{12}. Instead, metallic behavior is completely lost when temperature exceeds the kinetic energy of QPs at the Brinkman–Rice scale k_{B}T_{BR} ≃ ZE_{F}^{9,22}, and ρ(T) resembles a thermally activated semiconductor above the resistivity maximum^{23,24}. Also in the optical conductivity we observe the transition from an insulator (dσ_{1}/dω > 0 at low frequencies) to a metal (dσ_{1}/dω ≤ 0), forming a QP peak at ω = 0 upon increasing x and lowering T (Fig. 2e, f). The phase diagram in Fig. 2g summarizes the characteristic crossovers in κSTF_{x} (black symbols), also including the quantum Widom line (QWL)^{13,21,25} that separates the Mott insulator with a welldefined spectral gap from the incoherent semiconductor at elevated temperatures.
The lowenergy response of Fermi liquids and the corresponding quadratic scaling laws are well explored theoretically^{2,3,4,5}. A scattering rate γ ∝ ω^{2} implies an inductive response characterized by σ_{2} > σ_{1}, where σ_{1} and σ_{2} are the real and imaginary part of the optical conductivity, respectively. This occurs in the socalled ‘thermal’ regime^{4}, ω > γ, delimited by semielliptical regions in T − ω domain, as recently reported in Febased superconductors^{26}. Our optical data on κSTF_{x} indeed reveal inductive behavior, signaled by characteristic semiellipses with a phase angle \(\arctan ({\sigma }_{2}/{\sigma }_{1})\;> \; 4{5}^{\circ }\) (Fig. 2l–o), and occurring at temperatures where ρ ∝ T^{2} is seen in dc transport (Fig. 2h–k), i.e., at T < T_{FL} (black squares). Concerning the ω^{2}/T^{2} scaling in Eq. (1), Fermiliquid theory predicts a ‘Gurzhi parameter’ p = 2 for the optical scattering rate γ(T, ω)^{2}, a quantity that can be extracted from the optical conductivity via extended Drude analysis. Experimentally, values both in the range 1 ≤ p ≤2^{26,27,28,29} and p ≥2^{30} have been found for γ(T, ω) in few selected materials. While purely inelastic scattering among QPs yields p = 2, deviations towards p = 1 have been assigned to elastic scattering off quasistatic impurities (dopants, felectrons), for instance^{5}. It remains an open question how the T and ω dependences, and the value of p, develop as correlations advance towards the Mott MIT.
In κSTF_{x}, the broadband response follows γ ∝ ω^{2} at low T (Fig. 3), as expected from Eq. (1), in all the compounds of the series (x = 0.28, 0.44, 0.78, and 1.00) that also show Fermiliquid behavior in ρ(T). The pronounced dip visible in the spectra around 1200 cm^{−1} stems from a vibration mode with Fanolike shape in σ_{1}(ω) and does not affect the relevant lowfrequency behavior (see Fig. 2e, f and Supplementary Fig. 5). Analogue to the increase of the slope A in Fig. 2h–k, the ω^{2} variation of the scattering rate also becomes steeper as correlations gain strength (Fig. 3g), i.e., the coefficient B increases as x is reduced. In both cases, the quadratic energy dependence (and any dγ/dω > 0) appears only below γ_{IRM} = 1000 cm^{−1}.
The stringent prediction Eq. (1) can be directly verified by adding the T^{2} and ω^{2}dependences of γ(T, ω) to a common energy scale. In Fig. 3c, f, the curves at different T do fall on top of each other upon scaling via a Gurzhi parameter p = 6 ± 1 for all κSTF_{x} (see inset of panel c and Supplementary Fig. 11e–h). Even more striking, multiplying the energy scale by \({\left({m}^{\star }/{m}_{b}\right)}^{2}\) collapses the data of all four substitutions on one universal line (Fig. 3h). This manifestation of the Kadowaki–Woods relation^{20}, \(B\propto {({m}^{\star })}^{2}\) (see inset), rules out any relevance of spinons near the Mott MIT^{19,31}, in accord with dynamical meanfield theory (DMFT) results^{32}. All in all, the observed scaling provides compelling evidence for the applicability of Landau’s Fermiliquid concept, in agreement with previous studies on unconventional superconductors^{26,28,29}. The Gurzhi parameter significantly exceeds the inelastic limit (p = 2)^{5}, indicating quasielastic backscattering processes (see Eqs. (2) and (3) below).
Having analyzed the QP properties and their dependence on electronic correlations, we now want to evaluate how they behave when scattering increases as we cross over from the Fermi liquid into a bad metal. Figure 4 displays σ_{1}(ω) at distinct positions in the T–x phase diagram (stars in panel a); note the similarity between κSTF_{x} (black symbols) and κ(BEDTTTF)_{2}Cu_{2}(CN)_{3} subject to pressure tuning (gray). For all x ≥ 0.28 and T < T_{FL}, the optical spectra feature a Drudelike peak centered at ω = 0, representing the QP response, together with a broad absorption centered at U = 2000 cm^{−1} originating from electronic transitions between the Hubbard bands^{13}, as shown in Fig. 4d (see also Fig. 2e, f and Supplementary Fig. 6). While the highenergy features show only weak dependence on x and T, a marked shift of spectral weight takes place within the lowfrequency region (see Supplementary Note 3). The fingerprints of mobile carriers evolve upon moving away from the Fermiliquid regime by either changing x (Fig. 4b–d) or increasing T (dh), until they completely disappear both in the Mott insulator (panel b) and in the incoherent semiconductor (panel h, T > T_{BR} = 166 K at x = 0.28).
Closer scrutiny reveals that this gradual evolution of the lowfrequency absorption is accompanied by the appearance of a dip at ω = 0, which occurs at T ≥ T_{FL}; this is also where the resistivity becomes anomalous, deviating from ρ ∝ T^{2}. The QP response then evolves into a finitefrequency peak, that steadily shifts to higher ω and broadens with increasing T/reducing x (arrows in Fig. 4e–g and triangles in panel i). Such a displaced Drude peak eventually dissolves into the Hubbard band at T ~ T_{BR}. From contour plots of σ_{1}(T, ω) in Supplementary Fig. 7 we can estimate the T–ω trajectory of the DDP above T_{FL} for the different substitutions: a peak frequency around 100 cm^{−1} (dashed line in Fig. 2g) coincides with the steepest increase of the resistivity, i.e., the largest values of the exponent β > 2.
The emergence and fading of the DDP at T_{FL} and T_{BR}, respectively, indicate that the observed behavior is tightly linked to the badmetal response in the resistivity, tracking the changes experienced by the QPs as the Fermi liquid degrades. This physical picture is reminiscent of the recently introduced concept^{9,10,11} of ‘resilient’ QPs, which persist beyond the nominal Fermiliquid regime, but with modified (e.g., Tdependent) QP parameters. Note that the DDP phenomenon observed here, that is not predicted by current theoretical descriptions of Mott systems^{9,33}, also impacts charge transport itself: for example, the values of σ_{1} and γ seen at finite frequency in our optical experiments, which yield correlation strengths U/W = 1.3 for x ≥ 0.28 (see Supplementary Fig. 6), are compatible with those computed by DMFT^{33}, but the measured dc resistivity increases way beyond the theory values — a natural consequence of the drop of σ_{1} at low frequencies upon DDP formation.
Building on the considerations above, we now show that our experimental observations can be explained by an incipient localization of the carriers in the bad metal, caused by nonlocal, coherent backscattering corrections to semiclassical transport^{34,35}. We note that related ideas have been invoked to explain the bad metallicity and DDP observed in liquid metals^{34} and various correlated systems, including organics^{36}, cuprates^{37,38}, and other oxides^{39}, but no systematic quantitative investigations have been provided to date.
In order to describe the experimental observations, we now introduce a model that assigns the modifications of the Drude peak to backscattering processes^{34,35,40,41}:
Here the first term between brackets represents the standard metallic response with the energydependent γ from Eq. (1), where ω_{p} is the plasma frequency. The second term represents the leading finitefrequency correction beyond semiclassical transport, caused by additional elastic or quasielastic processes. Its sign is opposite to that of the semiclassical Drude response, leading to a dippeak structure in σ_{1}(ω) as illustrated in the inset of Fig. 1. The resulting peak frequency, \({\omega }_{peak}\simeq \sqrt{\gamma (0){\gamma }_{b}}\), gives a direct measure of the backscattering rate γ_{b}. Physically, the “localization” corrections embodied in Eq. (2) represent nonlocal interference processes, which can be viewed as finitefrequency precursors of a disorderinduced boundstate formation.
We have used Eq. (2) to fit the finitefrequency spectra at the substitution x = 0.28, where the DDP is most clearly identified in the experiment. The Fermiliquid response has been extracted from Fig. 3, setting a constant B = 6.7 × 10^{−4} cm at all temperatures up to T = 100 K. Importantly, \({\omega }_{p}^{2}\) is also kept constant, compatible with the fact that the spectral weight associated with the QPs is conserved from the Fermi liquid to the badmetallic region. The fits accurately describe the experimental data, as demonstrated by magenta lines for σ_{1}(T, ω) in Fig. 4d–g and for γ(T, ω) in Fig. 3a, b. Similar to the direct determination from the extended Drude analysis, the extracted γ(0) shows an initial T^{2} dependence which is lost at T ≥ T_{FL}, as illustrated in Fig. 4i. The parameter γ_{b} ≪ γ(0) shows a similar trend.
To get further microscopic insight, we isolate explicitly the anomalous scattering contributions by writing
where γ(ω, T) is the measured scattering rate, which has the general form Eq. (1), and γ_{FL,2} is the strict Fermiliquid prediction, i.e., Eq. (1) with p = 2. Direct comparison with Eq. (1) yields \(\delta \gamma (\omega ,T)=B({p}^{2}{2}^{2}){(\pi {k}_{B}T/\hslash)}^{2}\), from which the following conclusions can be drawn. First, the fact that we find a frequencyindependent correction directly confirms the assumed (quasi)static nature of the anomalous scatterers. Second, the fact that p = 6 ± 1 is almost constant for all substitutions (Fig. 3c, inset) means that the strength of δγ (in particular its variation with x) is governed by the QP scale embodied in the parameter B, i.e., \(\delta \gamma \sim {(\frac{{m}^{\star }}{{m}_{b}})}^{2} \sim {Z}^{2}\). This observation stresses the key role of strong correlation effects in the vicinity of the Mott point. Third, p ≫ 2 implies that the anomalous contribution δγ is dominant over the inelastic term, which consistently ensures that the corresponding localization effects are robust against the dephasing effects originating from inelastic QP scattering: whenever observed, the peak frequency ω_{peak} is much larger than the calculated dephasing term, ~ Bω^{2}.
Discussion
The κ[(BEDTSTF)_{x}(BEDTTTF)_{1−x}]_{2}Cu_{2}(CN)_{3} series studied here realizes a continuous tuning through the genuine Mott MIT near T → 0 that was previously not accessible by experiments applying physical pressure. Our systematic investigation of the electron liquid from the weakly interacting limit to the Mott insulator establishes Landau’s QPs as the relevant lowenergy excitations throughout the metallic phase. While demonstrating the universality of Landau’s QP picture, the foregoing analysis also reveals an enhanced elastic scattering channel that fundamentally alters the QP properties in these materials. This is best visible within the badmetallic regime, where it conspires with electronic correlations in causing a progressive shift of the Drude peak to finite frequencies, indicative of dynamical localization of the QPs. Our analysis also suggests that the same elastic processes may already set in within the Fermiliquid regime, causing deviations from the predicted ω^{2}/T^{2} scaling laws of QP relaxation. These conclusions are largely based on a straightforward analysis of experimental data by a general theoretical model describing the optical response of charge carriers in the presence of incipient localization. We now discuss possible scenarios to elucidate the possible microscopic origins. The key feature that requires explanation is the pronounced elastic scattering near the Mott point.
One firmly established example leading to DDP behavior and anomalously high resistivities is the “transient localization” phenomenon found in crystalline organic semiconductors^{42}. There, soft lattice fluctuations provide a strong source of quasielastic randomness at room temperature, causing coherent backscattering at low frequency and DDPs^{41,43}. In the present κSTF_{x} compounds the Debye temperature for the relevant intermolecular phonons, T_{D} ~ 30K, is similar to that of organic semiconductors, compatible with transient localization at high T. However, this picture is difficult to reconcile with the observed DDP at very low T close to the MIT that exhibits strong substitution dependence, indicating instead a clear connection with the Mott phenomenon. Similar caveats would apply if lattice fluctuations were replaced by other soft bosons unrelated to the Mott MIT, such as charge/magnetic collective modes. While the latter can also give rise to finitefrequency absorption peaks, our clear assignment of the DDP to metallic QP rules out such a situation in the present case^{44,45}.
An alternative possibility, that could reconcile the different experimental observations, is the physical picture of weakly disordered Fermi liquids^{46,47}, motivated by the unavoidable structural disorder that accompanies chemical substitution^{16}. Although a complete theory for such a situation is still not available, existing studies^{47} show that disorder directly affects the Fermi liquid, making its coherence scale T_{BR} spatially inhomogeneous with a broad distribution of local QP weights. In this case, one expects local regions with low T_{FL} to ‘drop out’ from the Fermi liquid and thus act essentially as vacancies — dramatically increasing the elastic scattering as temperature is raised. While providing a plausible physical picture for p > 2, this scenario would also be consistent with the observed scaling of γ with \({(\frac{{m}^{\star }}{{m}_{b}})}^{2}\) upon approaching the Mott point, reflecting the gradual buildup of correlations in the disordered Fermi liquid.
Finally, we argue that longrange Coulomb interactions, that are usually neglected in theoretical treatments of correlated electron systems, could actually play a key role both in the present compounds as well as in other bad metals where DDPs have been reported^{45}. The ability of nonlocal interactions in providing an effective disordered medium for lattice electrons has been recognized recently^{48,49,50,51}, with direct consequences on badmetallic behavior^{52}. The additional scattering channel associated with longrange potentials could well be amplified at the approach of the Mott transition, due to both reduced screening and collective slowing down of the resulting randomness, possibly causing DDP behavior as observed here.
Since the gradual demise of quasiparticles is a general phenomenon in poor conductors, displaced Drude peaks likely occur in many of them^{45}. In light of the present experiments, studying the interplay between electronic correlations and (selfinduced) randomness appears to be a very promising route for understanding how good metals turn bad.
Methods
Experimental
Platelike single crystals of κ[(BEDTSTF)_{x}(BEDTTTF)_{1−x}]_{2}Cu_{2}(CN)_{3} were grown electrochemically^{16} with a typical size of 1 × 1 × 0.3 mm^{3}; here BEDTTTF stands for bis(ethylenedithio)tetrathiafulvalene and BEDTSTF denotes the partial substitution by selenium according to Fig. 2a. The composition of 0 ≤ x ≤ 1 was determined by energydispersive Xray spectroscopy^{16}. The dc resistivity was recorded by standard fourpoint measurements; superconductivity was probed by magnetic susceptibility studies of polycrystalline samples using a commercial SQUID and magnetoresistance measurements on single crystals. We performed complementary pressuredependent transport experiments of the parent compound (x = 0), shown in Supplementary Fig. 2b, providing the gray data points in Fig. 4a. Since the compounds are isostructural, they retain the highly frustrated triangular lattice and do not exhibit magnetic order down to lowest temperatures; a hallmark of the spinliquid state. Using Fouriertransform infrared spectroscopy, the optical reflectivity at normal incidence was measured in the frequency range from 50 to 20000 cm^{−1} from T = 5 K up to room temperature; here also the visible and ultraviolet regimes were covered up to 47,600 cm^{−1} by a Woollam ellipsometer. The complex optical conductivity \(\hat{\sigma }(\omega)={\sigma }_{1}(\omega)+{\rm{i}}{\sigma }_{2}(\omega)\) is obtained via the Kramers–Kronig relations using standard extrapolations. Since the optical properties of both crystal axes provide similar information, we focus on the spectra acquired for the polarization along the crystallographic caxis.
Extended drude analysis
The frequencydependent scattering rate and effective mass are calculated via the extended Drude model^{53,54}
where \({\omega }_{p}=\sqrt{N{e}^{2}/{\epsilon }_{0}{m}_{b}}\) is the plasma frequency, comprising the chargecarrier density N and band mass m_{b}; ϵ_{0} is the permittivity of vacuum and e the elementary charge. ω_{p} is determined from the maximum of the dielectric loss function around 4000 cm^{−1} (see Supplementary Fig. 9).
Data availability
The authors declare that the data supporting the findings of this study are available within the paper and its Supplementary Information. Further information can be provided by A.P., M.D., or S.F.
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Acknowledgements
We acknowledge fruitful discussions with D.L. Maslov, A.V. Chubukov, A. Georges, and E. van Heumen. The project was supported by the Deutsche Forschungsgemeinschaft (DFG) via the projects DR228/393, DR228/411, DR228/481, and DR228/521. A.P. acknowledges support by the Alexander von HumboldtFoundation through the Feodor Lynen Fellowship. Work in Florida was supported by the NSF Grant No. 1822258, and the National High Magnetic Field Laboratory through the NSF Cooperative Agreement No. 1157490 and the State of Florida.
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A.P. and M.D. guided and conceived the experimental work. Optical experiments were conducted by A.P. and M.S.A., supported by Y.S. Dc transport measurements were performed by Y.S. and A.L. Crystals were grown by Y.S. and A.K. Circumstantial analysis of all results was carried out by A.P., with support from S.F. and in exchange with V.D. and M.D. Theoretical work was performed by S.F., with contributions from V.D. A.P., V.D., M.D., and S.F. discussed the data, interpreted the results, and wrote the paper with input from all authors.
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Pustogow, A., Saito, Y., Löhle, A. et al. Rise and fall of Landau’s quasiparticles while approaching the Mott transition. Nat Commun 12, 1571 (2021). https://doi.org/10.1038/s4146702121741z
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DOI: https://doi.org/10.1038/s4146702121741z
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