*Review* **Simultaneous Control of Bandfilling and Bandwidth in Electric Double-Layer Transistor Based on Organic Mott Insulator** κ**-(BEDT-TTF)2Cu[N(CN)2]Cl**

**Yoshitaka Kawasugi 1,\* and Hiroshi M. Yamamoto 2,\***


**Abstract:** The physics of quantum many-body systems have been studied using bulk correlated materials, and recently, moiré superlattices formed by atomic bilayers have appeared as a novel platform in which the carrier concentration and the band structures are highly tunable. In this brief review, we introduce an intermediate platform between those systems, namely, a band-filling- and bandwidth-tunable electric double-layer transistor based on a real organic Mott insulator κ-(BEDT-TTF)2Cu[N(CN)2]Cl. In the proximity of the bandwidth-control Mott transition at half filling, both electron and hole doping induced superconductivity (with almost identical transition temperatures) in the same sample. The normal state under electric double-layer doping exhibited non-Fermi liquid behaviors as in many correlated materials. The doping levels for the superconductivity and the non-Fermi liquid behaviors were highly doping-asymmetric. Model calculations based on the anisotropic triangular lattice explained many phenomena and the doping asymmetry, implying the importance of the noninteracting band structure (particularly the flat part of the band).

**Keywords:** organic conductor; Mott insulator; electric double-layer transistor; uniaxial strain

#### **1. Introduction**

The Mott transition, one of the core subjects in condensed matter physics, allows for the observation of intriguing phenomena, such as high-temperature superconductivity, exotic magnetism, pseudogap, and bad-metal behavior [1]. Although the Hubbard model is thought to include the essential physics of these phenomena, a detailed comparison of the model and real materials is lacking because the pristine Mott state is commonly obscured by a complicated band structure. In addition, the control parameter in a real Mott insulator is usually limited to either bandfilling or bandwidth. Moiré superlattices formed by atomic bilayers have recently emerged as a novel platform for correlated electron systems. Twisted bilayer graphene exhibits superconductivity and correlated insulating states [2], and transition metal dichalcogenide heterobilayers provide a correlation-tunable, Mott-insulating state on the triangular lattice [3]. These artificial systems are a powerful tool to understand the fundamental physics of quantum many-body systems. However, the electronic energy scale of these systems is quite different from that of bulk correlated materials, and an intermediate platform between the artificial and highly tunable moiré superlattices and the bulk correlated materials, such as high-*T*<sup>C</sup> cuprates, is invaluable.

In this brief review, we introduce bandfilling- and bandwidth-control measurements in a transistor device based on an organic antiferromagnetic Mott insulator (Figure 1) [4–6]. We fabricated an electric double layer (EDL) transistor [7], which is a type of field-effect transistor, using an organic Mott insulator. Gate voltages induced extra charges on the Mott insulator surface, which resulted in bandfilling shifts. On the other hand, by bending the EDL transistor, the Mott insulator was subjected to strain, which resulted in bandwidth changes. The (gate voltage)-(strain) phase diagram corresponded to the conceptual phase

**Citation:** Kawasugi, Y.; Yamamoto, H.M. Simultaneous Control of Bandfilling and Bandwidth in Electric Double-Layer Transistor Based on Organic Mott Insulator κ-(BEDT-TTF)2Cu[N(CN)2]Cl. *Crystals* **2022**, *12*, 42. https:// doi.org/10.3390/cryst12010042

Academic Editor: Andrej Pustogow

Received: 4 December 2021 Accepted: 26 December 2021 Published: 28 December 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

diagram of the Mott insulator in bandfilling-bandwidth 2D space. We experimentally mapped the insulating, metallic, and superconducting phases in the phase diagram. The experimental phase diagram showed that the superconducting phase surrounded the insulating phase with a particularly doping-asymmetric shape. The asymmetry was partly reproducible by calculations based on the Hubbard model on an anisotropic triangular lattice, implying the importance of the noninteracting band structure. We also showed that the normal states in the doped Mott-insulating state exhibited non-Fermi liquid behaviors, probably due to the partial disappearance of the Fermi surface (FS), similarly to the high-*T*<sup>C</sup> cuprates.

**Figure 1.** (**a**) Conducting BEDT-TTF layer in κ-(BEDT-TTF)2Cu[N(CN)2]Cl. (**b**) Conceptual phase diagram based on the Hubbard model [1]. The vertical axis denotes the strength of the electron correlation. κ-(BEDT-TTF)2Cu[N(CN)2]Cl is located near the tip of the insulating region. (**c**) Schematic side view of the device structure. Doping concentration and bending strain are controlled by EDL gating and substrate bending with a piezo nanopositioner, respectively.

#### **2. Materials and Methods**

#### *2.1. Subject Material: Organic Mott Insulator κ-(BEDT-TTF)2Cu[N(CN)2]Cl*

κ-(BEDT-TTF)2Cu[N(CN)2]Cl (BEDT-TTF: bisethylenedithio-tetrathiafulvalene, abbreviated κ-Cl hereinafter) is a quasi-two-dimensional molecular conductor in which the conducting (BEDT-TTF)2 <sup>+</sup> layer and the insulating Cu[N(CN)2]Cl<sup>−</sup> layer are stacked alternately [8]. The unit cell contains four BEDT-TTF molecules forming four energy bands based on the molecular orbital approximation. Two electrons are transferred to the anion layer, resulting in a 3/4 filled system [9]. However, the BEDT-TTF molecules are strongly dimerized, and the upper two energy bands are sufficiently apart from the remaining two bands, resulting in an effective half-filled system. If we regard the two BEDT-TTF dimers in the unit cell (with different orientations) as equivalent, κ-Cl can be modeled as a half-filled, single-band Hubbard model on an anisotropic triangular lattice: *t* - /*t* = −0.44 [10], where *t* is the nearest-neighbor hopping, and *t* is the next-nearest-neighbor hopping. Similar to the high-*T*<sup>C</sup> cuprates, the sign of *t* - /*t* is negative so that the van Hove singularity lies below the Fermi energy (hole-doped side). However, *t* exists only for one diagonal of the dimer sites and accordingly, the FS is elliptical (Figure 2).

**Figure 2.** Unit cells, Brillouin zones, band structure, and single-particle spectral functions of κ-Cl. Note that the calculations are based on the one-band model. However, the band structure and the spectral functions are shown in the two-site Brillouin zone [blue shaded area in (**b**)] because the adjacent BEDT-TTF dimers are not completely equivalent in the material. Accordingly, the X, Z, and M points in (**c**) and (**d**) correspond to points (π/2, −π/2), (π/2, π/2), and (π, 0) in the Brillouin zone of the one-site unit cell. (**a**) Schematic of the anisotropic triangular lattice of <sup>κ</sup>-Cl. Translational vectors *<sup>e</sup>***<sup>1</sup>** and *<sup>e</sup>***<sup>2</sup>** (*<sup>a</sup>* and *<sup>c</sup>*) are represented by red (blue) arrows. The red (blue) shaded region represents the unit cell containing one site (two sites). The ellipses on the sites denote the conducting BEDT-TTF dimers. (**b**) The momentum space for the anisotropic triangular lattice. The Brillouin zones of the one- (two-) site unit cell are represented by the red (blue) shaded region bounded by the red solid (blue-dashed) lines. The solid gray line indicates the FS. (**c**) Noninteracting, tight-binding band structure along highly symmetric momenta and density of states (DOS) of κ-Cl (*t* - /*t* = −0.44 with *t* = 65 meV). The Fermi level for half filling is set to zero and denoted by the dashed lines. (**d**) Single-particle spectral functions and DOS of κ-Cl at half filling in the antiferromagnetic state at zero temperature, calculated by variational cluster approximation [5]. The Fermi level is denoted by the dashed lines at zero energy. Reproduced with permission from [6].

Because of the narrow bandwidth and the half filling, κ-Cl (*U*/*t* = 5.5 [10]) is an antiferromagnetic Mott insulator at low temperatures due to the on-site Coulomb repulsion [11]. The material is in close proximity to the bandwidth-control Mott transition. When low hydrostatic pressure is applied (∼20 MPa), κ-Cl exhibits the first-order Mott transition to a Fermi liquid/superconductor (*T*<sup>C</sup> ~ 13 K). κ-(BEDT-TTF)2Cu[N(CN)2]Br (*U*/*t* = 5.1 [10]), which is a derivative with a slightly larger *t*, is also a Fermi liquid/superconductor. The transition has been thoroughly investigated using precise pressure control, such as continuously controllable He gas pressure [12] and chemical pressure by deuterated BEDT-TTF inκ-(BEDT-TTF)2Cu[N(CN)2]Br [13]. As *T*<sup>C</sup> is relatively high for the low Fermi temperatures, the bandwidth-control Mott/superconductor transition in κ-Cl is sometimes regarded as a counterpart to the bandfilling-control Mott-insulator/superconductor transitions in the high-*T*<sup>C</sup> cuprates [14].

κ-Cl has a few hole-doped derivatives. κ-(BEDT-TTF)4Hg2.89Br8 has a large *U*/*t* (nearly twice that of κ-Cl) but shows metallic conduction and superconductivity because of ~11% hole doping [15]. The transport properties [16,17] are reminiscent of high-*T*<sup>C</sup> cuprates; they show linear-in-temperature resistivity above the Mott–Ioffe–Regel limit (bad-metal behavior) and Hall coefficients inconsistent with the noninteracting FS. Applying pressure reduces *U*/*t*, and the temperature dependence of the resistivity approaches that of a Fermi liquid. If the doping concentration is precisely controllable, we would be able to obtain the desired bandwidth–bandfilling phase diagram. However, the doping concentration is fixed, and the doped derivatives are limited (only 11% [15] and 27% [18]).

#### *2.2. Experimental Method for Bandfilling Control: EDL Doping*

To control the bandfilling of κ-Cl, we employed a doping method based on the EDL transistor [Figure 3a]. The EDL transistor is a type of field-effect transistor in which the gate-insulating film is replaced by a liquid electrolyte such as an ionic liquid. EDL doping enables a higher doping concentration than the typical field-effect doping using a solid gate insulator due to the strong electric fields by the liquid electrolyte. First, we prepared polyethylene terephthalate (PET) substrates and patterned Au electrodes (source, drain, voltage-measuring electrodes, and side-gate electrodes) using photolithography. Next, we synthesized thin single crystals of κ-Cl by electrolysis of a 1,1,2-trichloroethane [10% (*v*/*v*) ethanol] solution in which BEDT-TTF (20 mg), TPP[N(CN)2] [tetraphenylphosphonium (TPP), 200 mg], CuCl (60 mg), and TPP-Cl (100 mg) were dissolved. We applied 8 μA current overnight and obtained tiny thin crystals of κ-Cl. However, we could not easily remove the thin crystals from the solution because the surface tension of the solution easily broke the crystals. We therefore moved the crystals together with a small amount of the solution by pipetting them into 2-propanol (an inert liquid). Then, using the tip of a hair shaft, we manipulated one crystal and placed it on the substrate in 2-propanol. After the substrate with the κ-Cl single crystal was taken out from 2-propanol and dried, the crystal tightly adhered to the substrate (probably via electrostatic force). The κ-Cl single crystal was shaped into a Hall bar using a pulsed laser beam at the wavelength of 532 nm [Figure 3b]. Lastly, we added a droplet of 1-ethyl-3-methylimidazolium 2-(2-methoxyethoxy) ethyl sulfate ionic liquid on the sample and the Au side-gate electrode and placed a 1.2-μm-thick polyethylene naphthalate (PEN) film on it to make the liquid phase thin. The thinning of the gate electrolyte using the PEN film reduced the thermal stress at low temperatures. We immediately cooled the sample to 220 K (~3 K/min), where the ionic liquid was less reactive. At lower temperatures, the ionic liquid solidified.

**Figure 3.** (**a**) Schematic view of device fabrication procedure. (**b**) Optical top view of an EDL transistor device. The κ-Cl crystal is laser-shaped into a Hall bar. (**c**) Gate voltage dependence of sheet resistivity at 220 K. (**d**) Gate-voltage dependence of accumulated charge density and doping concentration.

We controlled the doping concentration of the κ-Cl crystal surface by varying the gate voltage, *V*G, at 220 K. Both the positive and negative gate voltages, corresponding to the electron and hole doping, reduced the sample resistance, implying the deviation of the bandfilling from 1/2 [Figure 3c]. According to the charge displacement current measurements [19], the doping concentration reached approximately ±20% at *V*<sup>G</sup> of ±0.5 V [Figure 3d]. Increasing the gate voltage (|*V*G| > 0.7 V) led to the irreversible increase of resistance, indicating sample degradation due to chemical reactions. The choice of ionic liquid was important; the crystal immediately disappeared when we employed ionic liquids that were too reactive or too good for the solubilization of κ-Cl. Diethylmethyl(2 methoxyethyl)ammonium bis(trifluoromethylsulfonyl)imide [DEME-TFSI], a typical ionic liquid for EDL doping, was suitable for electron doping but not for hole doping at low temperatures. At the moment, 1-ethyl-3-methylimidazolium 2-(2-methoxyethoxy) ethyl sulfate is the best choice. We focused on the doping effect on this ionic liquid.

#### *2.3. Experimental Method for Bandwidth Control: Uniaxial Bending Strain via Substrate*

We usually control the bandwidth of a molecular conductor by applying hydrostatic pressure using a pressure medium oil and a pressure cell. However, because the heterogeneous device structure was unsuitable for hydrostatic pressure application, we adopted the strain effect caused by substrate bending, as shown in Figure 4. This method required no liquid pressure medium (the ionic liquid is already on the crystal) and enabled precise strain control by fine tuning the piezo nanopositioner that bent the substrate (Figure 4). Assuming that the bent substrate is an arc of a circle (angle: 2*θ*, curvature radius: *r*), strain *S* is estimated as

$$S = \frac{2\theta(r + d/2) - 2\theta r}{2\theta r} = \frac{\theta d}{l}$$

where *d* and *l* are the thickness and the length of the substrate, respectively. The relationship between the sides of the shaded triangle in Figure 4 gives

$$r\sin\theta \sim r\theta = \sqrt{r^2 - \left(r - x\right)^2}, \therefore \,\theta = \frac{4lx}{l^2 + 4x^2}$$

using the small angle approximation, where *x* is the displacement of the piezo nanopositioner. As a result,

$$S = 4dx/\left(l^2 + 4x^2\right).$$

**Figure 4.** Schematic illustration for the application of uniaxial tensile strain to the κ-Cl sample.

We employed PET substrates with *d* = 177 μm and *l* = 12 mm, and *x* was up to 2.5 mm so that the typical value of *S* was ~1%. Note that the strain in this experimental setup was tensile and uniaxial. As the strain generated by bending was tensile, we employed a PET substrate with a large thermal expansion coefficient to start the bandwidth scanning from the superconducting region. Biaxial compression of the κ-Cl crystal by the substrate at low temperatures resulted in the superconducting state without bending strain. Therefore, we applied the bending tensile strain to enhance *U*/*t* and induce the bandwidth-control Mott transition from the metallic/superconducting side to the insulating side. The strain effect should be dependent on the strain direction. However, we leave the detailed direction dependence to future work because the strain-induced Mott transition could be observed regardless of the strain direction at the moment.

#### **3. Results**

First, we introduced the superconducting phase transitions around the tip of the Mottinsulating state in the bandwidth–bandfilling phase diagram in Section 3.1 [4]. Then, we showed the transport properties under EDL doping at a large *U*/*t* in Section 3.2 [5,6] (we did not apply the bending strain shown in Section 2.3 here).

#### *3.1. Superconducting Phase around the Mott-Insulating Phase*

#### 3.1.1. Strain Effect without Gate Voltage

First, we showed the strain effect without the gate voltage (Figure 5). As mentioned in the Methods section, κ-Cl became a superconductor without bending strain owing to the thermal contraction of the substrate. *T*<sup>C</sup> (~12 K) was similar to that of the bulk κ-Cl crystal under low hydrostatic pressure. The superconducting state disappeared upon applying the uniaxial tensile strain, *S*, and the insulating state appeared at low temperatures. Despite the uniaxiality, the temperature dependence of the resistivity was qualitatively similar to that in the bulk κ-Cl crystal under hydrostatic pressure [Figure 5c]. The slope of the metallic/insulating phase in the phase diagram implied that the insulating state at the lowest temperatures had low entropy and was the antiferromagnetic Mott-insulating state. Thus, we could control the bandwidth (and consequently *U*/*t*) of the sample across the bandwidth-control Mott transition at half filling.

**Figure 5.** (**a**) Resistivity vs. temperature plots under different tensile strains at gate voltage *V*<sup>G</sup> = 0 V, and (**b**) contour plots of the resistivity data. (**c**) Pressure-temperature phase diagram of bulk κ-Cl. Reproduced with permission from [12].

#### 3.1.2. Doping Effect at Fixed Strain

Next, we fixed the uniaxial tensile strain at the very tip of the insulating state in the bandwidth–bandfilling phase diagram (*S* = 0.41%) and applied gate voltages (with warming of the sample to 220 K, changing *V*G, and cooling again). Both electron and hole doping reduced the resistivity and induced the superconducting state, as shown in Figure 6. *T*<sup>C</sup> was similar (~12 K) among the electron-doped and hole-doped states (and the undoped metallic state). However, the doping effect was highly asymmetric against the polarity of *V*G. By hole doping, the resistivity monotonically decreased, and superconductivity emerged for *V*<sup>G</sup> ≤ −0.3 V [approximately 10% hole doping according to Figure 3d]. On the other hand, the resistivity abruptly dropped, and a superconducting state emerged with low electron doping (+0.14 V ≤ *V*<sup>G</sup> ≤ +0.22 V, approximately 4 ∼ 7% electron doping). At the phase boundary, the resistivity discretely fluctuated [5]. After further electron doping, the resistivity increased again, and superconductivity disappeared. Interestingly, the normal-state resistivity at *T* > *T*<sup>C</sup> also decreased first and increased thereafter with electron doping.

**Figure 6.** Sheet resistivity vs. temperature plots under (**a**) hole doping and (**b**) electron doping at tensile strain *S* = 0.41%. The dashed line indicates pair quantum resistance *h*/4*e*2. (**c**) Contour plots of the data in (**a**,**b**). h-SC and e-SC denote hole-doped superconductor and electron-doped superconductor, respectively.

#### 3.1.3. Gate Voltage vs. Strain Phase Diagram

After obtaining the resistivity vs. gate voltage data at the fixed strain, we slightly increased the strain and repeated the same cycles, as shown in Figure 7a. These measurements resulted in the gate voltage vs. strain phase diagram at low temperatures, which corresponded (although not proportionally) to the bandfilling–bandwidth phase diagram, as shown in Figure 7b. The insulating phase was triangular on the hole-doped side (left), similar to the conceptual phase diagram of a Mott insulator, and the superconducting phase surrounded the insulating phase. On the other hand, the superconducting phase appeared to "penetrate" into the insulating phase on the electron-doped side (right).

As shown in Figure 8, the doping asymmetry was qualitatively reproduced by variational cluster approximation (VCA) calculations of the antiferromagnetic and superconducting order parameters in a Hubbard model on an anisotropic triangular lattice. At low *U*/*t*, the superconducting (antiferromagnetic) order parameter more drastically increased (decreased) by electron doping than hole doping [Figure 8a]. The doping dependence of the chemical potential was nonmonotonic only on the electron-doped side [Figure 8b], implying the possibility of a phase separation between the Mott-insulating and superconducting phases [Figure 8c]. The nonmonotonic behavior of the chemical potential seemed to have originated from the flat part at the bottom of the upper Hubbard band (along the Z–M axis), which was originally located below the Fermi energy in the noninteracting energy band. The flat part of the energy band was caused by the absence of *t* along the crystallographic *a*-axis, namely, by the nature of the triangular lattice.

Notice that the experimental results are not understood uniquely within the half-filled band scenario. Calculations on a more detailed quarter-filled band model for the κ-BEDT-TTF salts also predict the doping-induced superconductivity, where the doping polarity alters the pairing symmetry (electron doping: extended *<sup>s</sup>* <sup>+</sup> *dx*<sup>2</sup>−*y*<sup>2</sup> , hole doping: *dxy*) [20]. In addition, many quantum Monte Carlo calculations on the Hubbard model indicate the absence of superconductivity at near half filling [21,22], while superconductivity is predicted near quarter filling [23].

**Figure 7.** (**a**) Contour plots of sheet resistivity, *ρ*, under tensile strains, *S,* of 0.35%, 0.39%, 0.41%, 0.44%, 0.50%, and 0.55% as a function of temperature and gate voltage. (**b**) Contour plots of sheet resistivity, *ρ*, at 5.5 K as a function of gate voltage and tensile strain (left) and the corresponding conceptual phase diagram (right). Black dots in all figures indicate the data points where the sheet resistivity was measured. The doping concentration estimated from the average density of charge accumulated in the charge displacement current measurement [Figure 3d] is shown for reference on the upper horizontal axis in (**b**).

**Figure 8.** VCA calculations. (**a**) Antiferromagnetic and *dx*<sup>2</sup> <sup>−</sup> *dy*<sup>2</sup> superconducting order parameters, *M* and *D*, respectively, vs. doping concentration, *δ*, for several values of *U*/*t*. *M* and *D* for metastable and unstable solutions (empty symbols) at *U/t* = 4 and 4.5 under electron doping (corresponding to positive d) are also shown. (**b**) Doping concentration, *δ*, vs. chemical potential, *μ*, relative to that at half filling (*μ*half) for several values of *U*/*t*. The results for metastable and unstable solutions at *U*/*t* = 4 and 4.5 are indicated by dashed lines, and the results obtained by the Maxwell construction are denoted by solid vertical lines. The results imply the presence of phase separation and a firstorder phase transition. It is noteworthy that there is a steep (nearly vertical) increase in *δ* with increasing *μ* for larger values of *U*/*t* under electron doping, suggesting a strong tendency toward phase separation. (**c**) Chemical potential, *μ*, vs. doping concentration, *δ*, for *U*/*t* = 4. *δ*<sup>1</sup> and *δ*<sup>2</sup> are the doping concentrations of the two extreme states in the phase separation. All results in (**a**) to (**c**) were calculated using VCA for the single-band Hubbard model on an anisotropic triangular lattice (*t* - /*t* = −0.44) with a 4 × 3 cluster.

#### *3.2. Non-Fermi Liquid Behaviors in the Normal State under Doping*

Non-Fermi liquid behaviors, such as the metallic-like resistivity above the Mott–Ioffe– Regel limit and the Hall coefficient inconsistent with the volume of the FS, are ubiquitous features of the normal state of many strongly correlated materials. Here we show that our Mott EDL transistor also exhibited such behaviors.

Due to the principle of the EDL transistor, only the sample surface was doped. However, the nondoped region of the sample was also conductive at high temperatures in κ-Cl. Therefore, to discuss the non-Fermi liquid behaviors at high temperatures, we extracted surface resistivity, *ρ*s, and surface Hall coefficient, *R*Hs. Assuming a simple summation of the conductivity tensors of two parallel layers (surface monolayer and remaining bulk layers), we derived *ρ*<sup>s</sup> and *R*Hs from

$$
\rho\_{\rm s} = \frac{L}{W} \left( \frac{1}{\rho\_{\rm measured}} - \frac{1}{\rho\_{\rm OV}} \times \frac{N - 1}{N} \right)^{-1}
$$

$$
R\_{\rm Hs} = \rho\_{\rm s}^2 \left( \frac{R\_{\rm H\,\rm measured}^2}{\rho\_{\rm measured}^2} - \frac{R\_{\rm H\,\rm UV}^2}{\rho\_{\rm OV}^2} \times \frac{N - 1}{N} \right)
$$

where *L*, *W*, and *N* denote the length, width, and number of conducting layers, respectively. Suffixes "measured" and "0 V" stand for the actual measured (combined) values and the values at 0 V (nondoped values), respectively. A sample with a larger *U*/*t* (using the PEN substrate that had less thermal contraction) than the previous superconducting sample was measured, and uniaxial tensile strain was not applied.

#### 3.2.1. Temperature Dependence of the Resistivity

Figure 9a shows the temperature dependence of the surface resistivity, *ρ*s, under electron doping. Without the gate voltage, the system was insulating at all measured temperatures (2–200 K). Upon low electron doping (*V*<sup>G</sup> ∼ 0.1 *V*, >3% electron doping), metallic-like conduction (*dρ*s/*dT* > 0) above the Mott–Ioffe–Regel limit, *<sup>ρ</sup>*MIR, (∼ *<sup>h</sup>*/*e*2, assuming a two-dimensional isotropic FS) appeared at high temperatures even though the system remained insulating at the lowest temperatures. Although the resistivity was not linear-in-temperature (between linear and quadratic) in this temperature range, this was a bad-metal behavior in the sense that the mean free path of carriers was shorter than the site distance. At *V*<sup>G</sup> = 0.34 V, the resistivity below 50 K also exhibited an insulator-metal crossover across *ρ*MIR. For *V*<sup>G</sup> > 0.5 V, the temperature dependence of *ρ*<sup>s</sup> approached a Fermi liquid (quadratic in temperature) below 20 K.

**Figure 9.** Temperature *T* dependence of surface resistivity, *ρ*s, under (**a**) electron doping and (**b**) hole doping.

Hole doping also induced the bad-metal behavior at high temperatures, as shown in Figure 9b. Although an accurate estimation of the power-law exponent was difficult, the temperature dependence of *ρ*<sup>s</sup> appeared more linear in temperature than in the case of electron doping. The temperature dependence was consistent with the linear-intemperature resistivity in the hole-doped compound, κ-(BEDT-TTF)4Hg2.89Br8, at high temperatures [16,17]. However, we could not observe metallic conduction or the Fermi liquid behavior at low temperatures down to *V*<sup>G</sup> of −0.6 V. Thus, at high temperatures, the bad-metal behavior emerged in a wide doping range except at *V*<sup>G</sup> = 0 V, whereas the Fermi liquid state at low temperatures appeared only under high electron doping.

#### 3.2.2. Hall Coefficient

In the case of a Fermi liquid with a single type of carrier, 1/*e*|*R*H| (*R*H: Hall coefficient) would denote the carrier density corresponding to the volume enclosed by FS [24] and should be independent of temperature. On the contrary, temperature-dependent *R*H, which was inconsistent with the volume of the noninteracting FS, often appeared in the normal state of strongly correlated materials. In the noninteracting single-band picture, κ-Cl had a large hole-like FS so that 1/*eR*<sup>H</sup> in the metallic state should be +*p*(1 − *δ*), where *p* and *δ* are the half-filled hole density per layer and the electron doping concentration, respectively. Figure 10b shows the *V*<sup>G</sup> dependence of 1/*eR*<sup>H</sup> at 40 K. Near the chargeneutrality point (Mott-insulating state), we could not observe the distinct Hall effect due to the high resistivity. Upon electron doping, the Hall coefficients became measurable and were positive despite the doped electrons. 1/*eR*<sup>H</sup> appeared to obey +*p*(1 − *δ*) under sufficient electron doping, indicating that the Mott-insulating state collapsed, and the system approached the metallic state based on the noninteracting band structure. However, upon hole doping, 1/*eR*<sup>H</sup> became much less than +*p*(1 − *δ*). The values also differed from

the externally doped hole density, −*pδ*, implying that the Mott-insulating state collapsed by hole doping, but the system approached a different electronic state with a smaller FS than the noninteracting case.

**Figure 10.** (**a**) Hall resistance vs. magnetic field at 40 K. (**b**) Gate-voltage dependence of the hole density per site (estimated from 1/*eR*H) at 40 K. The dashed line denotes the hole density per site estimated from the volume bounded by the noninteracting FS assuming that doping concentration is proportional to *V*<sup>G</sup> (20% doping at 0.5 V). The center of the shaded insulating region corresponds to the charge neutrality point (resistivity peak). (**c**) Temperature dependence of *R*Hs. The solid line indicates the value where the hole density per site becomes one (half filling).

The temperature dependence of *R*Hs also revealed the peculiarity of the hole-doped state, as shown in Figure 10c. *R*Hs was almost temperature-independent under electron doping, as expected for a conventional metal. By contrast, *R*Hs under hole doping monotonically decreased with an increasing temperature, approaching values similar to those under electron doping.

#### 3.2.3. Resistivity Anisotropy

In-plane conductivity anisotropy also reflected the anomalous state under hole doping. Figure 11 shows the in-plane anisotropy of the surface resistivity, *ρc*/*ρa*, up to 200 K. Here, *ρ<sup>c</sup>* (*ρa*) denotes the surface resistivity along the *c* axis [*a* axis; the short axis of the elliptical FS is parallel to the *c* axis, as shown in Figure 2b]. Under electron doping, the resistivity was almost isotropic ( *ρc*/*ρ<sup>a</sup>* ∼ 1) and independent of temperature. Under hole doping, by contrast, *ρc*/*ρ<sup>a</sup>* was distinctly larger than one and increased with cooling. The conduction

along the c-axis diminished in the hole-doped state, and its origin was weakened at high temperatures.

**Figure 11.** Temperature dependence of in-plane anisotropy of surface resistivity (note that both *a* and *c* axes are parallel to the conducting plane in this material). Data are missing at low temperatures and low doping (white region in the right panel) due to the high resistance.

#### 3.2.4. Single-Particle Spectral Functions

We compared the above experimental results with model calculations. Figure 12 shows the single-particle spectral functions (corresponding to the DOS) of the Hubbard model on an anisotropic triangular lattice at 30 K using the cluster perturbation theory (CPT). The Mott-insulating state [Figure 12b,e] was reproduced at half filling, where the energy gap opened at all the k-points. When 17% of the electrons were doped [Figure 12c,f], the noninteracting-like FS emerged. On the other hand, the topology of FS under 17% hole doping [Figure 12a,d] appeared different from the noninteracting case. The spectral weight near the Z point was strongly suppressed (pseudogap), and a lens-like small hole pocket remained. The partial disappearance of FS was notable in the lightly hole-doped cuprates.

The calculated FS provides insights into the origin of the doping asymmetry of the Hall effect and the resistivity anisotropy. Sufficient electron doping reconstructed the noninteracting-like large hole FS, resulting in 1/*eR*<sup>H</sup> ∼ *p*(1 − *δ*). On the other hand, hole doping induced the partially suppressed lens-like small FS. In this state, Luttinger's theorem seemed violated, and *R*<sup>H</sup> could no longer be simply estimated. However, it was possible that *R*<sup>H</sup> was predominantly governed by quasiparticles with a relatively long lifetime (bright points of the spectral function in the reciprocal space in Figure 12), resulting in similar values of 1/*eR*<sup>H</sup> corresponding to the area of the lens-like hole pocket. In addition, the large resistivity anisotropy under hole doping could be simply explained by the suppression of the quasi-one-dimensional FS along the Z–M line, which contributed to the conduction along the *c*-axis. As shown in Figure 12g, the spectral density on the Z–M line was recovered at high temperatures, consistent with the tendency in the Hall and anisotropy measurements.

The suppression of the spectral weight near the Z–M line under hole doping seemed to be related to the van Hove singularity (the van Hove critical points lie on the Z–M axis). With the doping of holes, FS approached the van Hove singularity, and the effect of the interaction was expected to be enhanced. It was also revealed that the spin fluctuation was stronger in the hole-doped state because of the van Hove singularity. By contrast, FS departed from the van Hove singularity with the doping of electrons, resulting in a weaker interaction effect and a more noninteracting-like FS.

**Figure 12.** Fermi surfaces and single-particle spectral functions of the Hubbard model on an anisotropic triangular lattice at 30 K, calculated using the cluster perturbation theory (CPT). (**a**–**c**) Fermi surfaces for (**a**) 17% hole doping, (**b**) half filling, and (**c**) 17% electron doping, determined by the largest spectral intensity at the Fermi energy. (**d**–**f**) Single-particle spectral functions for (**d**) 17% hole doping, (**e**) half filling, and (**f**) 17% electron doping. The Fermi energy is located at *ω* = 0 and the parameter set of this model is *t* - /*t* = −0.44, *U*/*t* = 5.5, and *t* = 65 meV. (**g**) Temperature evolution of the spectral density at 17% hole doping. The suppression near the Z–M line diminishes at high temperatures.

#### **4. Summary**

We fabricated bandfilling- and bandwidth-tunable EDL transistor devices using a flexible organic Mott insulator with a simple band structure. As shown in Section 3.1, drastic resistivity changes ranging from superconducting to highly insulating (*ρ* > 10<sup>9</sup> Ω) states occurred in the proximity of the tip of the Mott-insulating phase in the phase diagram. The superconducting phase surrounded the Mott-insulating phase in the bandfilling−bandwidth phase diagram. The superconducting transition temperature, *T*C, was almost identical among the electron-doped, hole-doped, and nondoped states, in contrast to those of the high-*T*<sup>C</sup> cuprates. Model calculations based on the anisotropic triangular lattice qualitatively reproduced the doping asymmetry on the doping levels for superconductivity and the tendency towards phase separation under electron doping, implying the significance of the flat part of the upper Hubbard band (originating from the noninteracting band structure). However, the calculations did not reproduce the reentrance into the slightly insulating state beyond the electron-doped superconducting state. One possibility was that a magnetic or charge-ordered state emerged at specific doping levels (for example, ~12.5%), as in the case of the stripe order in the cuprates [25–28].

At high *U*/*t* where no superconductivity was observed, the transport properties exhibited non-Fermi liquid behaviors, as shown in Section 3.2. At high temperatures (above *T* ∼ 100 K), the metallic-like conduction above the Mott–Ioffe–Regel limit (the bad-metal behavior) widely emerged regardless of the doping polarity, supporting the universality of the bad-metal behavior near the Mott transitions. At lower temperatures, the anomalously large, temperature-dependent Hall coefficient and in-plane resistivity anisotropy appeared under sufficient hole doping. Model calculations of the spectral density explained the anomaly under hole doping in terms of the partial disappearance of FS (pseudogap) due to the approach of the Fermi energy to the flat part of the energy band (same part as the flat part of the upper Hubbard band before doping). The pseudogap state also appeared in the lightly doped high-*T*<sup>C</sup> cuprates. However, the location of the pseudogap in k-space differed, owing to the difference of the noninteracting band structure, resulting in the different doping asymmetry.

The experimental methods shown here are applicable to other molecular conductors, including κ-(BEDT-TTF)2Cu2(CN)3, which is a genuine Mott insulator without antiferromagnetic ordering [29]. The same experiments on this material may reveal more universal behaviors of a doped Mott insulator. Similar experiments on the molecular Dirac fermion system [30] are also possible and of great interest.

**Author Contributions:** Conceptualization, Y.K. and H.M.Y.; methodology, Y.K. and H.M.Y.; data curation, Y.K.; writing—original draft preparation, Y.K.; writing—review and editing, H.M.Y.; visualization, Y.K.; supervision, H.M.Y.; funding acquisition, Y.K. and H.M.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by MEXT and JSPS KAKENHI, Grant Numbers JP16H06346, JP19K03730, JP19H00891.

**Data Availability Statement:** All data needed to draw the conclusions in the paper are presented in the paper. Additional data related to this paper may be requested from the authors.

**Acknowledgments:** We would like to acknowledge R. Kato, K. Seki, S. Yunoki, J. Pu, T. Takenobu, S. Tajima, N. Tajima, and Y. Nishio for the collaborations and valuable discussions. The VCA and CPT calculations shown here were performed by K. Seki and S. Yunoki. This research was supported by MEXT and JSPS KAKENHI, Grant Numbers JP16H06346, JP19K03730, JP19H00891.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


#### *Article* **A Discrepancy in Thermal Conductivity Measurement Data of Quantum Spin Liquid** β*-* **-EtMe3Sb[Pd(dmit)2]2 (dmit = 1,3-Dithiol-2-thione-4,5-dithiolate)**

**Reizo Kato \*, Masashi Uebe, Shigeki Fujiyama and Hengbo Cui**

Condensed Molecular Materials Laboratory, RIKEN, Wako, Saitama 351-0198, Japan; masaliam020902@gmail.com (M.U.); fujiyama@riken.jp (S.F.); hengbocui@gmail.com (H.C.) **\*** Correspondence: reizo@riken.jp

**Abstract:** A molecular Mott insulator β- -EtMe3Sb[Pd(dmit)2]2 is a quantum spin liquid candidate. In 2010, it was reported that thermal conductivity of β- -EtMe3Sb[Pd(dmit)2]2 is characterized by its large value and gapless behavior (a finite temperature-linear term). In 2019, however, two other research groups reported opposite data (much smaller value and a vanishingly small temperature-linear term) and the discrepancy in the thermal conductivity measurement data emerges as a serious problem concerning the ground state of the quantum spin liquid. Recently, the cooling rate was proposed to be an origin of the discrepancy. We examined effects of the cooling rate on electrical resistivity, low-temperature crystal structure, and 13C-NMR measurements and could not find any significant cooling rate dependence.

**Keywords:** molecular conductors; quantum spin liquid; thermal conductivity; cooling rate; electrical resistivity; low-temperature crystal structure; 13C-NMR

**Citation:** Kato, R.; Uebe, M.; Fujiyama, S.; Cui, H. A Discrepancy in Thermal Conductivity Measurement Data of Quantum Spin Liquid β- -EtMe3Sb[Pd(dmit)2]2 (dmit = 1,3-Dithiol-2-thione-4,5-dithiolate). *Crystals* **2022**, *12*, 102. https://doi.org/10.3390/ cryst12010102

Academic Editor: Andrej Pustogow

Received: 11 December 2021 Accepted: 11 January 2022 Published: 13 January 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

#### **1. Introduction**

Quantum spin liquid (QSL) in a strongly frustrated spin system on a triangular lattice is characterized by the absence of long-range magnetic order or valence bond solid order among entangled quantum spins even at zero temperature [1,2]. Although theoretical works indicated that the ideal nearest-neighbor Heisenberg antiferromagnet on the triangular lattice has the long-range Néel ordered ground state (120-degree-structured state), a possibility of this third fundamental state for magnetism in the general S = 1/2 antiferromagnetic triangular lattice systems has attracted much attention. Indeed, the number of QSL candidates of real materials is increasing since the beginning of the 2000 [2].

An isostructural series of anion radical salts of a metal complex Pd(dmit)2 (dmit = 1,3-dithiol-2-thione-4,5-dithiolate), β- -Et*x*Me4−*xZ*[Pd(dmit)2]2 (Et: C2H5, Me: CH3, *Z* = P, As, and Sb; *x* = 0–2), are Mott insulators at ambient pressure [3]. In crystals of β- -Pd(dmit)2 salts with the space group *C*2/c, Pd(dmit)2 anion radicals are strongly dimerized to form a dimer with spin 1/2, [Pd(dmit)2]2 − (Figure 1). The dimers are arranged in an approximately isosceles-triangular lattice parallel to the *ab* plane, which leads to a frustrated *S* = 1/2 Heisenberg spin system. The anion radical layers and the non-magnetic cation layers are arranged alternately along the *c* axis. The ground state of the Pd(dmit)2 salts is found to change among antiferromagnetic long-range order (AFLO), QSL, and charge order (CO), depending on the anisotropy of the triangular lattice that can be tuned by the choice of the counter cation [3]. The cation effect on the degree of frustration is associated with the arch-shaped distortion of the Pd(dmit)2 molecule [4]. The QSL phase found in β- -EtMe3Sb[Pd(dmit)2]2 is situated between AFLO and CO phases [5,6].

**Figure 1.** Crystal structure of β- -EtMe3Sb[Pd(dmit)2]2. The cation on the two-fold axis shows an orientational disorder.

In β- -EtMe3Sb[Pd(dmit)2]2, no magnetic order is detected down to a very low temperature (~19 mK) that corresponds to *J*/12,000, where *J* (~250 K) is the nearest-neighbor spin interaction energy [7]. Although 13C-NMR spectra show an inhomogeneous broadening at low temperatures, the observed local static fields are too small to be explained by AFLO or spin glass state. Low-energy excitations in the QSL state of the β- -EtMe3Sb[Pd(dmit)2]2 are open to debate even now. Heat capacity and magnetization indicate gapless fermion-like excitations, while 13C-NMR indicates an existence of a nodal gap [7–9]. The thermal conductivity *κ* is free from the contribution from the rotation of the methyl (Me-) group that disturbs the heat capacity analysis below1K[8]. In addition, thermal conductivity measurements can detect the spin-mediated heat transport. In 2010, Yamashita and Matsuda reported that *κ*/*T* is finite as temperature *T* goes to zero, which indicates the presence of gapless excitations [10]. The finite *T*-linear term as well as largely enhanced *κ* values led to a proposal of contributions across the spinon Fermi surface. In 2019, however, two research groups reported that *κ* values are much smaller and *κ*/*T* is vanishingly small at 0 K, which caused a serious problem concerning the ground state of QSL [11,12].

In order to explain this sharp discrepancy in the thermal conductivity measurement data, Yamashita claimed that there were two kinds of crystals (large-*κ* and small-*κ* groups) in [13] published earlier than [11,12]. Yamashita pointed out the domain formation associated with the cation disorder or the micro cracks as an origin. It should be noted that in the context of "two kinds of crystals", the words "domain" and "micro cracks" are read as intrinsic properties that emerge in a crystal growth process or in a low-temperature phase, that is, they should be distinguished from extrinsic ones induced by improper sample handling. Although Yamashita did not disclose experimental evidence to justify the claim in [13], the claim had enough impact [14,15]. In response to the Yamashita's claim, the existence of two kinds of crystals was verified using X-ray diffraction (XRD), scanning electron microscope, and electrical resistivity measurements. The conclusion is that there is only one kind of crystal [11,12]. For example, no difference was found between the small-*κ* sample (sample G1) in [11] and the large-*κ* sample (sample C) in [13], both of which come from the very same growth batch (No. 752).

Meanwhile, in 2020, Yamashita et al. reported that one kind of crystal gives different results in the *κ* measurements depending on an experimental condition, "cooling rate" [16,17]. In their measurements, very slow cooling (−0.4 K/h) led to a finite linear residual thermal conductivity. In contrast, when the sample was cooled down rapidly

(−13 K/h), *κ*/*T* vanished at the zero-temperature limit, and the phonon thermal conductivity was strongly suppressed. These results suggest the existence of random scatterers introduced during the cooling process as another origin of the discrepancy. This proposal has raised a problem about effects of the newly proposed experimental parameter on other kinds of measurements. Herein, we investigated effects of the cooling rate on electrical resistivity, low-temperature crystal structure, and 13C-NMR measurements with relevance to the discrepancy in the thermal conductivity data.

#### **2. Materials and Methods**

For electrical resistivity and XRD measurements, we used single crystals from the same growth batch (No. 898) as that used in [13] (small-*κ* samples E and F) and [16] (crystals 1–3). The procedure of the crystal growth was as follows: (EtMe3Sb)2[Pd(dmit)2] (60 mg) was dissolved in acetone (100 mL). After addition of acetic acid (9.5 mL), the resultant solution was allowed to stand at −11 ◦C for 3 months. The β- -type crystals (black hexagonal plates) were obtained as a single phase. The 13C-enriched dmit ligand for 13C-NMR measurements was synthesized from tetrachloroethylene-13C1 (99 atom %, Sigma-Aldrich) that was converted to tetrathiooxalate for the reaction with CS2 [18]. The 13C-enriched single crystals of <sup>β</sup>- -EtMe3Sb[Pd(dmit)2]2 (No. 899 for slow cooling and No. 923 for rapid cooling) were also obtained by the above-mentioned procedure.

The temperature-dependent electrical resistivities along the *a* and *b* axes (*ρ*//*<sup>a</sup>* and *ρ*//*b*) were measured by the standard four-probe method from room temperature to 1.8 K using a physical property measurement system (Quantum Design Inc., San Diego, CA, USA). The electric leads were *φ*10 μm gold wires connected by carbon paste. Probe sizes are 325 × (1100 × 100) <sup>μ</sup>m3 for *<sup>ρ</sup>*//*<sup>a</sup>* and 150 × (1350 × 80) <sup>μ</sup>m<sup>3</sup> for *<sup>ρ</sup>*//*b*, respectively. For each current direction, the pristine crystal was cooled down to 1.8 K with different cooling rates of −0.6 K/h, −1.2 K/h, −6 K/h, and −150 K/h, in this order. In each thermal cycle, the warming rate was +6 K/h, except for the final cycle (+150 K/h).

Single crystal X-ray diffraction data were collected by a Weisenberg-type imaging plate system (R-AXIS RAPID/CS, Rigaku Corp., Tokyo, Japan) with monochromated Mo Kα radiation (UltraX6-E, Rigaku Corp., Tokyo, Japan). Low-temperature experiments were carried out in the cryostat cooled by a closed-cycle helium refrigerator (HE05/UV404, ULVAC CRYOGENICS Inc., Chigasaki, Japan). The temperature was controlled by Model 22C Cryogenic Temperature Controller (Cryogenic Control Systems Inc., Rancho Santa Fe, CA, USA). The pristine crystal was cooled down with a rate of −0.6 K/h. After the data collection at 5 K, the crystal was warmed up to room temperature with a rate of +60 K/h. The next cooling process was performed with a cooling rate of −120 K/h. All the diffraction data were processed using the CrystalStructure 3.8 crystallographic software package [19]. The structures were solved by the direct method (SIR92) [20] and refined by the full-matrix least-squares method (SHELXL-2018/3) [21]. The H atom coordinates were placed on calculated positions and refined with the riding model. Due to the orientational disorder of the EtMe3Sb cation on the two-fold axis (Figure 1), we assumed that the ethyl group and the corresponding methyl group share two equivalent positions with 50% occupation factor for the refinements.

13C-NMR spectra and nuclear relaxation rates were obtained by standard pulse Fourier transform technique using a single crystal. The magnetic field was applied along the direction 11 degrees tilted from the *c*\* axis to avoid an accidental cancellation of hyperfine fields originating from 2*p* and 2*s* electrons of 13C. The temperature was controlled by Model 32 Cryogenic Temperature Controller. The crystals were cooled at −40 K/h (rapid cooling) and −0.6 K/h (slow cooling), respectively.

#### **3. Results**

#### *3.1. Electrical Resistivity*

The electrical resistivity is sensitive to the crack formation. In addition, when the emergence of an electronic phase depends on the cooling rate as is the case of θ-(BEDT- TTF)2RbZn(SCN)4 (BEDT-TTF = Bis(ethylenedithio)tetrathiafulvalene) that possesses the charge-glass-forming ability, the electrical resistivity can detect a change of the electronic state [22]. Figure 2 shows temperature-dependent resistivities along the *a* and *b* axes measured with four different cooling rates, −0.6, −1.2, −6, and −150 K/h in this order. β- - EtMe3Sb[Pd(dmit)2]2 was a semiconductor and the resistivity became too high to measure below 28 K. Anisotropy within the *ab* plane was small, including the activation energy (~41 meV). As shown in Figure 2, temperature-resistivity curves for each cooling rate overlap almost completely, which means that there was no cooling rate dependence. In addition, no crack formation (indicated by an abrupt jump of the resistivity) and no thermal cycle dependence was observed.

**Figure 2.** Temperature-dependent resistivities measured along the *a* axis (*ρ*//*a*) and the *b* axis (*ρ*//*b*) with four different cooling rates for β- -EtMe3Sb[Pd(dmit)2]2. The inset is a photo of a single crystal. For both *ρ*//*<sup>a</sup>* and *ρ*//*b*, four *ρ*–*T* curves (cooling process) overlap almost completely.

#### *3.2. Low-Temperature Crystal Structure*

Using the same single crystal, the crystal structure of β- -EtMe3Sb[Pd(dmit)2]2 at 5 K was determined with two different cooling rates, −0.6 (slow) and −120 (rapid) K/h in this order. In both cases, the space group remained *C*2/*c* and no additional diffraction peak was observed. Determined crystal structures were identical with the previous result [23], and did not show any significant effects of the cooling rate on temperature factors and differential Fourier synthesis (Table 1).


**Table 1.** Crystal data for β- -EtMe3Sb[Pd(dmit)2]2 at 5 K obtained using two different cooling rates.

In the crystal, the EtMe3Sb+ cation is located on the 2-fold axis (//*b*). Since the cation does not have the 2-fold symmetry, the cation has two possible orientations (1 and 2 in Figure 1), which could work as an origin of the domain formation. However, our analysis, where the ethyl group is assumed to be overlaid with the methyl group with 50% occupation factor, did not find significant difference in the average cation structure for both cooling rates (Figure 3).

**Figure 3.** Average structure of the cation at 5 K viewed from the *b* axis (the two-fold axis) for two different cooling rates: (**a**) Rapid cooling (−120 K/h); and (**b**) slow cooling (−0.6 K/h).

#### *3.3. 13C-NMR*

13C-NMR enables us to investigate the microscopic electronic states of a crystal. Compared with the electrical resistivity, both NMR spectra and nuclear relaxations are insensitive to microcracks in a crystal. However, they would be able to detect possible domain formations in a cooling procedure.

Figure 4a shows the spectra at 5K. The 13C atom is introduced into one of the carbon sites in the C=C bond in the dmit ligand. The four independent 13C sites at which hyperfine coupling constants distribute up to 10% cause asymmetric spectra [24]. Two spectra with contrasting cooling rates nearly overlap with each other, by which we conclude negligible cooling rate variation in the static electronic states.

**Figure 4.** 13C-NMR of β- -EtMe3Sb[Pd(dmit)2]2 at 5 K with rapid and slow cooling rates. (**a**) Spectra; and (**b**) nuclear magnetization curves.

Nuclear magnetization (*M*z) curves of the 1/*T*<sup>1</sup> (nuclear spin-lattice relaxation rate) measurements at 5 K by the rapid and slow cooling procedures are shown in Figure 4b. Two curves agree with each other, and we conclude that the spin dynamics of the quantum spin liquid state is insensitive to the cooling rate.

It should be noted that the previous 13C-NMR data obtained using randomly orientated crystals suggest no cooling rate dependence either [5,7]. In [5], the cooling process at a rate of ca. −10 K/h and measurements at a constant temperature (during 1–2 days) were repeated alternately, and the sample was cooled from room temperature down to 1.4 K spending about one month. In [7], on the other hand, the sample was cooled from room temperature down to 1.8 K within 10 h before the measurements in very low-temperature region (1.8 K–20 mK). The 13C-NMR data with these two different experimental conditions coincide with each other in the same temperature region [25].

#### **4. Discussion**

In this work, we could not observe the effect of the cooling rate on resistivity, lowtemperature crystal structure, and 13C-NMR. Of course, we must be careful in discussing the relation between these physical/structural properties and the thermal conductivity. The problem we are facing is the thermal conductivity below 1 K. In the low temperature region, the electrical resistivity of the present material is very high, and the mean free path of a charge carrier becomes shorter than lattice lengths. In such a case, conduction electrons would be unaffected by an event in the whole crystal. On the other hand, the low-temperature crystal structures we determined are average ones and do not provide direct information about the possible domain or defect formation.

Nevertheless, the cooling rate engages in a process in the whole temperature region. If random scatterers are generated during the cooling process, they would be detected by physical/structural properties other than the thermal conductivity even in the high temperature region. In addition, local changes in a crystal could affect an average crystal structure and 13C-NMR. As we mentioned before, the cooling rate dependence was not detected in the high temperature region (>~5 K) in this work. In the lower temperature region, the smaller heat capacity provides more homogeneous temperature distribution, and thus it is less plausible that the cooling rate plays an important role.

In this sense, the results of this work suggest that further analysis is necessary before concluding that the cooling rate is an essential experimental condition. Let us reconsider two different sets of thermal conductivity data from [10,16] (Figure 5). Measurements with different cooling rates indicated that the slower cooling rate gave the larger *κ* and finite *κ/T* [16]. However, even with the slowest cooling rate (−0.4 K/h), the *κ* values are much smaller than the first reported ones [10]. That is, the large-*κ* data have never been reproduced. In addition, the measurements in [10] performed with the rapid cooling rate of −10 K/h show larger *κ* values than those for Crystal 1 measured with the slowest cooling rate of −0.4 K/h in [16]. This is quite puzzling and suggests that the cooling rate is not essential.

In conclusion, the present situation is that one kind of crystal provides two different thermal conductivity data and a role of the newly proposed experimental parameter, cooling rate, remains to be seen. In addition, the large-*κ* data have never been reproduced. It is an urgent matter to clarify the intrinsic thermal conductivity. From this point of view, the description "*the crystals often do not recover to the initial state after a thermal cycle*" in [16] suggests that the stress from experimental environments including a setup may enhance or suppress the thermal conductivity. Indeed, crystals used in [10,16] frequently fell apart when the leads were removed by rinsing out the paste with diethyl succinate. This suggests the existence of the stress from leads on a crystal. In contrast, we did not observe any thermal cycle dependence in this work. In order to clarify this point, monitoring of electrical resistivity during thermal conductivity measurements will be valuable, because the resistivity is sensitive to the crack formation and pressure in the wide range of temperature (>~28K).

**Figure 5.** Two different sets of the thermal conductivity (*κ*) data for β- -EtMe3Sb[Pd(dmit)2]2 from [10,16]. The cooling rate in each measurement is indicated in a parenthesis. The inset is an enlarged view of the data from [16]. The behavior of *κ* reported in [11,12] is similar to that of crystal 3 (−13 K/h) in [16].

**Author Contributions:** Conceptualization, funding acquisition, project administration, sample preparation, and writing—original draft preparation, R.K.; investigation and data curation, M.U., S.F. and H.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partially supported by JSPS KAKENHI [grant number JP16H06346].

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Crystallographic information files are available from the CCDC, reference numbers 2125994-2125995. These data can be obtained free of charge via https://www.ccdc. cam.ac.uk/structures/.

**Acknowledgments:** We deeply thank Daisuke Hashizume (RIKEN Center for Emergent Matter Science) for technical help with the crystal structure analysis of β- -EtMe3Sb[Pd(dmit)2]2 at 5 K.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Perspective* **Are Heavy Fermion Strange Metals Planckian?**

**Mathieu Taupin and Silke Paschen \***

Institute of Solid State Physics, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040 Vienna, Austria; taupin@ifp.tuwien.ac.at

**\*** Correspondence: paschen@ifp.tuwien.ac.at; Tel.: +43-1-58801-13716

**Abstract:** Strange metal behavior refers to a linear temperature dependence of the electrical resistivity that is not due to electron–phonon scattering. It is seen in numerous strongly correlated electron systems, from the heavy fermion compounds, via transition metal oxides and iron pnictides, to magic angle twisted bi-layer graphene, frequently in connection with unconventional or "high temperature" superconductivity. To achieve a unified understanding of these phenomena across the different materials classes is a central open problem in condensed matter physics. Tests whether the linear-in-temperature law might be dictated by Planckian dissipation—scattering with the rate ∼ *k*B*T*/¯*h*—are receiving considerable attention. Here we assess the situation for strange metal heavy fermion compounds. They allow to probe the regime of extreme correlation strength, with effective mass or Fermi velocity renormalizations in excess of three orders of magnitude. Adopting the same procedure as done in previous studies, i.e., assuming a simple Drude conductivity with the above scattering rate, we find that for these strongly renormalized quasiparticles, scattering is much weaker than Planckian, implying that the linear temperature dependence should be due to other effects. We discuss implications of this finding and point to directions for further work.

**Keywords:** heavy fermion compounds; strange metals; Planckian dissipation; quantum criticality; Kondo destruction

#### **1. Introduction**

A first step in understanding matter is to delineate the different phases in which it manifests. To do so, a characteristic that uniquely identifies a phase must be found, and using its order has worked a long way. How this classification should be extended to also incorporate topological phases [1] is a matter of current research. Here, we focus on topologically trivial matter and thus take order-parameter descriptions [2] as a starting point and consider the case of second-order phase transitions. As an order parameter develops below a transition (or critical) temperature, the system's symmetry is lowered (or broken). Cornerstones are the power law behavior of physical properties near the critical temperature, with universal critical exponents, and the associated scaling relationships. Combined with renormalization-group ideas [3], this framework is now referred to as the Landau–Ginzburg–Wilson (LGW) paradigm. It has also been extended to zero temperature. Here, phase transitions—now called *quantum* phase transitions [4]—can occur as the balance between competing interactions is tipped. To account for the inherently dynamical nature of the *T* = 0 case, a dynamical critical exponent needs to be added. This increases the effective dimensionality of the system, which may then surpass the upper critical dimension for the transition, so that the system behaves as noninteracting, or "Gaussian". Interestingly, however, cases have been identified where this expectation is violated [5–8], evidenced for instance by the observation of dynamical scaling relationships [9] that should be absent according to the above rationale. We will refer to this phenomenon as "beyond order parameter" quantum criticality. It appears to be governed by new degrees of freedom specific to the quantum critical point (QCP). This is a topic of broad interest both in condensed matter physics and beyond, but a general framework is lacking. We will here

**Citation:** Taupin, M.; Paschen, S. Are Heavy Fermion Strange Metals Planckian? *Crystals* **2022**, *12*, 251. https://doi.org/10.3390/ cryst12020251

Academic Editor: Andrej Pustogow

Received: 24 December 2021 Accepted: 10 February 2022 Published: 12 February 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

discuss it from the perspective of heavy fermion compounds, where it can manifest as Kondo destruction quantum criticality [5,6]. We will in particular discuss materials that display linear-in-temperature "strange metal" electrical resistivity, as well as the proposed relation [10,11] to Planckian dissipation. We will allude to similar phenomena in other material platforms and point to directions for further research to advance the field.

#### **2. Simple Models for Strongly Correlated Electron Systems**

Strongly correlated electron systems host electrons at the brink of localization. The simplest model that can capture this physics is the Hubbard model

$$H = -t\sum\_{\langle i\rangle,\sigma} (d\_{i\sigma}^{\dagger}d\_{j\sigma} + d\_{j\sigma}^{\dagger}d\_{i\sigma}) + lI\sum\_{i} d\_{i\uparrow}^{\dagger}d\_{i\uparrow}d\_{i\downarrow}^{\dagger}d\_{i\downarrow} \,. \tag{1}$$

The hopping integral *t* transfers electrons from site to site and thus promotes itineracy, whereas the onsite Coulomb repulsion *U* penalizes double occupancy of any site, thereby promoting localization. Thus, with increasing *U*/*t*, a (Mott) metal–insulator transition is expected. This simple model is suitable for materials where transport is dominated by one type of orbital with moderate nearest neighbor overlap, leading to one relatively narrow band. Well-known examples are found in transition metal oxides, for instance the cuprates. Here, the relevant orbitals are copper *d* orbitals, kept at distance by oxygen atoms. The creation and annihilation operators are called *d* and *d*† here.

If two different types of orbitals interplay—one much more localized than the other—a better starting point for a theoretical description is the (periodic) Anderson model that, for the one-dimensional case, reads [12,13]

$$H = \sum\_{\mathbf{k}, \sigma} \epsilon\_k \mathbf{c}\_{k\sigma}^\dagger \mathbf{c}\_{kr} + \sum\_{\mathbf{j}, \sigma} \epsilon\_f f\_{\mathbf{j}\sigma}^\dagger f\_{\mathbf{j}\sigma} + \mathcal{U} \sum\_{\mathbf{j}} f\_{\mathbf{j}\uparrow}^\dagger f\_{\mathbf{j}\uparrow} f\_{\mathbf{j}\downarrow}^\dagger f\_{\mathbf{j}\downarrow} + \sum\_{\mathbf{j}, \mathbf{k}, \sigma} V\_{\mathbf{jk}} (e^{i\mathbf{k}\mathbf{x}} f\_{\mathbf{j}\sigma}^\dagger \mathbf{c}\_{k\sigma} + e^{-i\mathbf{k}\mathbf{x}} c\_{\mathbf{k}\mathbf{r}}^\dagger f\_{\mathbf{j}\sigma}) \tag{2}$$

Orbitals with large overlap, with the associated creation and annihilation operators *c* and *c*†, form a conduction band with dispersion *k*. Orbitals with vanishing overlap situated at the positions *xj* are associated with the operators *f* and *f* †. They are assumed to be separated by a distance greater than the *f* orbital diameter and thus no hopping between them is considered. However, the hybridization term *V* allows the *f* electrons to interact. This model is particularly well suited for the heavy fermion compounds, which contain lanthanide (with partially filled 4 *f* shells) or actinide elements (with partially filled 5 *f* shells) in addition to *s*, *p*, and *d* electrons. For the so-called Kondo regime, where *f* orbitals effectively act as local moments, the Anderson model can be transformed into the Kondo (lattice) model

$$H = \sum\_{k,\sigma} \epsilon\_k \mathbf{c}\_{kr}^\dagger \mathbf{c}\_{kr} - J \sum\_i \vec{S}\_i \cdot \mathbf{c}\_{i,\sigma}^\dagger \vec{\mathcal{F}}\_{\sigma,\sigma'} \mathbf{c}\_{ir'} \tag{3}$$

where the interaction between the localized and itinerant electrons is expressed in terms of an antiferromagnetic exchange coupling *J*. *S* is the local magnetic moment of the *f* orbital and*σσ*,*σ* are the Pauli spin matrices. One of the possible ground states of this model is a paramagnetic heavy Fermi liquid with a large Fermi surface, which contains both the local moment and the conduction electrons. The resonant elastic scattering at each site generates a renormalized band at the Fermi energy. Its width is of the order of the Kondo temperature *T*K, which can be orders of magnitude smaller than the noninteracting band width. In the (typically considered) simplest case (with a uniform and *k* independent hybridization), this band extends across essentially the entire Brillouin zone.

In popular terms, this heavy fermion band could be seen as the realization of a nearly perfect "flat band" (an early description of an interaction-driven truly flat band, with zero energy, is given in [14] and its relevance for strange metal physics is discussed in [15,16]). Flat bands have also been predicted [17] and later identified in magic angle twisted bi-layer graphene (MATBG) [18] as a result of moiré band formation, and are expected in lattices

with specific geometries [19,20] such as the kagome lattice [21,22] through destructive phase interference of certain hopping paths. Whereas the theoretical description of these latter flat band systems may be simpler than solving even the simplest Hamiltonians for strongly correlated electron systems, such as (1)–(3), the inverse might be true for the challenge on the experimental side. Heavy fermion compounds with a large variety of chemical compositions and structures [23–25] can be quite readily synthesized as high-quality (bulk) single crystals; the heavy fermion "flat bands" are robust (not fine tuned), naturally extend essentially across the entire Brillouin zone, and are pinned to the Fermi energy. Albeit, they form in the Kondo coherent ground state of the system, which is typically only fully developed at low temperatures. To realize such physics via a complementary route that might bring these properties to room temperature is an exciting perspective. Bringing together these different approaches bears enormous potential for progress. Indeed, for both twisted trilayer graphene [26] and MATBG [27] the connection to heavy fermion physics has very recently been pointed out. Another topic discussed across the various platforms is "strange metal" physics, which we address next.

#### **3. Strange Metal Phase Diagrams**

Metals usually obey Fermi liquid theory, even in the limit of strong interactions. This is impressively demonstrated by the large body of heavy fermion compounds that, at sufficiently low temperatures, display the canonical Fermi liquid forms of the electronic specific heat

$$\mathcal{C}\_p = \gamma T \, , \tag{4}$$

the Pauli susceptibility

*χ* = *χ*<sup>0</sup> , (5)

and the electrical resistivity

*ρ* = *ρ*<sup>0</sup> + *AT*<sup>2</sup> , (6)

where *ρ*<sup>0</sup> is the residual (elastic) resistivity. Theoretically, the prefactors *γ*, *χ*0, and *A* all depend on the renormalized electronic density of states *N*<sup>∗</sup> = *N*/*N*0, or the related renormalized (density-of-states) effective mass *m*<sup>∗</sup> = *m*/*m*<sup>0</sup> ∼ *N*∗, to first approximation as *<sup>γ</sup>* ∼ *<sup>m</sup>*∗, *<sup>χ</sup>*<sup>0</sup> ∼ *<sup>m</sup>*∗, and *<sup>A</sup>* ∼ (*m*∗)2. *<sup>N</sup>*<sup>0</sup> and *<sup>m</sup>*<sup>0</sup> are the free electron quantities. Indeed, in double-logarithmic plots of *γ* vs. *χ*<sup>0</sup> (Sommerfeld-Wilson) and *A* vs. *γ* (Kadowaki-Woods), experimental data of a large number of heavy fermion compounds fall on universal lines, thereby confirming the theoretically expected universal ratios [28]. The scaling works close to perfectly if corrections due to different ground state degeneracies [29] and effects of dimensionality, electron density, and anisotropy [30] are taken into account.

More surprising, then, was the discovery that this very robust Fermi liquid behavior can nevertheless cease to exist. This can have multiple reasons, but the predominant and best investigated one is quantum criticality [4,25,31,32]. In the standard scenario for quantum criticality of itinerant fermion systems [33–35], a continuously vanishing Landau order parameter (typically of a density wave) governs the physical properties. Its effect on the electrical resistivity is expected to be modest because (i) the long-wavelength critical modes of the bosonic order parameter can only cause small-angle scattering, which does not degrade current efficiently, and (ii) critical density wave modes only scatter those areas on the Fermi surface effectively that are connected by the ordering wavevector. Fermions from the rest of the Fermi surface will short circuit these hot spots [36]. For itinerant ferromagnets, *<sup>ρ</sup>* ∼ *<sup>T</sup>*5/3 is theoretically predicted [4] and experimentally observed [37]. For itinerant antiferromagnets, this type of order-parameter quantum criticality should result in *<sup>ρ</sup>* <sup>∼</sup> *<sup>T</sup>* with 1 <sup>≤</sup> <sup>≤</sup> 1.5, depending on the amount of disorder [36]. Whereas this may be consistent with experiments on a few heavy fermion compounds, a strong dependence of with the degree of disorder has, to the best of our knowledge, not been reported. More importantly, for relatively weak disorder, the current is dominated by the contributions from the cold regions of the Fermi surface which stay as quasiparticles and the resistivity would have the *T*<sup>2</sup> dependence of a Fermi liquid [38].

Instead, a number of heavy fermion compounds exhibit a linear-in-temperature electrical resistivity

*ρ* = *ρ*- <sup>0</sup> + *A*- *T* , (7)

a dependence dubbed "strange metal" behavior from the early days of high-temperature superconductivity on [39]. In Figure 1a–d we show four examples, in the form of temperature– magnetic field (a,b,d) or temperature–pressure (c) phase diagrams with color codings that reflect the exponent of the temperature-dependent inelastic electrical resistivity, Δ*ρ* ∝ *T*, determined locally as = *∂*(ln Δ*ρ*)/*∂*(ln *T*). In all cases, fans of non-Fermi liquid behavior ( = 2) appear to emerge from QCPs, with close to 1 in the center of the fan and extending to the lowest accessed temperatures (at least in a,c,d).

**Figure 1.** *Cont*.

**Figure 1.** Color-coded phase diagrams featuring strange metal behavior in various materials platforms. (**a**) YbRh2Si2 (**left**) and YbRh2(Si0.95Ge0.05)2 (**right**), from [40]. (**b**) CeRu2Si2, from [41]. (**c**) CeRhIn5, from [42]. (**d**) Ce3Pd20Si6, from [43]. (**e**) SrRu3O7. Note that the temperature scale is cut at 4.5 K. At lower temperatures, deviations from linear behavior towards larger powers are observed; from [44]. (**f**) La2−*x*Sr*x*CuO4, from [45]. (**g**) BaFe2(As1−*x*P*x*)2, from [46]. (**h**) Magic-angle twisted bi-layer graphene, adapted from [47].

The most pronounced such behavior is found in YbRh2Si2 (Figure 1a, left). Below 65 mK, the system orders antiferromagnetically [48]. As magnetic field (applied along the crystallographic *c* axis) continuously suppresses the order to zero at 0.66 T [40], linear-in-temperature resistivity, with *A*- = 1.8 μΩcm/K and *ρ*- <sup>0</sup> = 2.43 μΩcm, extends from about 15 K [48] down to the lowest reached temperature (below 25 mK) [40]. Recently, this range was further extended down to 5 mK, showing *A*- = 1.17 μΩcm/K for a higher-quality single crystal (*ρ*- <sup>0</sup> = 1.23 μΩcm) [49], thus spanning in total 3.5 orders of magnitude in temperature. This happens in a background of Fermi liquid behavior away from the QCP. A linear-in-temperature resistivity is also seen in the substituted material YbRh2(Si0.95Ge0.05)2. Its residual resistivity is about five times larger than that of the stoichiometric compound. That this sizeably enhanced disorder does not change the power indicates that the order-parameter-fluctuation description of an itinerant antiferromagnetic quantum critical point [36] is not appropriate here. This point will be further discussed in Section 7.

For CeRu2Si2 (Figure 1b), the situation is somewhat more ambiguous. Linear-intemperature resistivity does not cover the entire core region of the fan; both above 2 K and below 0.5 K, crossovers to other power laws can be seen [41]. In CeRhIn5 (Figure 1c), at the critical pressure of 2.35 GPa, linear-in-temperature resistivity extends from about 15 K down to 2.3 K, the maximum critical temperature of a dome of unconventional superconductivity [42]. That the formation of emergent phases such as unconventional superconductivity tends to be promoted by quantum critical fluctuations is, of course, of great interest in its own right even if, pragmatically, it can be seen as hindering the investigation of the strange metal state. Finally, Ce3Pd20Si6 exhibits two consecutive magnetic field-induced QCPs, with linear-in-temperature resistivity emerging from both [43]. Other heavy fermion systems show similar behavior, though color-coded phase diagrams may not have been produced. A prominent example is CeCoIn5. Its electrical resistivity was first broadly characterized as being linear-in-temperature below 20 K down to the superconducting transition temperature of 2.3 K [50]. Both magnetic field [51,52] and pressure [53] suppress the linear-in-temperature dependence and stabilize Fermi liquid behavior, in agreement with temperature over magnetic field scaling of the magnetic Grüneisen ratio indicating that a quantum critical point is situated at zero field [54]. Indeed, small Cd doping stabilizes an antiferromagnetic state [55].

In Figure 1e–h, we show resistivity-exponent color-coded phase diagrams of other classes of strongly correlated materials, a ruthenate, a cuprate, an iron pnictide, and a schematic phase diagram of MATBG. Extended regions of linear-in-temperature resistivity are also observed. Before we discuss this strange metal behavior in more detail in Section 5, we take a closer look at the Fermi liquid regions of the heavy fermion phase diagrams.

#### **4. Fermi Liquid Behavior near Quantum Critical Points**

The low energy scales and associated low magnetic ordering temperatures typically found in heavy fermion compounds call for investigations of these materials at very low temperatures. Indeed, since early on, measurements down to dilution refrigerator temperatures have been the standard. Because scattering from phonons is strongly suppressed at such low temperatures, this is ideal to study non-Fermi liquid and Fermi liquid behavior alike. The phase diagrams in Figure 1a–d all feature Fermi liquid regions, at least on the paramagnetic side of the QCPs. The fan-like shape of the quantum critical regions dictates that the upper bound of the Fermi liquid regions shrinks upon approaching the QCP. Nevertheless, high-resolution electrical resistivity measurements still allow to extract the evolution of the Fermi liquid *A* coefficient upon approaching the QCP. In Figure 2 we show such dependencies for four different heavy fermion compounds. In all cases, the *A* coefficient is very strongly enhanced towards the QCP. In fact, within experimental uncertainty, the data are even consistent with a divergence of *A* at the QCP, as indicated by the power law fits, *<sup>A</sup>* ∼ 1/(*<sup>B</sup>* − *<sup>B</sup>*c)*a*, with *<sup>a</sup>* close to 1, in Figure 2a,c,d, suggesting that the effective mass diverges at the QCP.

**Figure 2.** Variation of the *A* coefficient of the Fermi liquid form of the electrical resistivity, *ρ* = *ρ*<sup>0</sup> + *AT*2, across QCPs in various heavy fermion compounds. (**a**) YbRh2Si2, from [40]. (**b**) CeRu2Si2, from [56]. (**c**) CeCoIn5, from [51]. (**d**) Ce3Pd20Si6, from [43].

This finding challenges the classification of heavy fermion compounds into lighter and heavier versions, that has been so popular in the early days of heavy fermion studies and that had culminated in the celebrated Kadowaki–Woods and Sommerfeld–Wilson plots, with each heavy fermion compound represented by a single point. Which *A* (*γ*, *χ*0) value should now be used in these graphs? In [32] the use of lines instead of points was suggested, using the largest and smallest actually measured values (and not extrapolations beyond them) as end points. The question that remains is whether there is a "background" value, away from a quantum critical point, that is characteristic of a given compound. We will get back to this question in the next section.

#### **5. Strange Metal Behavior and Planckian Dissipation**

The occurrence of fans or, in some cases, differently shaped regions of linear-intemperature resistivity in the phase diagrams of a broad range of correlated electron systems, as highlighted in Figure 1, raises the question whether a universal principle may be behind it. A frequently made argument is that linear-in-temperature resistivity is a natural consequence of the systems' energy scales vanishing at a quantum critical point and thus temperature becoming the only relevant scale. However, both the experimental observation of power laws <sup>Δ</sup>*<sup>ρ</sup>* <sup>∼</sup> *<sup>T</sup>* with <sup>=</sup> 1 in quantum critical heavy fermion compounds [57–60] and predictions from order-parameter-fluctuation theories of such laws [36] tell us that this argument cannot hold in general. We thus have to be more specific and ask whether for quantum critical systems that *do* exhibit linear-in-temperature resistivities and, apparently, require description beyond this order-parameter framework, a universal understanding can be achieved.

A direction that is attracting considerable attention [10,11,61] is to test whether the transport scattering rate 1/*τ* of such systems may be dictated by temperature via

$$\frac{1}{\tau} = \alpha \frac{k\_{\rm B}T}{\hbar} \tag{8}$$

with *α* ≈ 1. Should this be the case and *τ* be the only temperature-dependent quantity in the electrical resistivity, then a linear-in-temperature resistivity would follow naturally. Conceptually, this roots in the insight, gained from the study of models without quasiparticles [4,62–65], that a local equilibration time (after the action of a local perturbation) of any many-body quantum system cannot be faster than the Planckian time

$$
\pi\_\mathrm{P} = \frac{\hbar}{k\_\mathrm{B}T} \tag{9}
$$

associated with the energy *k*B*T* via the Heisenberg uncertainty principle [65]. The question then is how to experimentally test this scenario. The simplest starting point is the Drude form for the electrical resistivity which, in the dc limit, reads

$$
\rho = \frac{m}{ne^2} \frac{1}{\pi} \,, \tag{10}
$$

with a temperature-independent effective mass *m* and charge carrier concentration *n*, and (8) for the scattering rate 1/*τ*, leading to

$$
\rho = \kappa \frac{m}{m\varepsilon^2} \frac{k\_\mathrm{B} T}{\hbar} \,\mathrm{.}\tag{11}
$$

Interpreting this as the inelastic part of the linear-in-temperature electrical resistivity (7), with *dρ*/*dT* = *A*- , one obtains

$$
\alpha = \frac{n}{m} \frac{c^2 \hbar}{k\_B} A'\tag{12}
$$

or, in convenient units format,

$$\alpha = 2.15 \cdot \frac{n(\text{nm}^{-3})}{m/m\_0} \cdot A'(\mu \Omega \text{cm}/\text{K}) \,, \tag{13}$$

where *m*<sup>0</sup> is the free electron mass. When this results in *α* ≈ 1, the dissipation is said to be "Planckian". Before looking at experiments, let's contemplate this for a moment. Relation (12) is based on the simple Drude model, and combines properties of well defined quasiparticles (*n* and *m*) with a property that characterizes a non-Fermi liquid (*A*- )—possibly one

without quasiparticles—that is unlikely to follow the Drude model. Furthermore, as shown in Section 4, the Fermi liquid *A* coefficient, which is a measure of *m*, varies strongly with the distance to the QCP. Another defining property of at least some of these strange metals are Fermi surface jumps at the QCP (see Section 7). This adds a nontrivial temperature and tuning parameter dependence to *n*. One should thus bear in mind that choosing a simple Drude model as starting point holds numerous pitfalls. If still doing so, it is unclear which *m* and *n* value to use.

In [10], published quantum oscillation data, in part combined with results from density functional theory (DFT), were used to estimate *m* and *n* for a range of different materials, including also "bad metals" (see Section 6) and simple metals in the regime where their resistivity is linear-in-temperature due to scattering from phonons. As an example, for Sr3Ru2O7, de Haas–van Alphen (dHvA) data [66] measured at dilution refrigerator temperatures on the low-field side of the strange metal fan (Figure 1e) were used. Contributions from the different bands, assumed as strictly 2D, were summed up as

$$
\sigma = \pi \frac{e^2}{\hbar} \sum\_{i} \frac{n\_i}{m\_i} \,\prime\,\tag{14}
$$

i.e., a constant relaxation time was assumed for all bands. Then, the heavy bands with small carrier concentration play only a minor role. In this way, *α* = 1.6 was obtained. The dHvA effective masses of all bands were found to be modest (at most 10*m*0) and essentially field-independent [66], even though the *A* coefficient increases by more than a factor of 8 on approaching the strange metal regime from the low field side [66]. The dHvA experiments may thus not have detected all mass enhancement [10,66]. As shown below, using a larger effective mass would reduce *α*.

Similar analyses were performed for the other materials [10] and we replot the results as black points in Figure 3. The *x* axis of this plot is the Fermi velocity *v*<sup>F</sup> which, for a 3D system, can be brought into the form

$$\upsilon\_{\rm F}(\rm m/s) = 3.58 \cdot 10^5 \cdot \frac{[n(\rm nm^{-3})]^{1/3}}{m/m\_0} \,\mathrm{.}\tag{15}$$

The *y* axis is the inverse of *v*<sup>F</sup> multiplied by *α* (13) which, again for a 3D system, can be written as *<sup>α</sup>*

$$\frac{\mu}{\upsilon\_{\text{F}}} (\text{s/m}) = 6.01 \cdot 10^{-6} \cdot A'(\mu \Omega \text{cm/K}) \cdot \left[ n \left( \text{nm}^{-3} \right) \right]^{2/3}. \tag{16}$$

To further assess how the results for *α* depend on the choice of the quasiparticle parameters *m* and *n*, we here take a different approach. Instead of quantum oscillation data, we use global (effective) properties, namely, the *A* coefficient and the Hall coefficient *R*H, and estimate *α* for a number of strange metal heavy fermion compounds. Because of the extreme mass renormalizations observed in this class of materials (see Section 4), it is particularly well suited for this test. Combining

$$\frac{m}{m\_0} \cdot n^{1/3} = \frac{\gamma\_{\text{mole}-\text{f.u.}}}{V\_{\text{f.u.}}} \frac{3\hbar^2}{N\_{\text{A}} m\_0 k\_{\text{B}}^2 (3\pi r^2)^{1/3}}\tag{17}$$

with the Kadowaki–Woods ratio *A*/*γ*<sup>2</sup> = 10−<sup>5</sup> μΩcm(mole K/mJ)2, which is known to be very well obeyed in heavy fermion compounds [28], we obtain

$$\frac{m}{m\_0} \cdot \left[ n \left( \text{nm}^{-3} \right) \right]^{1/3} = 3.26 \cdot 10^4 \frac{\sqrt{A \left( \mu \Omega \text{cm} / \text{K}^2 \right)}}{V\_{\text{f.u.}} \left( \text{\AA}^3 \right)} \,. \tag{18}$$

The rationale for using *A* instead of *γ* is that precise resistivity measurements are most abundant in the literature (also under challenging conditions such as high pressure and magnetic field) and that the resistivity is much less sensitive to extra contributions from phase transitions than the specific heat. In addition, and unlike *γ*, the *A* coefficient picks up effective mass anisotropies, which further improves our analysis. In all cases where reliable *γ* values were available [43,67–69], the agreement with our *A* coefficient *γ* was satisfactory.

A note is due on the determination of the charge carrier concentration *n*. It is commonly extracted from the Hall coefficient *R*H, using the simple one-band relation *R*<sup>H</sup> = 1/*ne*. Heavy fermion compounds are typically multiband systems, and thus compensation effects from electron and hole contributions can occur [70]. To limit the effect of anomalous Hall contributions, low-temperature data should be used [71]. Quantum oscillation experiments can determine the carrier concentration of single bands. However, heavy bands are hard to detect and it is unclear how to sum up contributions from different bands. An alternative is to determine *n* via the superfluid density [72], as was done previously [49,73], using the relation (in cgs units)

$$m = \left(\frac{\xi\_0 \cdot T\_\odot \cdot \gamma}{7.95 \cdot 10^{-24}}\right)^{3/2},\tag{19}$$

where *ξ*<sup>0</sup> is the superconducting coherence length, *T*<sup>c</sup> is the superconducting transition temperature, and *γ* is the normal-state Sommerfeld coefficient, which can be rewritten as

$$n(\text{nm}^{-3}) = 3020 \cdot \left( \frac{\text{ ${}\_{0}$ }(\text{nm}) \cdot {}\_{\text{c}}(\text{K}) \cdot \gamma(\text{J} \text{mol}^{-1} \text{K}^{-2})}{V\_{\text{f.u.}}(\text{Å}^{\text{3}})} \right)^{3/2}. \tag{20}$$

This may be used as a lower bound of the carrier concentration in the normal state.

Table 1 lists the materials we inspected, with their *A* coefficients (or, when unavailable, *γ*), the best estimate of the charge carrier concentration *n* following the above discussion (see Table 2 for details), and the strange metal *A* coefficient. *m*/*m*<sup>0</sup> as calculated via (18), or (17), is also listed.

**Table 1.** Parameters used for Figures 3 and 4. The red (or blue) square represents the largest *A* coefficient (measured closest to the QCP), the shaded red (or blue) lines the range of *A* coefficient measured upon moving away from the QCP. The Sommerfeld coefficient *γ* is estimated from *A* via the Kadowaki–Wood ratio, unless *A* data are unavailable. The charge carrier concentrations *n* and their error bars (where applicable) are taken from Table 2. For CeCoIn5, several values are listed because the *A* coefficient is different for in-plane (*Ha*) and out-of-plane (*Hc*) field, and the *A*- coefficient is different for in-plane (*ja*) and out-of-plane (*jc*) currents. For YbAgGe, the *A* coefficient changes with field; the two extreme *A* values are denoted by the two red squares. For CeCoIn5 (*j* ⊥ *c*), Figure 3 shows the range *A*- = (0.8 ± 0.2) μΩcm/K from [74]. Data for Ce3Pd20Si6 refer to the second QCP (near 2 T, see Figure 1d) because for the lower field QCP no full data set on single crystals is published [43,75].


All these data are then included in Figure 3 in the following way. The *v*<sup>F</sup> (15) and *α*/*v*<sup>F</sup> (16) value resulting from the largest measured *A* coefficient (or *γ* value) for each compound is shown as red square. The shaded red lines represent the published ranges of *A* coefficient (or *γ* value). The error bars represent uncertainties in the determination of the charge carrier concentration (see Table 1). Lines for *α* = 1, 0.1, and 0.01 are also shown. It is clear that none of the shaded red lines overlaps with the *α* = 1 line. The discrepancy with the points extracted from quantum oscillation experiments [10] is quite striking.

**Table 2.** Charge carrier concentrations (in nm−3) determined as follows: (i) *n*sc from the superconducting coherence length *ξ*0, the superconducting transition temperature *T*c, and the normal-state Sommerfeld coefficient *γ*, all in zero field, via (20); (ii) *n*<sup>H</sup> from the Hall coefficient at the lowest temperatures, where anomalous contributions are minimal, via *R*<sup>H</sup> = 1/*ne*; (iii) *n*qo from quantum oscillation experiments reviewed in [10], by summing up the carrier concentrations from all detected bands. For CeCoIn5, the *γ* coefficient is taken at 2.5 K, without taking into account the logarithmic divergence. The error bar in *n* used for CeCoIn5 (*j* ⊥ *c*) in Figure 3 reflects the range of the parameters given in [74]. YbRh2Si2 is close to being a compensated metal, resulting in a strong sensitivity of *n* to small differences in the residual resistivity. The largest reported *R*<sup>H</sup> value, which corresponds to *n*<sup>H</sup> = 26.0 [71], has the lowest compensation and is thus most accurate. Nevertheless, the *R*<sup>H</sup> value of LuRh2Si2 is even larger, corresponding to *n*<sup>H</sup> = 11.6 nm−<sup>3</sup> [70], suggesting that there is still some degree of compensation in the sample of [71]. We list the average of both values, 18.8 nm<sup>−</sup>3, as best *n*<sup>H</sup> estimate. For the plots, we use the approximate average of *n*sc and *n*H, i.e., 10 nm−3, with an asymmetric error bar *<sup>δ</sup>n*<sup>+</sup> <sup>=</sup> 10 nm−<sup>3</sup> and *<sup>δ</sup>n*<sup>−</sup> <sup>=</sup> <sup>−</sup>5 nm−<sup>3</sup> (see Table 1). Similar compensation effects are also encountered in UPt3 [81]. Bold fonts indicate the values used for the *α* estimates (see Table 1).


**Figure 3.** Planckian dissipation plot of [10] revisited. Double-logarithmic plot of Fermi velocity *v*<sup>F</sup> vs. *ne*2/(*k*B*k*F)(*dρ*/*dT*) = *α*/*v*<sup>F</sup> with data from [10] (black points) and data of the heavy fermion compounds listed in Table 1 and analyzed here. The red squares result from the largest measured *A* coefficient (or *γ* value) for each compound near the strange metal regime, the shaded red lines from the published ranges of *A* coefficient (or *γ* value), and the error bars from uncertainties in the determination of the charge carrier concentration *n* and sometimes other parameters (see Table 1). The full, dashed, and dotted line represent *α* = 1, 0.1, and 0.01, respectively.

In Figure 4 we present these results in a different form, as *α* vs. (*m*/*m*0)/*n*. The red squares and red shaded lines have the same meaning as in Figure 3. The dashed lines are extrapolations of the shaded lines to *α* = 1. We can thus directly read off the values of (*m*/*m*0)/*n* for which a given compound would, in this simple framework, be a Planckian scatterer. In all cases, this is for effective masses significantly smaller than even the smallest measured ones in the Fermi liquid regime.

What are the implications of this finding? We first comment on the discrepancy with the results from [10]. Apparently, averaging the contributions from different bands detected in quantum oscillation experiments via (14) leads to sizeably larger Fermi velocities (sizeably smaller effective masses) than our *A* coefficient approach. In heavy fermion compounds, a coherent heavy fermion state forms at low temperatures, and the Fermi liquid *A* coefficient is known to be a pertinent measure thereof. It is thus either the use of (14) that should be reconsidered or the reliance in quantum oscillation experiments to detect the heaviest quasiparticles. Clearly, if dissipation in strange metal heavy fermion compounds is to be Planckian, this would hold only for the very weakly renormalized quasiparticles, as argued for in [88]. To us, this is a rather puzzling result as heavy fermion bands get successively renormalized with decreasing temperature and thus one would have expected that the "background" to effects of quantum critical fluctuations already contains a sizeable non-critical Kondo renormalization.

**Figure 4.** No Planckian dissipation from heavy quasiparticles in heavy fermion compounds. Doublelogarithmic plot of *α* vs. (*m*/*m*0)/*n* for various strange metal heavy fermion compounds, as given in Table 1. Red squares and shaded lines have the same meaning as in Figure 3. The dashed lines are to help reading off the values of (*m*/*m*0)/*n* for which the linear-in-temperature electrical resistivity in these compounds could be governed by Planckian dissipation. Note that in all cases the "Planckian dissipation" effective masses obtained in this way are sizeably smaller than even the smallest values experimentally accessed by tuning the systems away from the strange metal regime (top end of full shaded lines).

#### **6. Strange Metal Behavior and the Mott–Ioffe–Regel Limit**

In a number of strongly correlated electron systems, including quasi-2D conductors such as the high-*T*<sup>c</sup> cuprates but also 3D transition metal oxides and alkali-doped fullerides, linear-in-temperature resistivity is observed beyond the Mott–Ioffe–Regel (MIR) limit [89,90]. At this limit, the electron mean free path approaches certain microscopic length scales such as the interatomic spacing or the wavelength 2*π*/*k*<sup>F</sup> [65,91–94]. Semiclassical transport of long-lived quasiparticles might then, at least naively, be expected not to exist and the resistivity should saturate, in 3D systems of interest to us here to

$$
\rho\_{\rm MIR} = \frac{\hbar}{\varepsilon^2} \cdot L \,\,\,\,\,\tag{21}
$$

where *L* is the relevant microscopic length scale. Using the Drude resistivity (10) with the Fermi velocity *v*<sup>F</sup> = *hk*¯ F/*m*, the Fermi wave vector *k*<sup>F</sup> = (3*π*2*n*)1/3, and the mean free path = *τv*<sup>F</sup> one obtains

$$
\rho = \frac{\hbar}{\varepsilon^2} \cdot \frac{3\pi}{2} \frac{1}{k\_{\text{F}}^2 \ell} = \frac{\hbar}{\varepsilon^2} \cdot L \cdot \mathbb{C} \,, \tag{22}
$$

where the value of the constant *C* depends on details of the electronic and crystal structure. Assuming *C* = 1, one gets

$$
\rho\_{\rm MIR}(\mu\Omega\text{cm}) = 258 \cdot L(\text{Å})\,,\tag{23}
$$

In heavy fermion compounds, linear-in-temperature resistivities are limited to low temperatures (Figure 1a–d) and the *A* coefficients (Table 1) typically result in inelastic resistivities of the order of 10 μΩcm at the upper bound of the linear regime. This is well below the MIR limit. For instance, for YbRh2Si2, using the lattice parameters *a* = 4.007 Å and *c* = 9.858 Å [48] for *L* in (23) gives *ρ*MIR ≈ 1000 μΩcm and ≈ 2500 μΩcm, respectively, much larger than even the total resistivity at 15 K (which is about 30 μΩcm for YbRh2Si2 [48]), the upper bound of linear-in-temperature resistivity for that compound. In this case, a confusion with a linear-in-temperature resistivity due to electron-phonon scattering [65,95] can be safely ruled out.

#### **7. Strange Metal Behavior and Fermi Surface Jumps**

In Section 5, a simple Drude form was used for the electrical resistivity and all temperature dependence was attributed to the scattering rate. Then, the question was asked which quasiparticles (with which *m*/*n*) to take if the scattering were to be Planckian. The answer was that this would have to be very weakly interacting quasiparticles, certainly not the ones close to the QCP from which the strange metal behavior emerges. Here we address another phenomenon that may challenge a Planckian scattering rate picture: Fermi surface jumps across these QCPs.

This phenomenon was first detected by Hall effect measurements on YbRh2Si2 [71,96] (Figure 5a). Let us first recapitulate the experimental evidence for a Fermi surface jump across a QCP, as put forward in these works. Hall coefficient *R*<sup>H</sup> (or Hall resistivity *ρ*H) isotherms are measured as function of a tuning parameter *δ* (in case of YbRh2Si2 the magnetic field) across the QCP. A phenomenological crossover function, *R*<sup>∞</sup> <sup>H</sup> <sup>−</sup> (*R*<sup>∞</sup> <sup>H</sup> − *<sup>R</sup>*<sup>0</sup> <sup>H</sup>)/[1 + (*δ*/*δ*0)*p*] [71], is fitted to *R*H(*δ*) [or to *dρ*H/*dB*(*δ*)] and its full width at half maximum (FWHM) is determined as a reliable measure of the crossover width. Only if this width extrapolates to zero in the zero-temperature limit a Hall coefficient jump is established. Of course, the jump size must remain finite in the zero temperature limit. To identify a Fermi surface jump, other origins of Hall effect changes must be ruled out, for instance anomalous Hall contributions from abrupt magnetization changes at a metamagnetic/first order transition [97]. All this was done for YbRh2Si2 [71,96]. For Ce3Pd20Si6, using a very similar procedure, two Fermi surface jumps were found at the two consecutive QCPs (Figure 1d) [43,75]. The crossover at the first QCP [75] is shown in Figure 5b. It is also important to remind oneself that no Fermi surface discontinuity is expected at a conventional antiferromagnetic QCP as described by the spin-density-wave/order-parameter scenario [6]. Band folding of the (even at *T* = 0) continuously onsetting order parameter can in that case only lead to a continuously varying Hall coefficient, as seen for instance in the itinerant antiferromagnet Cr upon the suppression of the order by doping or pressure (see [98] for more details and the original references).

**Figure 5.** Fermi surface jumps as evidenced by Hall effect measurements in several strange metals. (**a**) YbRh2Si2, from [32,96]. (**b**) Ce3Pd20Si6, from [75]. (**c**) Substitution series of three high-*T*<sup>c</sup> cuprates, from [99]. (**d**) MATBG, from [100].

These jumps are understood as defining signatures of a Kondo destruction QCP, first proposed theoretically [5,6,101] in conjunction with inelastic neutron scattering experiments on CeCu5.9Au0.1 [9]. At such a QCP, the heavy quasiparticles, composites with *f* and conduction electron components, disintegrate. The Fermi surface jumps because the local moment, which is part of the Fermi surface in the paramagnetic Kondo coherent ground state [102], drops out as the *f* electrons localize. As such, Kondo destruction QCPs are sometimes referred to as *f*-orbital selective Mott transitions. More recently, THz time-domain transmission experiments on YbRh2Si2 thin films grown by molecular beam epitaxy revealed dynamical scaling of the optical conductivity [103]. This shows that the charge carriers are an integral part of the quantum criticality, and should not be seen as a conserved quantity that merely undergo strong scattering (as in order-parameterfluctuation descriptions with intact quasiparticles). We also note that a Drude description of the optical conductivity fails rather drastically in the quantum critical regime [103]. It is thus unclear how this physics could be captured by the simple Planckian scattering approach described above.

Interestingly, Hall effect experiments in other strange metal platforms also hint at Fermi surface reconstructions. Two examples are included in Figure 5: a series of substituted high-*T*c cuprates [99] (panel c) and MATBG as function of the total charge density induced by the gate [100] (panel d). Evidence for related physics has also been found in the pnictides [104]. The physics here appears to be related to the presence of *d* orbitals with a different degree of localization, with one of them undergoing a Mott transition, such as described by multiorbital Hubbard models [105,106]. It may well be that Fermi surface jumps are an integral part of strange metal physics, and should be included as a starting point in its description.

#### **8. Summary and Outlook**

We have revisited the question whether the strange metal behavior encountered in numerous strongly correlated electron materials may be the result of Planckian dissipation. For this purpose, we have examined strange metal heavy fermion compounds. Their temperature–tuning parameter phase diagrams are particularly simple: Fans of strange

metal behavior emerge from quantum critical points, in a Fermi liquid background. This, together with the extreme mass renormalizations found in these materials, makes them a particularly well-suited testbed.

As done previously, we use the Drude form of the electrical conductivity as a starting point, but complementary to a previous approach based on quantum oscillation data, we here rely on the Fermi liquid *A* coefficient as precise measure of the quasiparticle renormalization. We find that for any of the measured *A* coefficients, the slope of the linear-in-temperature strange metal resistivity *A* is much smaller than the value expected from Planckian dissipation. We also propose a new plot that allows to read off the ratio of effective mass to carrier concentration that one would have to attribute to the quasiparticles for their scattering to be Planckian. It corresponds to very modest effective masses. While this could be something like a smooth background to quantum critical phenomena, the fact that the strange metal regime occurs entirely below the temperature for the initial onset of the dynamical Kondo correlations suggests that this background should already incorporate the non-critical Kondo correlations and thus correspond to a relatively heavy mass.

We have also pointed out that several heavy fermion compounds exhibit Fermi surface jumps across strange metal quantum critical points and that this challenges the Drude picture underlying the Planckian analysis. Indications for such jumps are also seen in other platforms and may thus be a common feature of strange metals. Further careful studies that evidence a sharp Fermi surface change in the zero temperature limit, such as providing for some of the heavy fermion compounds, are called for. On the theoretical side, approaches that discuss the electrical resistivity as an entity and do not single out a scattering rate as the only origin of strangeness, are needed.

**Author Contributions:** Conceptualization and original draft preparation, S.P.; data analysis, M.T. and S.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research has received funding from the European Union's Horizon 2020 Research and Innovation Programme under Grant Agreement no 824109, and from the Austrian Science Fund (FWF Grant 29296-N27).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** No new data were created in this study. Data sharing is thus not applicable to this article.

**Acknowledgments:** Open Access Funding by the Austrian Science Fund (FWF). We acknowledge fruitful discussions with Joe Checkelsky, Piers Coleman, Pablo Jarillo-Herrero, Stefan Kirchner, Jose Lado, Patrick Lee, Xinwei Li, Doug Natelson, Aline Ramires, T. Senthil, Vasily R. Shaginyan, Qimiao Si, Chandra Varma, and Grigory Volovik.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

