Mott-driven BEC-BCS crossover in a doped spin liquid candidate, kappa-(BEDT-TTF)4Hg2.89Br8

The pairing of interacting fermions leading to superfluidity has two limiting regimes: the Bardeen-Cooper-Schrieffer (BCS) scheme for weakly interacting degenerate fermions and the Bose-Einstein condensation (BEC) of bosonic pairs of strongly interacting fermions. While the superconductivity that emerges in most metallic systems is the BCS-like electron pairing, strongly correlated electrons with poor Fermi liquidity can condense into the unconventional BEC-like pairs. Quantum spin liquids harbor extraordinary spin correlation free from order and the superconductivity that possibly emerges by carrier doping of the spin liquids is expected to have a peculiar pairing nature. The present study experimentally explores the nature of the pairing condensate in a doped spin-liquid candidate material and under varying pressure, which changes the electron-electron Coulombic interactions across the Mott critical value in the system. The transport measurements reveal that the superconductivity at low pressures is a BEC-like condensate from a non-Fermi liquid and crosses over to a BCS-like condensate from a Fermi liquid at high pressures. The Nernst-effect measurements distinctively illustrate the two regimes of the pairing in terms of its robustness to the magnetic field. The present Mott tuning of the BEC-BCS crossover can be compared to the Feshbach tuning of the BEC-BCS crossover of fermionic cold atoms.


A. Crossover from non-Fermi liquid to Fermi liquid
To precisely associate the nature of electron pairing with that of the normal-state fluidity above T c , we performed resistivity measurements under finely incremented pressures [25]. To avoid resistivity jumps due to microcracking of the crystal encountered in the ambient-pressure measurements, we had to apply finite pressures to obtain jump-free resistivity data. As seen in Fig. 2(a), the temperature dependence of the in-plane resistivity ρ(T ) varies systematically with pressure; the linear temperature dependence changes into superlinear dependences at elevated pressures. Fitting the form of ρ n (T )=ρ 0 +AT α to the normal-state resistivityρ(T ) below 10 K measured under 9 T yields the value of the residual resistivity ρ 0 for each pressure. The logarithmic plot of (ρ(T )ρ 0 ) versus T (Fig. 2(b)) highlights the pressure evolution of the temperature-dependent part of resistivity. The slopes of these curves, given by α(T ) ≡ d log(ρ(T )ρ 0 )/d logT , are defined as the exponent at temperature T . The contour plot of the deduced α(T ) values on the pressure-temperature plane ( Fig. 2(c)) shows that at low temperatures, α(T ) is approximately unity before the pressure reaches 0.4 GPa and sharply switches to the value of 2 at higher pressures, demonstrating that an NFL crosses over to an FL at approximately 0.35-0.40 GPa. This finding reproduces the previous results [29] and additionally reveals that the crossover is so sharp, as theoretically suggested [37], that it is reminiscent of a quantum phase transition [38][39][40][41]. This sharp crossover is regarded as a correspondence to the Mott transition in κ-Cu 2 (CN) 3 , as implied by Fig. 1(c); at the crossover, the interaction strength crosses a Mott critical value at which the double occupancies of holes on a site, namely, Mottness, drastically changes [29], as theoretically suggested [30]. Furthermore, the NFL state does not only occur at the quantum critical point but extends to lower pressures suggesting a stable quantum critical phase [42].
Such an NFL phase is also argued in a heavy electron system presumably hosting a QSL [43]. Charge carriers moving around in QSL phases may not achieve sufficient coherence, losing Fermi degeneracy and resulting in NFL phases. Theoretically, the temperature-linear resistivity observed in κ-HgBr at low pressures is suggested for bosonic charge excitations in a spin-charge separated doped spin liquid [17].
B. Upper critical field and superconducting coherence length : BEC-BCS crossover Next, we explore how the nature of the superconducting condensate changes upon crossover from the anomalous metal to the ordinary metal by pressure. To assess the BEC or BCS nature, we evaluate the in-plane Ginzburg-Landau coherence length ξ , which measures the Cooper-pair size, albeit not strictly in the BEC regime [44], from the perpendicular upper critical field, H c2⊥ . To determine H c2⊥ , we traced the in-plane resistivity upon sweeping the magnetic field normal to the conducting layers shown at fixed temperatures [25]. Figure 3(a) shows the resistivity evolution upon the destruction of superconductivity by magnetic field at 0.32 GPa as an example (see [25] for the resistivity data at other pressures, 0.24, 0.27, 0.32, 0.39, 0.52, 0.58, 0.73, 0.83 and 1.0 GPa). As the normal state shows no appreciable magnetoresistance as seen in the figure, the field dependence of the resistivity stems from the mobile vortices and/or superconducting fluctuations. The resistivity behavior in the magnetic field-temperature (H-T ) plane at each pressure is shown in the contour plot of the reduced resistivity ρ(T , H)/ρ n (T ), where ρ n (T ) is the normal-state resistivity determined above, in Figs. 3(b)-3(j). In the blue area, the resistivity vanishes or is small.
As the magnetic field increases, the system enters the mixed state with mobile vortices, ρ(T , H)/ρ n (T ) takes finite values (light blue-yellow area), and eventually, the normal-state resistivity is restored, i.e., ρ(T , H)/ρ n (T ) ≈ 1 (red area). In the low-pressure regime, there is a wide region with finite values in ρ(T , H)/ρ n (T ), indicating that mobile vortices or superconducting fluctuations are developed even at high fields. In contrast, at high pressures, the superconductivity is fully suppressed by low magnetic fields, and the transition to the normal state is sharp, as in conventional superconductors. Overall, superconductivity is robust to magnetic fields at low pressures.
The coherence length ξ is evaluated from the temperature derivative of H c2⊥ near T c through T c [dH c2⊥ (T )/dT ] T =Tc = φ 0 /2πξ 2 , where φ 0 is the flux quantum. In highly twodimensional layered superconductors, finite resistivity is caused by vortex flow under perpendicular magnetic fields, as observed in this system [45]; therefore, the definition of H c2⊥ is not straightforward. In such a case, H c2⊥ is better characterized by an initial drop in resistivity. Here, we adopt as H c2⊥ the 80% and 90% transition points (ρ(T , H)/ρ n (T ) = 0.8 and 0.9), which are indicated by bold contours in Fig. 3 . The values of [dH c2⊥ (T )/dT ] T =Tc for the the 80% and 90% transition lines are shown in Fig. 4. Using these values, we determined the ξ values, whose pressure dependence is also shown in Fig.   4. At pressures below 0.4 GPa, ξ ∼ 3 nm, whereas, it increases steeply with pressure and reaches ∼ 20 nm at 1 GPa. Assuming a cylindrical Fermi surface, we evaluated the Fermi wavenumber k F from the Hall coefficient under pressure [25] and obtained the k F ξ value, an index of the BEC/BCS nature or, roughly speaking, the degree of overlap of Cooper pairs. The pressure dependence of k F ξ is shown along with T c in Fig. 5. At high pressures, e.g., 1 GPa, k F ξ reaches ∼ 50, pointing to highly overlapping pairs as in the conventional BCS regime, whereas at low pressures, k F ξ is as small as ∼ 3; the size of the Cooper pair may become even smaller than the evaluated ξ value when approaching the BEC regime according to a theoretically suggestion [44]. The k F ξ value of the order of unity signifies that the pairing is BEC-like, while the value of several tens points to BCS-like pairing, thus demonstrating that pressure induces a BEC-to-BCS crossover. The field-robust vortex liquid state and fluctuations at low pressures are consistent with the short coherence length and small density of the bosonic pairs inherent in BEC-like pairing, as shown theoretically [46].

C. Nernst effect and superconducting fluctuations
We further verified the preformed nature and magnetic field robustness of the pairing by examining the Nernst effect, in which a thermal flow generated by a temperature gradient ∇ x T under a perpendicular magnetic field induces a transverse electric field E y (see the inset of Fig.6). This effect, characterized by e N ≡ ±E y /∇ x T (with the sign depending on the field direction), is known to be enhanced by phase fluctuations arising from vortex liquid or preformed pairs in the superconducting state under magnetic fields, as observed in real materials such as cuprate and organic superconductors [10,[47][48][49][50][51]. Figure 6 shows the temperature variation of e N at several fixed magnetic fields under a pressure of 0.3 GPa. The e N value, which is small in the normal state, starts to increase at temperatures well above T c upon cooling, signaling superconducting preformation; it is remarkable that e N shows a field-insensitive universal temperature variation at high magnetic fields. are extended vertically and even tend to go beyond the experimental maximum field, 9 T. In general, quasi-two-dimensional superconductivity is easily suppressed by the perpendicular magnetic field far less than the Pauli limiting field, which is ∼ 10 T at 0.3 GPa, due to the orbital depairing. No such symptom up to 9 T and higher evidences unusual robustness of the pairing to the magnetic field, in line with the view of the BEC condensate. At higher pressures, the ridge fields become suppressed along with T c and inclined, as in conventional superconductors. To see the pressure-temperature profile of e N , we make a contour plot of the T c values at the maximum experimental field, 9 T (Fig. 8), which illustrates that the field-robust Nernst signal and its persistence at high temperatures far beyond T c is specific to the low-pressure phase. This is the Nernst behavior typical of BEC-like condensates; the effect of preformed pairs is expected to set in at higher temperatures in the Nernst signal than in the resistivity [10], consistent with the present observations. The systematic change of the Nernst profile with pressure lends support to the BEC-BCS crossover of the superconducting condensate.
It is reported that the 13 C NMR relaxation rate divided by temperature, 1/T 1 T , shows a decrease below 7-9 K [52] despite of T c ∼ 4.2 K at ambient pressure and the decrease becomes less prominent at higher pressures. This behavior is a likely manifestation of the preformed pairs in the BEC-like regime at ambient pressure and its crossover to the BCS regime at high pressures. We also note that the recent torque measurements suggest the fluctuating superconductivity that sets in at around 7 K at ambient pressure, implying the preformation of the Cooper pairs [53].

III. CONCLUDING REMARKS
The electron transport and Nernst effect in the doped triangular system, κ-HgBr, revealed that the nature of the superconductivity crosses over from BEC-like to BCS pairing with increasing pressure. The present BEC-BCS crossover is associated with an NFL-FL crossover, at which the Coulombic repulsive energy relative to the band width exceeds a critical value for a Mott transition that would occur unless doped. To explore the nature of fermionic pairing that encompasses the BEC and BCS regimes, it is crucial to vary the control parameter linked to the interparticle interaction across some critical value at which the nature of the interaction dramatically changes. In cold fermionic atoms, the controlling parameter is the magnetic field, which varies interatomic interactions across a critical value through the Feshbach resonance. In the present case of interacting electrons, pressure is a parameter that varies the Coulombic interactions across the Mott critical value at which the double occupancies of carriers are critically allowed or forbidden [30]. This Mott-driven BEC-BCS crossover is a novel addition to the physics of the pairing instability of interacting fermions.

Measurements of resistivity
The in-plane resistivity was measured by the conventional four-probe method; four gold wires were attached on the planar surface of a crystal with carbon paste. The Quantum Design Physical Property Measurement System (PPMS) was used for the resistivity measurements. To investigate the perpendicular upper critical field, H c2⊥ , we traced the resistivity with the magnetic field applied perpendicular to the conducting layers up to 9 T. In organic conductors, the rapid cooling often causes unwanted crystal cracking and/or conformational disorder of terminal ethylene groups in BEDT-TTF. To avoid suffering from these, the cooling rate was kept below 0.5 K/min.

Measurements of the Nernst effect
The Nernst voltage, e N , was generated in a direction perpendicular to both the directions of an externally created temperature gradient and an applied magnetic field (see the inset of Fig. 6). The magnetic field was applied perpendicular to the conducting layers. The temperature difference across the crystal, ∆T , along the two-dimensional plane was kept less than 0.5 K in the field-sweep measurements and less than T /10 in the temperature- a heater attached on one side of the crystal and the sample temperature was defined by the mean value of the temperatures of the hot and cold sides. As in the resistivity measurements, the cooling rate was kept below 0.5 K/min. The thermopower was also measured along with the Nernst voltage for the identical sample in order to know the T c of the sample used.

Pressurization
To apply hydrostatic pressures to the sample, a dual structured clamp-type pressure cell made from BeCu and NiCrAl cylinders was used with the Daphne 7373 oil as a pressuretransmitting medium. The pressure values quoted in this article are the internal pressures at low temperatures determined with the manganin pressure gauge or the tin manometer.
According to Ref. [54], the clumped pressure gradually decreases by 1.5-2.0 kbar on cooling from 300 K to 50 K and is nearly constant below that. The family of organic charge-transfer salts, κ-(BEDT-TTF) 2 X, where BEDT-TTF and X stand for bis(ethylenedithio)tetrathiafulvalene and anions, respectively, have layered structures composed of BEDT-TTF conducting layers and insulating anion layers, both of which have monoclinic sublattices with the space group C2/c and C2/m, respectively [55]. In the conducting layers, the BEDT-TTF molecules form dimers, which constitute a quasitriangular lattice characterized by two kinds of transfer integrals, t and t ′ , between the adjacent dimer molecular orbitals ( Fig. 1(b)). According to the molecular orbital calculations, the ratio, t ′ /t, in κ-(BEDT-TTF) 4 Hg 2.89 Br 8 is 1.02 [56], which is compared to the value in the Mott-insulating spin liquid candidate, κ-(BEDT-TTF) 2 Cu 2 (CN) 3 , 1.06 [57] (0.83 [58] and 0.83-0.99 [59] according to the first-principles calculations). surability precisely determined by x-ray diffraction [24,55] results from chemistry between BEDT-TTF, Hg and Br. Such non-stoichiometry occurs in other Hg-containing compounds as well [61]. The band filling slightly deviated from a half is evidenced by the valence of BEDT-TTF determined by the Raman spectroscopy [62]. This band-filling deviation from a half corresponds to a 11% hole doping to the half-filled band.

Determination of the upper critical field
The upper critical field is usually defined as the magnetic field at which the vanishingly small resistivity transitions to the normal-state one. In highly two-dimensional layered superconductors like the present material, however, resistivity is induced by mobile vortices or, broadly speaking, superconducting phase fluctuations under perpendicular field even when the amplitude of the superconducting order parameter is developed. Figure 9 shows the magnetic field dependence of the in-plane resistivities at each pressure, where the magnetic field was applied normal to the conducting plane. As expected, the resistive transition against field variation is gradual, in particular, for low pressures. Therefore, we defined the perpendicular upper critical filed H c2⊥ by the field at which the resistivity drops by 10% and 20% from the normal-state value, ρ n (T ), of the form, ρ n (T ) = ρ 0 + AT α , fitting the normal state resistivity; namely, H c2⊥ is defined in two ways as the fields of ρ(T, H)/ρ n (T ) = 0.8 and 0.9, where ρ(T, H) is the resistivity at temperature T and magnetic field H. The contour curves of ρ(T, H)/ρ n (T ) = 0.8 and 0.9 are shown by the bold lines in Fig. 3(b)-(j).

Evaluation of k F ξ
The in-plane coherence length ξ was calculated from the slope of H c2⊥ near T c , dH c2⊥ /dT , through the formula, ξ = φ 0 2πTc|dH c2⊥ /dT | The experimental dH c2⊥ /dT and T c values are shown in Fig. 4 and Fig. 5 in main text. The Fermi wavenumber, k F , was estimated from the Hall coefficient data [29] using the relation R H = 1/ne and assuming the twodimensional cylindrical Fermi surface, where n is the carrier density and e is the elementary charge. The pressure dependences of the R H value at 10 K [29] and the calculated k F values are shown in Fig. 10: the k F values at pressures studied here were obtained by interpolating the points in Fig. 10(b).  Fig. 2(c)). The purple area in κ-(BEDT-TTF) 2 Cu 2 (CN) 3 depicts the Mott insulating spin liquid phase [35,36]. T c in κ-(BEDT-TTF) 2 Cu 2 (CN) 3 is taken from Ref. [35], and T c in κ-(BEDT-TTF) 4      The magnetic field was applied perpendicular to the conducting layer. Temperature was fixed at intervals of 0.3 K below T c . For reference data at a high temperature well above T c , the resistivity curves at 10 K were measured.