Universal linear-temperature resistivity: possible quantum diffusion transport in strongly correlated superconductors

Abstract

The strongly correlated electron fluids in high temperature cuprate superconductors demonstrate an anomalous linear temperature (T) dependent resistivity behavior, which persists to a wide temperature range without exhibiting saturation. As cooling down, those electron fluids lose the resistivity and condense into the superfluid. However, the origin of the linear-T resistivity behavior and its relationship to the strongly correlated superconductivity remain a mystery. Here we report a universal relation $d\mathit{\rho }/\mathit{dT}=({\mathit{\mu }}_{\mathbf{0}}{\mathit{k}}_{\mathit{B}}/\mathbf{\hslash }){\mathit{\lambda }}_{\mathit{L}}^{\mathbf{2}}$, which bridges the slope of the linear-T-dependent resistivity (dρ/dT) to the London penetration depth λ _L at zero temperature among cuprate superconductor Bi₂Sr₂CaCu₂O_8+δ and heavy fermion superconductors CeCoIn₅, where μ ₀ is vacuum permeability, k _B is the Boltzmann constant and ħ is the reduced Planck constant. We extend this scaling relation to different systems and found that it holds for other cuprate, pnictide and heavy fermion superconductors as well, regardless of the significant differences in the strength of electronic correlations, transport directions, and doping levels. Our analysis suggests that the scaling relation in strongly correlated superconductors could be described as a hydrodynamic diffusive transport, with the diffusion coefficient (D) approaching the quantum limit D ~ ħ/m*, where m* is the quasi-particle effective mass.

Introduction

In quantum mechanics, the uncertainty principle gives rise to quantum fluctuations of the system that may impose some universal bound on its physical properties. Calculations based on the AdS/CFT (Anti de-Sitter/Conformal Field Theory) have suggested a lower bound for the liquid viscosity, η/s ≥ ħ/4πk _B ¹, where η is the shear viscosity and s the entropy. Recent experiments also revealed a quantum bound D _s ≥ ħ/m for the spin diffusivity D _s in a strongly interacting Fermi gas^{2, 3}. Here ħ is the reduced Planck constant and m is the mass of particles. It is therefore interesting to ask if such a lower bound may be realized in the electron transport of strongly correlated quantum critical systems.

One of the distinguished features of strongly correlated cuprate superconductors is the linear-temperature (T) dependent resistivity⁴, which could extend to very high temperature⁴ and violate the Mott-Ioffe-Regel (MIR) limit⁵. The linear relationship has also been observed in some heavy fermion superconductors^{6, 7}, starting at around the superconducting transition temperature T _c and extending to high temperatures of about 10–20 times of T _c. Many different mechanisms have been proposed to explain the microscopic origin of the linear-T resistivity behavior including quantum critical theories and the more exotic AdS/CFT calculations. On the other hand, recent experiment suggested that the linear-T resistivity in different materials may share a similar scattering rate⁸.

In the present work, we investigated the linear-T resistivity in a number of strongly correlated superconductors and demonstrate a connection between its coefficient and the superfluid density responsible for the charge carrying in the superconducting state. We show that this can be understood by a diffusion transport of heavy quasi-particles whose diffusion coefficient approaches the quantum limit D = ħ/m*, where m* is the effective mass of the quasi-particles.

Results

We start with the heavy fermion superconductor CeCoIn₅. Among all strongly correlated superconductors, CeCoIn₅ is remarkably similar to the high T _c cuprate superconductors in several aspects⁹. For example, it has also a two-dimensional Fermi surface^{10, 11}, its superconducting phase is near to an antiferromagnetic phase^12–14, and its superconducting gap has d-wave symmetry^15–17. Besides, CeCoIn₅ is one of the purest strongly correlated superconductors^{6, 7}, with a tunable linear-T resistivity under modest applied pressure^18–21. To examine its transport properties, we have therefore grown high quality CeCoIn₅ single crystal samples by an indium self-flux method⁶ and performed detailed transport measurements under pressure to avoid disorder related effects.

Figure 1(a) demonstrates the T-dependent resistivity curve of CeCoIn₅ under pressure from 0 GPa to 1.0 GPa. All of them exhibit a perfect linear-in-T resistivity from around T _c to about 20 K as indicated by the dashed lines. The inset of Fig. 1(a) shows the T-dependent resistivity of CeCoIn₅ up to 300 K. For comparison, we also plot in Fig. 1(b) the resistivity of Bi₂Sr₂CaCu₂O_8+x from underdoped to overdoped regime with the oxygen contents from x = 0.2135 to 0.27²². Figure 1(c) demonstrates the dρ/dT versus ${\lambda }_L^2$ for both compounds, using the experimental results for the penetration depth measured previously by muon spin spectroscopy²³ and ac susceptibility²⁴. We see remarkably that all the investigated samples fall on the same straight line described by $d\rho /dT=({\mu }_0{k}_B/\hslash ){\lambda }_L^2$, with a coefficient that is determined entirely by the fundamental constants (μ ₀: the vacuum permeability; k _B: the Boltzmann constant; ħ: the reduced Planck constant). This indicates a universal origin for the charge transport in both compounds.

Figure 1.

(a) T dependent resistivity ρ of CeCoIn₅ under the pressure 0, 0.3, 0.55, 1.0 GPa. The arrow points to the increase in pressures. Inset to 1(a) is the ρ of CeCoIn₅ up to 300 K. (b) T dependent resistivity ρ of oxygen doped Bi₂Sr₂CaCu₂O_8+x with x = 0.2135, 0.217, 0.22, 0.24, 0.245, 0.255, 0.26, 0.27, respectively. The arrow points to the Bi2212 from underdoped to overdoped. The resistivity data of Bi₂Sr₂CaCu₂O_8+x are taken from the literature²². (c) Linear scale plot of dρ/dT vs. ${\lambda }_L^2$ for CeCoIn₅ (red triangles) and Bi₂Sr₂CaCu₂O_8+x (Bi2212) (black circles). dρ/dT is the slope of linear-temperature-dependent resistivity, and λ _L is the London penetration depth of superconductors at zero temperature. The straight line corresponds to $d\rho /dT=({\mu }_0{k}_B/\hslash ){\lambda }_L^2$, where ρ is in the unit of μΩcm, λ _L is in μm, and T is in K. See Table 1 for details, including errors.

Table 1.

Transport parameters and London penetration depth at zero temperature.

Material	label	T c(K)	dρ/dT(μΩcmK −1)	T-linear range in calculation (K)	Refs	λL(nm)	Refs
c-axisYBa2Cu3O6.93	1	92	12.3 ± 0.07	250–400	49	923	26
c-axisYBa2Cu3O6.88	2	92	12.8 ± 0.17	300–400		1400
c-axisYBa2Cu3O6.78	3	82	30 ± 0.84	320–400		2900
a-axis YB2Cu3O6.93	YBa2Cu3Ox	92	0.78 ± 0.001	105–300		160	50, 51
b-axisYBa2Cu3O6.93			0.37 ± 0.004	150–300		100
a-axis YB2Cu3O7			0.95 ± 0.002	110–300	52	160
b-axisYBa2Cu3O7			0.43 ± 0.003	255–300		100
Tl2Ba2CuO6+x	Tl2Ba2CuO6+x	80	1.55 ± 0.001	120–300	53	240 ± 20	54
c-axis Tl2Ba2CuO6+x		85	2681 ± 3.4	220–300	55	17500 ± 2500	55–57
Bi2Sr2CaCu2O8+x (x = 0.2135)	Bi2Sr2CaCu2O8+x	72 ± 2	2.68  ±  0.01	200–300	22, 58	447 ± 50	24
Bi2Sr2CaCu2O8+x (x = 0.217)		79.3	2.36 ± 0.02	200–300		370 ± 20
Bi2Sr2CaCu2O8+x (x = 0.22)		81.9	1.78 ± 0.003	180–300		338 ± 20
Bi2Sr2CaCu2O8+x (x = 0.24)		87.9	1.5 ± 0.002	160–300		290 ± 20
Bi2Sr2CaCu2O8+x (x = 0.245)		89	1.28 ± 0.002	155–300		271 ± 20
Bi2Sr2CaCu2O8+x (x = 0.255)		87.8	1.14 ± 0.001	150–300		243 ± 20
Bi2Sr2CaCu2O8+x (x = 0.26)		86	1.03 ± 0.001	125–300		227 ± 20
Bi2Sr2CaCu2O8+x (x = 0.27)		84	0.78 ± 0.01	105–300		216 ± 20
SrFe2(As0.7P0.3)2	SrFe2(As0.7P0.3)2	25	0.7 ± 0.01	100–300	59	270 ± 10	59
BaFe2(As0.67P0.33)2	BaFe2(As0.67P0.33)2	29.5	1.16 ± 0.001	31–150	60	315 ± 15	61
NaFe0.97Co0.03As	NaFe0.97Co0.03As	21.8	1.46 ± 0.004	50–250	62	375 ± 15	62
FeSe	FeSe	8	5.84 ± 0.01	20–80	63	425 ± 20	64
c-axis La1.9Sr0.1CuO4	4	27	160 ± 0.5	500–800	65	5908 ± 400	66
c-axis La1.9Sr0.12CuO4	3	30	195 ± 0.2	300–800		5345 ± 400
c-axis La1.88Sr0.15CuO4	2	35.8	154.8 ± 0.3	300–800		3816 ± 280
c-axis La1.8Sr0.2CuO4	1	31.7	147.6 ± 0.2	320–800		2441 ± 200
La1.85Sr0.15CuO4	La2−xSrxCuO4 - Hussey		1.7 ± 0.4	The slope are directly taken from Hussey et al.67,		249 ± 20
La1.84Sr0.16CuO4			1.5 ± 0.4			229 ± 20
La1.83Sr0.17CuO4			1.4 ± 0.4			213 ± 20
La1.82Sr0.18CuO4			1.2 ± 0.4			203 ± 15
La1.81Sr0.19CuO4			1.1 ± 0.4			198 ± 15
La1.8Sr0.2CuO4			1.1 ± 0.4			197 ± 15
La1.79Sr0.21CuO4			1.05 ± 0.4			198 ± 15
La1.78Sr0.22CuO4			0.9 ± 0.4			199 ± 15
La1.77Sr0.23CuO4			1.08 ± 0.4			199 ± 15
La1.76Sr0.24CuO4			1.0 ± 0.4			199 ± 15
La1.93Sr0.07CuO4	La2−xSrxCuO4	12.3	4.79 ± 0.018	200–400	68	497 ± 37
La1.92Sr0.08CuO4		22	3.96 ± 0.016	200–400		377 ± 30
La1.91Sr0.09CuO4		24.5	3.21 ± 0.02	200–400		314 ± 30
La1.9Sr0.1CuO4		27.5	2.75 ± 0.003	250–400		286 ± 30
La1.89Sr0.11CuO4		29.6	2.34 ± 0.002	250–400		282 ± 30
La1.88Sr0.12CuO4		30.4	2.05 ± 0.002	250–400		280 ± 20
La1.87Sr0.13CuO4		34.6	1.72 ± 0.001	200–400		277 ± 20
La1.86Sr0.14CuO4		36.3	1.54 ± 0.006	200–400		268 ± 20
La1.85Sr0.15CuO4		39.3	1.44 ± 0.001	180–400		249 ± 20
La1.84Sr0.16CuO4		36.6	1.26 ± 0.004	130–400		229 ± 20
La1.83Sr0.17CuO4		35.7	1.18 ± 0.004	200–400		213 ± 20
La1.82Sr0.18CuO4		36	1.05 ± 0.003	200–400		203 ± 15
La1.81Sr0.19CuO4		33	0.99 ± 0.006	50–400		198 ± 15
La1.8Sr0.2CuO4		30.3	0.96 ± 0.001	150–400		197 ± 15
La1.79Sr0.21CuO4		28.5	0.92 ± 0.001	220–400		198 ± 15
La1.78Sr0.22CuO4		25.5	0.88 ± 0.001	260–400		199 ± 15
Bi2Sr1.8La0.2CuO6+δ	Bi2Sr2−xLaxCuO6+δ	28.1	1.24 ± 0.004	250–300		319 ± 25	69
Bi2Sr1.6La0.4CuO6+δ		29	1.62 ± 0.004	250–300		297 ± 20
Bi2Sr1.4La0.6CuO6+δ		12	3.25 ± 0.01	250–300		553 ± 40
UPt3	UPt3	0.5	9.2 ± 0.2	5–10	70	715	70
c-axis Upt3		0.5	3.3 ± 0.12	5–10		422
CeCoIn5 (0 Gpa)	CeCoIn5	2.3	1.61 ± 0.008	3–20	Our data	350 ± 12	23
CeCoIn5 (0.3 Gpa)		2.51	1.36 ± 0.007	3–20		300 ± 12
CeCoIn5 (0.55 Gpa)		2.58	1.20 ± 0.002	3–20		280 ± 12
CeCoIn5 (1 Gpa)		2.63	0.97 ± 0.004	3–20		262 ± 12

The above relation between dρ/dT and ${\lambda }_L^2$ can be extended to various other strongly correlated superconductors with linear-T resistivity. The data are summarized in Fig. 2 on a log-log scale. Most resistivity data were taken from experimental results on high-quality single crystal samples in order to obtain the intrinsic linear-in-T coefficient. The values of the penetration depth were obtained by muon spin spectroscopy²⁵, optical conductivity measurement^{26, 27} and some other techniques. Note that for superconducting thin films, the experimental magnetic penetration depth generally deviates from the London penetration depth λ _L due to structural disorders in the films^{28, 29}. Even in high quality ultrathin films, there is a large difference in superfluid density between the film and the bulk materials with same T _c ^{30, 31}. Consequently all the data of the London penetration depth shown in Fig. 2 were taken only from bulk materials. It is worth noting that Fig. 2 also includes the transport data for cuprate superconductors along different transport directions, e.g., YBa₂Cu₃O_6.93 along the a, b and c-axis. Cuprates generally exhibit a metallic in-plane resistivity but an insulating-like resistivity along the c-axis below certain temperature, which reflects the two-dimensional nature of the system. Correspondingly, the penetration depth along the c-axis is determined by a Josephson-coupling between superconducting layers^32–34, which is different from the in-plane one²⁶. Thus it is amazing to observe that the same scaling relation holds true for both directions. Combining the data for all the strongly correlated superconductors summarized here, we see that the scaling, $d\rho /dT=({\mu }_0{k}_B/\hslash ){\lambda }_L^2$, spans over several orders of magnitude. Note that the in-plane LSCO data in the extremely underdoped regime 0.07 ≤ p ≤ 0.12 demonstrates a systematic deviation from the scaling relationship as shown in Fig. 2. The deviation could be understood in terms of the complex competing phase, like charge density wave and pseudogap, which become significant in the underdoped regime.

Figure 2.

Log-log plot of dρ/dT vs. ${\lambda }_L^2$ for various strongly correlated superconductors. The orange line is the scaling relation $d\rho /dT=({\mu }_0{k}_B/\hslash ){\lambda }_L^2$. See Table 1 for details, including errors.

The above scaling relation is consistent with several well-known experimental facts. First, considering the special case at T = T _c and neglecting the residual resistivity, the scaling relation $d\rho /dT\propto {\lambda }_L^2$ gives the well-known Homes’s law, ${\sigma }_c{T}_c\propto {\lambda }_L^{-2}$, where σ _c is the dc conductivity at T _c ²⁶. Second, the Drude formula³⁵ is often used to describe the resistivity of conventional metals, ρ = m*/n _n e ² τ, where m* is the effective mass of the quasi-particles, n _n is the carrier density of quasi-particles, e is the charge of electrons, and τ is the relaxation time. If we naively match the Drude formula with the above scaling relation for a non-quasiparticle system and assume that the normal fluid and the superfluid are composed of the same charge carriers, λ _L = (m*/μ ₀ n _n e ²)^1/2, we obtain immediately a material-independent scattering rate τ ⁻¹ = k _B T/ħ for all these strongly correlated superconductors. This is consistent with the universal scattering rate recently observed in the linear-in-T resistivity region among good and “bad” metals⁸. However, one can not take it for granted that the normal fluid in Drude model and superfluid in London equation are always the same. Actually, experiments showed that only part of normal carriers condensate into superfluid³⁶. In addition, the measurements of the London moment already revealed the mass of Cooper pairs are undressed and have twice of the electron’s bare mass, regardless the conventional metal superconductors³⁷, heavy fermion superconductors³⁸ or cuprates³⁹, which is different from the effective mass in the Drude formula. These results suggest that the mass and carrier density of the superfluid (n _s) and the normal fluid (n _n) are different in strongly correlated superconductors. So one can not directly obtain the universal scaling relation simply by replacing ρ with Drude model and λ _L with London equation. The universal scaling relation $d\rho /dT=({\mu }_0{k}_B/\hslash ){\lambda }_L^2$ has much deeper physics, which directly links the superfluid at zero temperature to the normal fluid responsible for the linear-in-T resistivity in strongly correlated superconductors. It reveals an underlying relation between the superfluid and normal carriers: n _s/m _e = n _n/m*. And indeed experimental evidence shows that about one of fourth normal carriers³⁶ condensates into superfluid in optimal doped cuprates while the effective mass of optimum cuprates is about 3–4 times of the electron free mass^{36, 40}, which validate n _s/m _e = n _n/m*.

The above result provides important information on the nature of the electron transport in the quantum critical regime. Recently, several experiments have shown that electrons in solid can exhibit hydrodynamic flows similar to a classical viscous liquid, if the electron fluid equilibrates by the electron-electron collisions^41–43. Thus the electron transport in strongly correlated superconductors, where electron-electron interactions play a major role in the scattering processes, might in principle have a hydrodynamic description. Consequently, its linear-in-T resistivity could be described by the well-known Einstein’s relation⁴⁴, an important law for the hydrodynamic transport, which states that the mobility (μ) of a particle in a fluid is related to its diffusion coefficient (D), namely, D = μk _B T. Hence we have ρ = k _B T/n _n e ² D and in the linear-in-T regime, the diffusion coefficient D must be a temperature-independent constant. Combining this and the scaling relation immediately yields D = ħ/m*, which is the quantum limit of the charge diffusion coefficient for the quasi-particles with an effective mass, m*. This is one of the most important consequence of our observations. Actually, the quantum limit of the diffusion coefficient was recently observed in cold fermionic atomic gases in the unitary limit of scattering^{2, 3}. It implies that quantum diffusion transport might be a universal property of strongly correlated fermionic systems where the electron scatterings are so strong that the transport becomes highly incoherent. In fact, it was proposed recently that the transport in an incoherent metal is controlled by the collective diffusion of energy and charge⁴⁵, supporting the proposed scenario of quantum diffusion transport in the present work. Thus, the obtained scaling relation suggests the superfluid could also be governed by the quantum diffusion, since it connects the ground state with the normal state in the strongly correlated superconductors.

Our results also provide some insights on the nature of strongly correlated superconductivity, which is often born out of strongly correlated normal fluid in the quantum critical regime. Since the latter already approaches the quantum diffusion limit before it transits into the superfluid state, it implies a zero-point motion of the superfluid. Some people considered the quantum diffusion as a necessary condition for the presence of superfluid^{46, 47}. In fact, the quantum diffusion might explain the Uemura results for superconducting transition temperatures. Y. Uemura et al. observed that the underdoped cuprate superconductors exhibit a Bose-Einstein-condensation (BEC)-like superconducting transition but with a reduced transition temperature^{25, 48}. Actually, the BEC generally occurs when the thermal de Broglie wavelength λ _dB is comparable to the distance between bosons, where λ _dB characterizes a length scale within which the bosons can be regarded as quantum mechanical wave-packets. However, the quantum diffusion gives a new length scale ${\xi }_{Th}=\sqrt{D\tau }$ with τ = ħ/k _B T, which characterizes the length scale that carriers can travel before losing their quantum coherence. Since the diffusion length ${\xi }_{Th}=\sqrt{{\hslash }^2/m^{⁎}{k}_{B}T}$ is less than ${\lambda }_{dB}=\sqrt{2\pi {\hslash }^2/2m^{⁎}{k}_{B}T}$ of electron pairs under certain temperature, it makes ξ _Th a new dephasing length to determine the BEC temperature. Thus the BEC temperature (T _B) is reduced to T _c/T _B = (ξ _Th/λ _dB)² = 1/π as observed in the Uemura plot^{25, 48}.

Conclusion

In summary, we observed a universal scaling relation $d\rho /dT=({\mu }_0{k}_B/\hslash ){\lambda }_L^2$, which connects linear-T-dependent resistivity to superconducting superfluid density at zero temperature in strongly correlated superconductors. Our analysis suggests that the quantum diffusion might be the origin of this scaling relation. In this case, the charge transport is viewed as a diffusion process of quasi-particles with a diffusion coefficient that approaches the quantum limit, D ~ ħ/m*.

Method

The high quality CeCoIn₅ single crystal samples are grown by an indium self-flux method⁶. High quality crystals were chosen to perform the transport measurements. Four leads were attached to the single crystal, with the current applied parallel to the crystallographic a axis. The resistivity was measured both in ambient pressure as well as under hydrostatic pressure P.

Author Contributions

T.H. Planned the research. Y.L. carried out the experiment. T.H. and H.X. wrote the manuscript. G.M. and Y.Y. give fruitful discussions. All authors were intensively involved in the research.

Competing Interests

The authors declare that they have no competing interests.