Vortex dynamics in the two-dimensional BCS-BEC crossover

The Bardeen–Cooper–Schrieffer (BCS) condensation and Bose–Einstein condensation (BEC) are the two limiting ground states of paired Fermion systems, and the crossover between these two limits has been a source of excitement for both fields of high temperature superconductivity and cold atom superfluidity. For superconductors, ultra-low doping systems like graphene and LixZrNCl successfully approached the crossover starting from the BCS-side. These superconductors offer new opportunities to clarify the nature of charged-particles transport towards the BEC regime. Here we report the study of vortex dynamics within the crossover using their Hall effect as a probe in LixZrNCl. We observed a systematic enhancement of the Hall angle towards the BCS-BEC crossover, which was qualitatively reproduced by the phenomenological time-dependent Ginzburg-Landau (TDGL) theory. LixZrNCl exhibits a band structure free from various electronic instabilities, allowing us to achieve a comprehensive understanding of the vortex Hall effect and thereby propose a global picture of vortex dynamics within the crossover. These results demonstrate that gate-controlled superconductors are ideal platforms towards investigations of unexplored properties in BEC superconductors.

The phase diagram of superconductors is usually drawn on the T -carrier density plane, whereas, in cold atom 37 systems, the phase diagram is often drawn on the plane of T -1/k F a s , where k F and a s denote the Fermi vector and 38 scattering length, respectively, and thus 1/k F a s represents the normalized interaction strength. Therefore, it has not 39 been possible to directly compare the two BCS-BEC crossover systems yet. Recently, we presented a phase diagram 40 on the T /T F -∆/E F plane for 2D superconductors [1], which is free from the parameters specific to superconductors. 41 On the other hand, in the 2D 6 Li system, the experimental determination of ∆/E F as a function of the interaction 42 strength was recently reported [2]. Combining the phase diagram on the same system published in 2015 [3], we 43 are able to draw a phase diagram on the T /T F -∆/E F plane for the 2D 6 Li system. This allows us to construct a 44 unified experimental phase diagram of the BCS-BEC crossover, which is displayed in Supplementary Fig. 1. Though 45 there remain discrepancies due to the difference in definition of each parameter, the phase diagram shows that the 46 data of Li x ZrNCl and 6 Li just overlap with each other and encourages us to consider the BEC limit from the BCS side.

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Supplementary Figure 1. A unified BCS-BEC crossover phase diagram from combined experimental data for cold atom superfluids 6 Li and density-controlled superconductors LixZrNCl. Starting from the BCS side, the data points for the 2D superconductor LixZrNCl [1], the system studied in this work, are drawn. Here T * is the gap-opening temperature and Tc is the critical temperature. Starting from the BEC side, the purple data points correspond to the cold atom superfluid 6 Li, taken from previous reports [2,3]. Drawing the phase diagram for both systems of superconductivity and cold atom superfluidity on a common scale reveals the achieved overlap of data points in the BCS-BEC crossover. This ultimately motivates the study of BEC superconductivty starting from the BCS regime in LixZrNCl.
Supplementary Note 3: Basic transport properties and doping level determination 49 To determine the doping level, i.e., Li content x, the Hall effect was used. The linear slope of Supplementary 50 Fig. 2a was used to determine x and a systematic dependency of the slope with changing Li ion concentration is 51 apparent. The device operation for intercalation was analogous to previously established work [4]. To compute the Li 52 content, the Hall coefficient at 150 K was measured and we assume that each Li ion supplies one electron to the ZrNCl 53 system. For the determined doping levels, superconductivity was achieved, and we show the longitudinal resistivity as 54 a function of temperature for each in Supplementary Fig. 2b. With decreasing doping level, the critical temperature 55 T c increased from 11.4 K to 16.8 K for x = 0.47 and 0.0040 respectively. T c was determined by the temperature at 56 which the resistivity is half of the normal-state value at 30 K. The superconducting transition is sharp for high doping 57 levels but is significantly broadened towards lower doping levels. This may be explained by the enhanced fluctuation 58 strength (β in Supplementary Figs. 6 and 7) towards the BCS-BEC crossover. In addition, the dimensional crossover 59 from an anisotropic 3D superconductor to a 2D superconductor can be relevant since the dimensional crossover occurs 60 around x ∼ 0.1, as discussed in previous works [1,4]. In the low doping regime, the transition is better described by  in Li x ZrNCl are shown. To calculate E F and k F from n, an ideal parabolic band dispersion in two-dimensions was 81 employed. k F = (4πn layer /ss ) 1/2 and E F = 2 k 2 F /2m * , with n layer the 2D carrier density per layer, s the spin degree 82 of freedom, s the valley degree of freedom, the reduced Planck constant and m * the effective electron mass. In 83 the case of Li x ZrNCl, s = s = 2 and m * = 0.9m 0 , as reported before [5]. Here m 0 denotes the free electron mass.  Figure 4. Doping dependence of the ratio of the mean free path l and superconducting coherence length ξ0. Red dots represent the data points from this work while black circles represent the data from the previous work [1]. The green area highlights the moderately clean regime where 0.5 < l/ξ0 < 5, above or below which the Hall anomaly is seemingly not observable as described by Hagen et al. [6]. The inset shows the Hall angle dependence on l/ξ0. Almost no modulation of the l/ξ0 ratio with doping is observed, which is also reflected in the weak dependence of the Hall angle on this ratio. This concludes that the system stays in the relevant range of l/ξ0 over the course of this work and the observed trend in the Hall angle vs doping is not dominated by a change of this ratio.
Supplementary Note 6: Theoretical determination of T c by Hartree approximation 89 As explained in Methods, the time-dependent Ginzuburg-Landau (TDGL) model is given by where γ = π/8T * , λ = −(1/2T * )∂T * /∂E F , ξ = φ 0 /2πB c2 (0), A(r) = Bxŷ, and ∆(r, t) is the superconducting 91 order parameter varying in space and time. 92 We consider the GL Hamiltonian corresponding to Eq. (S8) as where a is another phenomenological parameter. Replacing |∆| 4 with 2 |∆| 2 |∆| 2 in Eq. (S9) by the Hartree approx-94 imation, we obtain the approximated Hamiltonian (S10) Here, the renormalized mass satisfies the self-consistent equation: where · · · is the canonical average using the Hamiltonian H GL and the temperature T . Expanding ∆(r) as Here, N and q are the Landau level index and its degeneracy index, 99 respectively, H N (z) is the N th Hermite polynomial, l := φ 0 /2πB, and the dimensionless magnetic field is defined 100 as h := (ξ/l) 2 = B/B c2 (0). Then, the self-consistent equation (S11) may be rewritten as where c is a cutoff parameter representing the limitation of the gradient expansion in the GL Hamiltonian (S9).

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We further rewrite Eq. (S12) as = ( Assuming large enough c/h and taking c/h → ∞ in [· · · ] in the right-hand side, we can obtain where ψ(z) is the digamma function, which satisfies ψ(z) = lim n→∞ [ln n − for z → ∞. Then, we can obtain the equation for zero magnetic field as (S14) Based on the right-hand side of Eq. (S14), we define the renormalized T c as 108 (S15) Note that the spontaneous symmetry breaking does not occur at finite temperatures since we consider a 2D system, 109 and T c characterizes a typical temperature for significant changes in physical quantities such as conductivity.

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Performing the time and space integrations and the canonical average in Eq. (S20), we can finally obtain the following 126 formulas: where µ N := + h + 2N h, and a cutoff c is introduced similarly to Eq. (S12).
Thus, for low enough temperatures, where + h h and σ V ab dominates over the normal-state conductivity σ N ab , 132 the resistivities are given as ρ xx : , and the Hall angle Θ H follows By explicitly setting γ = π/8T * and λ = −(∂T * /∂E F )/2T * in (S25), we finally obtain the opposite sign to that in the normal state.

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To gain further insight into the dynamics of the condensate, it would be helpful to to rewrite ∆(r, t) as |∆(r, t)|e iχ(r,t) and decompose TDGL equation (S8) multiplied by e −iχ(r,t) into real and imaginary parts The GL Hamiltonian (S9) is introduced in eq. (S27a), which describes the relaxation dynamics of the condensate. In 152 eq. (S27b), we introduce the notation: the latter of which is nothing but (S18). We can regard Eq.
which follows from Eqs. (S27a) and (S27b), and the Ampere-Maxwell equation (see derivation of [9]). Let us see the physical meaning of each term in order to confirm that this equation is really regarded as the momentum balance relation. We start with the right-hand side. Here the electric magnetic fields are denoted by ε = −∂A/∂t and h = ∇ × A. The first two terms in the right-hand side represent the electromagnetic Lorentz force. The third term in the right-hand side is the dissipation force due to the time variation of the modulus of |∆| (This mechanism was first pointed out by Tinkham [10]). The last term in the right-hand side in Eq. (S30) is the other dissipation force due to conversion between the superfluid and normal components. This disspative force is caused by the time-variation of the phase of ∆. Thus these two terms show that the vortex motion is the source of the dissipative force. In the left-hand side, P represents the hydrodynamic momentum flux tensor, which is given in the present case by which coincides with the momentum flux tensor in the London equation [11]. We then finally identify −ρ s Q in the 163 first term in the left-hand side with the superfluid component of the momentum density. We see that this term has 164 the same sign as that of ρ s j s . When λ < 0, the momentum density is antiparallel to the electric current density j s and 165 thus the condensate corresponds to the positive electron density and the dynamics is similar to the electron motion.

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When λ > 0, on the other hand, the momentum density is parallel to the electric current density j s and thus the 167 condensate corresponds to deficit of electron density and the dynamics is similar to the hole motion. In this section, 168 we argue that the dynamics of the condensate described by the TDGL equation is similar to that of electron (hole)   Figure 5. Gap-opening temperature T * and superconducting critical temperature Tc as previously established [1].
The black dashed line is the fitting curve of T * (EF).
Supplementary Note 10: Mean-field critical temperature T *

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We discuss the difference between ∂T * /∂µ and ∂T * /∂E F for the 2D Fermi gas model (S39) within the mean-field 202 approximation. Using the two-particle binding energy E B instead of the coupling constant g, we obtain the equation 203 to determine T * for a given µ [17, 18]:  If we further reduce the doping x of Li x ZrNCl than in the present experiment, the distance between conduction 219 electrons can become larger than the range of the effective attractive interaction. Then, the system may be described 220 by the 2D Fermi gas model with a contact attractive interaction (g > 0): where ψ σ (r) and ψ † σ (r) are the Fermion field operators. In this model, the crossover from the BCS regime to the BEC where θ(z) is the Heaviside step function, P(· · · ) means the Cauchy principal value, µ * is the mean-field chemical

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When vortices flow parallel to the superflow, the transport current J tr and v v are anti-parallel and thus E and J tr ×B 251 are parallel, i.e. σ xx = 0 and σ xy < 0 for B parallel to z-axis. It then follows that Θ H = −π/2.