Transition of a prestine Mott insulator to a correlated Fermi liquid: Pressure-dependent optical investigations of a quantum spin liquid

TL;DR

This study investigates how applying pressure transforms a two-dimensional organic quantum spin liquid from a Mott insulator into a correlated Fermi liquid, revealing electronic, lattice, and anisotropic changes.

Contribution

It provides the first detailed pressure-dependent infrared analysis of the Mott insulator-metal transition in a quantum spin liquid, highlighting the coupling between lattice vibrations and electronic states.

Findings

Suppression of insulating state with increasing pressure

Emergence of coherent quasiparticles above 0.6 GPa

Strong electron-lattice coupling evidenced by vibrational mode shifts

Abstract

Systematic pressure- and temperature-dependent infrared studies on the two-dimensional organic quantum spin-liquid $\beta^{\prime}$-EtMe$_3$Sb[Pd(dmit)$_2$]$_2$ disclose the electronic and lattice evolution across the Mott insulator-metal transition. Increasing hydrostatic pressure continuously suppresses the insulating ground state; for $p>0.6$~GPa, a Drude-like component develops indicating the appearance of coherent quasiparticles at the Fermi level. In the vicinity of the Mott transition, not only the electronic state changes rapidly, but also the vibration modes exhibit a jump both in frequency and Fano constant, underlining the strong coupling between lattice and electrons. The anisotropy of the in-plane optical response becomes inverted above 0.6~GPa. The findings are discussed in detail and summarized in a phase diagram comprising different experimental approaches.

Equations

$$\frac{m_{\mathrm{SW}}^{\ast }}{m_b}=\frac{\int {\sigma }_1^{\mathrm{intra}}(\omega )\mathrm{d}\omega }{\int {\sigma }_1^{\mathrm{Drude}}(\omega )\mathrm{d}\omega }.$$

$$\frac{m^{\ast }(\omega )}{m_b}=\frac{{\omega }_p^2}{4\pi \omega }\frac{{\sigma }_2(\omega )}{{\sigma }_1^2(\omega )+{\sigma }_2^2(\omega )},$$

Full text

I Introduction

The physics governing the Mott metal-insulator transition is of paramount importance for understanding strongly correlated electron systems. It is based on the concept that the electrons in a half-filled system – i.e. one charge per lattice site – tend to localize when electron-electron interactions become strong. From the viewpoint of the simple Hubbard model, the key parameter to control the metal-insulator transition is the ratio of Coulomb repulsion $U$ to bandwidth $W$. Despite enormous progress in dynamical mean-field theory (DMFT) (Georges et al., 1996; Kotliar et al., 2006) and decades of investigating typical Mott insulators, such as V2O3 McWhan et al. (1973); Imada et al. (1998); Limelette et al. (2003); Arcangeletti et al. (2007), their ground state and the correlation-driven phase transition remain a challenge to condensed matter physicists because antiferromagnetic order – commonly observed in these compounds at low temperatures Imada et al. (1998) – obscures the genuine Mott state.

The solution is offered by quantum spin liquids, i.e. Mott insulators that do not exhibit any sign of long-range magnetic order despite strong antiferromagnetic coupling Norman (2016); Savary and Balents (2017); Zhou et al. (2017); Dressel and Pustogow (2018). Several quasi-two-dimensional molecular crystals came under particular scrutiny as they form highly frustrated triangular arrangements of dimers occupied by a single charge with $S=\frac{1}{2}$. In $\kappa$-(BEDT-TTF)2Cu2(CN)3 or $\beta^{\prime}$-EtMe3Sb[Pd(dmit)2]2, for instance, $J\approx 200$ K but no ordering occurs down to a few mK Kanoda and Kato (2011); Zhou et al. (2017). When temperature and correlations are scaled by the bandwidth: $T/W$ and $U/W$, a generic phase diagram of the Mott insulator-metal transition is unveiled, with the quantum Widom line indicating the crossover from bad metal to Mott insulator, a critical endpoint of the phase boundary followed by a Pomeranchuk-like back-bending and metallic quantum fluctuations in a coexistence regime Pustogow et al. (2018). However, not much is known about how the dynamics of charge carriers develops as correlations advance across the metal-insulator transition. What is the nature of the metallic state that gradually evolves in these highly frustrated compounds?

For the rather soft compounds, hydrostatic pressure is the most suitable way to tune across the insulator-metal phase boundary. When pressurized by only 0.6 GPa, for instance, $\beta^{\prime}$-EtMe3Sb[Pd(dmit)2]2 becomes metallic at low temperatures without signs of superconductivity Itou et al. (2017). Optical spectroscopy is the superior method for investigating the electronic properties of organic conductors, because it allows us to analyse the dynamics of the charge carriers and directly extract the Coulomb repulsion $U$ and the bandwidth $W$ as well as the coherent quasiparticle contribution to the dynamical conductivity. Here we want to explore in detail the transition from the Mott insulating to the metallic state by performing pressure- and temperature-dependent optical investigations on $\beta^{\prime}$-EtMe3Sb[Pd(dmit)2]2 single crystals. We see how the Mott gap is suppressed on the insulating side and how the effective mass evolves on the metallic side. A thorough analysis of the vibrational features provides interesting insight into the electron-phonon coupling and changes upon crossing the insulator-metal transition.

II Experimental Details

High quality single crystals of $\beta^{\prime}$-EtMe3Sb[Pd(dmit)2]2 are prepared by an aerial oxidation method as described previously Kato et al. (2012); here EtMe3Sb stands for ethyl-trimethyl-stibonium and dmit is 1,3-dithiole-2-thione-4,5-dithiolate. As shown in Fig. 1(a) the organic conductor $\beta^{\prime}$-EtMe3Sb[Pd(dmit)2]2 consists of alternating conducting layers of [Pd(dmit)2]${}_{2}^{-}$ dimers and insulating layers of EtMe3Sb+ cations. The quasi-two-dimensional electronic properties stem from the conduction band formed by the dominant $p$-orbital of the sulphur atom in [Pd(dmit)2]2. The bands of EtMe3Sb are well below the Fermi level Scriven and Powell (2012); Nakamura et al. (2012); Ueda et al. (2018).

Specimens with shiny and flat surfaces are selected for optical measurements; the large size up to 0.5 mm allows measuring the low-energy response down to far-infrared frequencies. High-pressure experiments from $p=0.23$ to 1.2 GPa were conducted in a clamp-type BeCu cell [Fig. 1(b)] with Daphne 7373-type oil as pressure medium Beyer and Dressel (2015). With the help of a home-built He-cryostat attached to a Fourier-transform spectrometer, temperature-dependent reflectivity measurements spanning from 150 to 8000 cm*-1* were performed down to $T=10$ K. The use of proper polarizers allow us to probe the response along different crystal directions. The optical conductivity is calculated via a Kramers-Kronig analysis with a Hagen-Rubens extrapolation for the metallic state and a constant low-frequency reflectivity in the insulating state Dressel and Grüner (2002).

III Results and Analysis

The temperature dependence of the optical properties measured with light polarized parallel to the $a$-axis is presented in Fig. 2 for selected pressures below and above the critical pressure $p_{c}$, defining the insulator-metal transition. A qualitatively similar temperature behavior is observed for $E\parallel b$; the complete set of spectra is presented in the Supplemental Materials SM . Under ambient conditions the optical conductivity contains a pronounced mid-infrared absorption around 2000 cm*-1* and a non-vanishing zero-frequency conductivity due to some incoherent bad metallic conductivity; a behavior rather typical for most families of organic metals Dressel and Drichko (2004); Pustogow et al. (2018). The two strong and sharp vibrational features observed around 1200 cm*-1* result from electron-molecular vibrational modes and have been extensively investigated by Yamamoto et al. Yamamoto et al. (2011a, b, 2017). As the temperature is reduced, the low-frequency conductivity first rises slightly, but below 200 K it gradually vanishes as the Mott gap develops. The overall spectrum and temperature behavior does not change substantially when pressure increases up to $p=0.6$ GPa.

From the false-color contour plots of the $p=0.23$ and 1.2 GPa data, we see the qualitatively different behavior in the insulating and metallic regimes. For a quantitative analysis of the Drude-like contribution, the temperature dependence of the spectral weight $SW=\int^{700}_{0}\sigma_{1}(\omega){\rm d}\omega$ is plotted in Fig. 2(g) and (h). In the insulating state $SW$ first rises upon lowering the temperature, but then it decreases as the Mott gap in the charge excitations gradually opens. This transition temperature is reduced with pressure, indicating a transition from a insulator-like to a metal-like behavior. Our conclusion is corroborated by pressure-dependent transport measurements Itou et al. (2017), where the change in slope from ${\rm d}\rho/{\rm d}T>0$ to ${\rm d}\rho/{\rm d}T<0$ determines the crossover from metal to insulator. Above $p_{c}=0.6$ GPa the spectral weight exhibits a completely different temperature behavior: the continuous increases down to $T=10$ K is a measure for the build-up of the coherent charge response. We do not see indication of a complete energy gap or pseudogap in our data, as inferred from the low-temperature upturn in the high-pressure resistivity Itou et al. (2017); future experiments at lower temperatures and smaller frequencies might clarify this controversy. It is interesting to note the recent results on a low-energy gap in the Dirac system $\alpha$-(BEDT-TTF)2I3 where strong correlations eventually become effective Liu et al. (2016); Uykur et al. (2018).

In order to demonstrate the pressure dependence of the optical conductivity more clearly, in Fig. 3 we present the spectra recorded at $T=295$, 150 and 10 K. At room temperature the mid-infrared absorption is barely affected by pressure, while the low-energy range increases slightly without developing a pronounced Drude peak; $\beta^{\prime}$-EtMe3Sb[Pd(dmit)2]2 remains in a bad metallic state over the whole pressure range. Our findings are in accord with dc measurements Itou et al. (2017) indicating a decrease of resistivity within factor of ten as pressure increases to 1.8 GPa. The difference between low and high-pressure spectra becomes more pronounced when the sample is cooled down to $T=150$ K, as shown in Fig. 3(b). In particular below 500 cm*-1* the behavior clearly forks: a sharp Drude peak develops for $p=0.9$ and 1.2 GPa, while the far-infrared conductivity is suppressed in the case of $p=0.23$ GPa due to the gradual opening of the Mott gap upon cooling; this is reflected in the temperature behavior of the spectral weight, plotted in Fig. 2(h).

When going to the lowest temperature, thermally excited charge carriers freeze out completely; at $T=10$ K a full gap has developed in the Mott state ($p_{c}<0.6$ GPa), while above $p_{c}$ a Drude peak is present that becomes enhanced with increasing pressure. The corresponding spectral-weight transfer is quantitatively analyzed in the inset of Fig. 3(a), where the Drude term and mid-infrared band $\int^{\omega_{u}}_{\omega_{l}}\sigma_{1}(\omega){\rm d}\omega$, (with the lower and upper bound $\omega_{l}=1000~{}{\rm cm}^{-1}$ and $\omega_{u}=3000~{}{\rm cm}^{-1}$, respectively) is plotted as a function of pressure. The small $SW$ at low-energies increases significantly only, when pressure exceeds $p_{c}$ and the Mott gap has closed. Concomitantly the mid-infrared $SW$ drops rapidly when the phase boundary is crossed. The transfer of $SW$ over such a large energy scale is taken as evidence of strong correlations Dressel and Grüner (2002); Basov et al. (2011). Fig. 3(d) illustrates that the change of optical conductivity at $T=10$ K is most severe when pressure increases from 0.53 to 0.6 GPa. Here the Mott gap closes and the compound becomes metallic for higher pressure.

IV Discussion

IV.1 Electron-Electron Interaction

In a next step we performed a Drude-Lorentz analysis of our spectra, in order to directly compare the results with theoretical predictions Georges et al. (1996); Kotliar et al. (2006); Merino et al. (2008). As examples, the low-temperature conductivity for the lowest and highest pressure is plotted in Fig. 4 together with the corresponding fit SM . For the insulating state ($p=0.23$ GPa) a satisfactory description is reached with one broad mid-infrared band (dark blue) centered at 1500 cm*-1* and an additional high-energy tail (cyan) accounting for the interband transitions. At $p=1.2$ GPa two additional contributions are necessary: a Drude term and an additional mode at 750 cm*-1*. According to single-band DMFT calculated in the realm of the single-band Hubbard model Rozenberg et al. (1995); Georges et al. (1996) these features are assigned to transitions between the lower and upper Hubbard bands separated by $U$; the width of the spectral feature corresponds to twice the bandwidth $2W$. The contributions of the coherent quasi-particles at the Fermi energy, and the excitations between the Hubbard band and the central peak at $U/2$ appear only when $U/W$ drops below a critical value $(U/W)_{c}$. In Fig. 4(e) the effective correlations $U/W$ are plotted as a function of pressure. While the onsite Coulomb repulsion $U$ varies only slightly with pressure, the bandwidth increases significantly SM . From our experiments we estimate $(U/W)_{c}\approx 1.1$ in excellent agreement with DMFT predictions Vučičević et al. (2013).

In the insulating phase, we can estimate the charge gap via linear extrapolation as depicted by the dashed line in Fig. 4(a). In a first approximation it linearly decreases with pressure from the ambient pressure value 2$\Delta_{0}=600~{}{\rm cm}^{-1}$ until it closes at $p_{c}=0.6$ GPa.

The degree of electronic correlation in the metallic state is commonly expressed by the effective mass $m^{*}$, which can be estimated from the analysis of the spectral weight $SW=\int\sigma_{1}(\omega){\rm d}\omega=\omega_{p}^{2}/8=\frac{\pi}{2}\frac{Ne^{2}}{m^{*}}$. Already the inset of Fig. 3(a) illustrates that the overall spectral weight is not recovered within the infrared range considered; this loss of $SW$ with decreasing pressure evidences the enhancement of correlations. For a quantitative analysis we consider the ratio of the zero-frequency term $\int\sigma_{\rm Drude}(\omega){\rm d}\omega$ related to the itinerant carriers and the intraband contribution obtained by subtracting the interband contributions from the total conductivity, $SW_{\rm intra}=SW_{\rm total}-SW_{\rm inter}$:

[TABLE]

The development of ${m^{*}_{\rm SW}}/{m_{b}}$ with pressure is plotted in Fig. 4(d) as black squares.

Another approach starts from Fermi-liquid theory and considers the energy-dependent scattering rate and effective mass. From the extended Drude analysis of the complex optical conductivity $\hat{\sigma}(\omega)=\sigma_{1}(\omega)+{\rm i}\sigma_{2}(\omega)$ we obtain Dressel and Grüner (2002)

[TABLE]

where the plasma frequency $\omega^{2}_{p}=8\int^{\omega_{c}}_{0}\sigma_{1}(\omega){\rm d}\omega$ is calculated up to a cutoff $\omega_{c}=3000~{}{\rm cm}^{-1}$, chosen to disregard interband transitions.

For $T=10$ K, we plot the frequency dependence of the effective mass $m^{*}(\omega)/m_{b}$ and scattering rate $\gamma(\omega)$ for different pressures, $p>p_{c}$, in Fig. 5. While at high energies the effective mass is basically energy independent, we see a prominent increase for frequencies below $600~{}{\rm cm}^{-1}$. This dispersion becomes more pronounced as the Mott insulator transition is approached with lowering pressure. Correspondingly the scattering rate $\gamma(\omega)$ decreases as we go down in frequency, as displayed in Fig. 5(b). The quality of the data and systematic errors do not allow us to draw quantitative conclusions on the functional dependence.

In Fig. 4(d) the effective mass is plotted as a function of pressure obtained from the spectral weight evaluation [Eq. (1)] and the $\omega\rightarrow 0$ limit of the extended Drude analysis [Eq. (2)]. Both results are in good accord as far as the pressure dependence and the absolute value is concerned. When $p$ exceeds the critical value $p_{c}$, the system becomes less correlated and $m^{*}/m_{b}$ decreases. The corresponding analysis of the $E\parallel b$ spectra yield a similar behavior. Our findings are also in accord with observations reported from optical investigations on $\kappa$-phase BEDT-TTF salts tuned by physical or chemical pressure Klehe et al. (2000); McDonald et al. (2003); Merino et al. (2008); Dumm et al. (2009). As the systems approach the metal-to-insulator transition from the metallic side by enhancing electronic correlations, the effective mass increases rapidly.

IV.2 Electron-Phonon Interaction

Ideally the Mott transition is supposed to represent a purely electronic phase transition; in a real crystals, however, a coupling to the underlying lattice is unavoidable. Optical spectroscopy provides the unique tool to investigate the vibrational response of the molecular and crystal structure at the phase transition in order to extract information on changes in structure and charge distribution. Fig. 6 displays the behavior of the two strongest infrared-active Ag vibrational modes of $\beta^{\prime}$-EtMe3Sb[Pd(dmit)2]2 as a function of temperature and pressure. The four C=C bonds in the Pd(dmit)2 molecular dimer vibrate such that they are in phase within one molecule but out of phase with respect to the sibling molecule (C-mode); the B-mode denotes the out-of-phase vibration within the molecules as depicted in panel (a) Yamamoto et al. (2017). Since the electric dipole is perpendicular to the molecular axis, the C-mode couples stronger to the electronic background compared to the B-mode where the dipoles point along the Pd(dmit)2 axes Yamamoto et al. (2011a, b, 2017). Due to thermal contraction, the vibrational features in general harden upon cooling; for lowest pressure, however, we observe a blue shift at the Mott transition and the Fano shape becomes more pronounced [Fig. 6(c)-(h)]. We interpret this behavior as indication that the electronic screening changes upon approaching the metal-insulator transition.

The low-temperature spectra exhibit a clear change in frequency and electronic coupling when the pressure drops below $p_{c}$, as depicted in Fig. 6(j)-(k). It is interesting to note that thermal expansion measurements on the related compound $\kappa$-(BEDT-TTF)2Cu[N(CN)2]Cl also reveal a jump of the expansion coefficient when tuned by pressure across the first-order insulator-metal line Gati et al. (2016).

Our optical investigations now give conclusive evidence that the strength of itinerant (inter-dimer) charge contribution increase with pressure but on the expense of the localized (intra-dimer) spectral weight. Above $p_{c}=0.6$ GPa the vibrational modes become decoupled from the electronic background.

IV.3 Electronic Anisotropy

$\beta^{\prime}$-EtMe3Sb[Pd(dmit)2]2 is a quasi-two-dimensional conductor with a small anisotropy at ambient conditions: $\sigma_{1a}>\sigma_{1b}$. While the ratio $\sigma_{1b}/\sigma_{1a}=0.8$ to 0.9 remains constant throughout the insulating state ($p<p_{c}$), the Drude term is found to increase more rapidly for $E\parallel b$ than for $E\parallel a$ when entering the metallic phase. In Fig. 7 we present the temperature and pressure dependence of the anisotropy $\sigma_{1b}/\sigma_{1a}$ as obtained from the $\omega\rightarrow 0$ data indicating the inversion for low temperature and high pressure. Similar results are obtained from the analysis of the spectral weight of the Drude term in both directions. Our findings infer the opening of a pseudogap along the $a$-axis. Beside investigations of the thermal expansion, we suggest more detailed studies of the Fermi surface of $\beta^{\prime}$-EtMe3Sb[Pd(dmit)2]2 for high pressure and low temperature where the Mott phase transition is crossed and the system turns from an insulator to a metal. Hall measurements indicate an electron-hole doping asymmetry in the antiferromagnetic Mott insulator $\kappa$-(BEDT-TTF)2Cu[N(CN)2]Cl Kawasugi et al. (2016). Recent Hall studies on the $\kappa$-salt spin-liquid Mott insulators $\kappa$-(BEDT-TTF)${}_{2}X$ yield evidence for Mott-Anderson localization Čulo et al. (2019). It would be of interest to extend these Hall investigations to the title compound $\beta^{\prime}$-EtMe3Sb[Pd(dmit)2]2.

V Summary and Conclusions

Employing temperature- and pressure-dependent optical investigations on the quantum spin liquid candidate $\beta^{\prime}$-EtMe3Sb[Pd(dmit)2]2, we could elucidate the electrodynamics at the Mott insulator-metal transition in detail. The results unambiguously confirm the realization of the pure bandwidth-controlled phase transition via applied pressure. From our optical data we can determine values for the onsite Coulomb repulsion $U$ and the bandwidth $W$. With increasing pressure the effective correlations $U/W$ are continuously reduced. The extracted parameters of $T/W$ and $U/W$ are in good agreement with DMFT calculations SM .

In the pressure-temperature phase diagram of Fig. 8 we summarize our optical experiments and complement them with transport data Itou et al. (2017). At ambient pressure $\beta^{\prime}$-EtMe3Sb[Pd(dmit)2]2 is a Mott insulator with a Mott-Hubbard gap of $600~{}{\rm cm}^{-1}$ that linearly decreases with pressure and vanishes at $p_{c}=0.6$ GPa. The high-temperature electronic state can be characterized as a rather poor metallic state with only a small amount of free charge carriers. When cooled down at ambient and weak pressure ($p<p_{c}$), the system becomes insulating at the quantum Widom line, i.e.  the spectral weight shifts to higher energies as the Mott gap opens in the optical spectrum. Due to the electron-phonon coupling, we also identify changed in the vibrational features (Fig. 6). With $p>p_{c}$ a metallic ground state is established at reduced temperatures with a pronounced zero-frequency component. When the Mott transition is approached from the metallic side, the increase of correlations strength is monitored quantitatively by the enhancement of the effective mass $m^{*}$ at $p_{c}$. Lower-temperature measurements are required for a more detailed characterization of this possibly Fermi-liquid state.

Acknowledgements.

We thank G. Untereiner for preparing the crystals for optical measurements and T. Biesner for the help during the data analysis. This work was partially supported by JSPS KAKENHI grant No. JP16H06346. We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) via DR228/52-1.