A method to estimate the volume fraction and shape of superconducting domains in organic superconductors
Kaushal K. Kesharpu, Vladislav D. Kochev, Pavel D. Grigoriev
aa r X i v : . [ c ond - m a t . s up r- c on ] D ec A method to estimate the volume fraction and shape of superconducting domains inorganic superconductors
Kaushal K. Kesharpu, ∗ Vladislav D. Kochev, † and Pavel D. Grigoriev
1, 2, 3, ‡ National University of Science and Technology MISiS, 119049, Moscow, Russia L. D. Landau Institute for Theoretical Physics, 142432, Chernogolovka, Russia P. N. Lebedev Physical Institute, RAS, 119991, Moscow, Russia
In highly anisotropic organic superconductor (TMTSF) ClO , superconducting (SC) phase coex-ists with metallic and spin density wave phases in the form of domains. Using the Maxwell-Garnettapproximation (MGA), we provide a method to calculate the volume ratio and the shape of theseembedded SC domains from resistivity data. Due to percolation of SC domains, the zero resistancecan be achieved even when the SC volume ratio φ = φ c ≪ . This percolation threshold φ c dependson the shape and size of SC domains and of the sample, and may be anisotropic. Using our theorywe find φ for various cooling rates of (TMTSF) ClO samples. We also analyze the effect of disorderon the shape of SC domains. We found that the SC domains have oblate shape, being the short-est along the interlayer z-axis. This contradicts the widely assumed filamentary superconductivityalong z-axis, used to explain the anisotropic superconductivity onset. We show that this anisotropicresistivity drop at the SC transition can be described by the analytical MGA theory with anisotropicbackground resistance, while the anisotropic T c can be explained[1] by considering a finite size andflat shape of the samples. Due to a flat/needle sample shape, the probability of percolation viaSC domains is the highest along the shortest sample dimension (z-axis), and the lowest along thesample length (x-axis). Our theory can be applied to other heterogeneous superconductors, wherethe size d of SC domains is much larger than the SC coherence length ξ , e.g. cuprates, iron basedor organic superconductors. It is also applicable when the spin/charge-density wave domains areembedded inside a metallic background, or vice versa. I. INTRODUCTION
Raising the superconducting transition temperature (Tc) has been the goal of active research for a century. Com-pounds like cuprates [2, 3], iron based superconductors [4], organic superconductors (hereafter denoted as OrS) [5]are some major classes of high-Tc superconductors. These materials have several common properties: (i) layeredcrystal structure and, hence, high conductivity anisotropy; (ii) interplay between various types of electron ordering,i.e. between spin/charge-density wave and superconductivity; (iii) spatial inhomogeneity. Therefore, many effects andmethods, both experimental and theoretical, are common for these materials. A recent and good review on all thesesystems is given by Stewart [6]. Our paper concerns the OrS and is devoted to two problems: (i) providing a method ofestimating superconducting volume fraction in coexistence regime of superconducting, metallic and spin/charge den-sity wave phases; (ii) analysis of corresponding experimental data[7, 8] in the organic superconductor (TMTSF) ClO to study the effect of cooling rate and of disorder on the volume fraction, shape and size of SC domains.(TMTSF) X series belongs to quasi-1D OrS and has been widely studied for 40 years [5, 9–11]. Many important ef-fects have been discovered and investigated on these compounds, e.g. angular magnetoresistance oscillations (AMRO)in quasi-1D metals [10–12], field-induced spin-density waves (FISDW) [13, 14], etc [10, 11]. An interesting and puz-zling property of these materials related to our subject is that, with the increase of pressure for (TMTSF) PF or ofanion ordering for (TMTSF) ClO , the superconductivity first appears along the least conducting z -axis, while alongthe most conducting x -direction only in the last turn [7, 8, 15, 16]. Recent experimental study on (TMTSF) PF [15–18] and (TMTSF) ClO [7, 8] has shown that spin density wave (SDW), superconducting (SC) and metallic phasecoexist in form of segregated domains. With the increase in pressure (for (TMTSF) PF ) [15, 16, 19] or in anionordering (for (TMTSF) ClO ) [7, 8] the volume fraction of SC or metallic phase increases.Few theories has been suggested to describe the coexistence regime in these materials, especially in (TMTSF) PF .One of them is the application of SO(4) symmetry that explains SDW and SC coexistence [20] but does not accountfor the observed hysteresis [17], for the strong enhancement of the upper critical field H c [21] and for the anisotropicSC onset [15, 16, 19] in the coexistence phase. Less exotic theories suggest a separation of SDW and SC in a coordinate[15–17, 21–25] or momentum space [24]. The momentum SC-SDW separation assumes a semi-metallic state in a SDW ∗ https://orcid.org/0000-0003-4933-6819 † https://orcid.org/0000-0003-2334-1628 ‡ [email protected]; https://orcid.org/0000-0002-4125-1215 phase, where, due to the imperfect nesting, small ungapped Fermi-surface pockets appear and become superconducting[24]. The H c enhancement can be explained in both scenarios [24, 25], but the observed hysteresis suggests a spatialSDW/SC separation[17]. To explain the H c enhancement [21] the SC domain width must not exceed the in-planeSC penetration depth λ ab ≡ λ . In (TMTSF) ClO the in-plane penetration depth is [26–28] λ ab ( T = 0) ≈ µm , andthe out-of-plane penetration depth is[29] λ bc ( T = 0 . K ) ≈ µm . One possible mechanism of the formation of suchnarrow domains in the SDW state could be the soliton phase [15, 22, 23, 25, 30]. It suggests that SDW order parameterbecomes non-uniform with metallic domain appearing perpendicular to the highest conducting x -axis. However, thewidth of these soliton-wall domains, being of the order of SDW coherence length ξ SDW ∼ nm, is too small to beconsistent with the recent observation of AMRO and FISDW in (TMTSF) PF [16] and in (TMTSF) ClO [31],suggesting the domain size d > µm . Moreover, the soliton-phase scenario accounts for the SC suppression along themost conducting x -axis only, but it could not explain why SC first appears along the least conducting z -axis, becausein this scenario the soliton walls are extended along both y and z -axes, which should result to SC along both thesedirections.A probable explanation of this anisotropic SC onset was proposed recently [1]. It is based on two ideas. First , inanisotropic media the isolated SC islands increase conductivity much stronger along the least conducting directionthan along the others, as observed in FeSe[32, 33] and described[32–34] using the Maxwell-Garnett approximation(MGA) [35] for small volume fraction φ of SC phase. However, this MGA theory cannot explain the anisotropiczero-resistance onset. For this we need the second idea, which takes into account the finite sample size L as comparedto the size d of SC grains. If the sample shape is very anisotropic, e.g. a thin plate, then the current percolation viathe SC grains, responsible for the zero-resistance onset, is most probable along the shortest sample dimension,[1] i.e.along the sample thickness, and least probable along the sample length. These two ideas were applied to explain[1]the experimental data[15, 16, 19] in PF6, taken on thin elongated samples with dimensions[15, 19] × . × . mm .As (TMTSF) ClO samples are usually flat shaped either[7, 8], similar ideas can be applied to analyze the resistivityexperimental data there too. In this paper using the MGA we find superconducting volume ratio φ and shape ofinclusions from available experimental data. We also investigate how disorder and cooling rate affect the inclusions’shape and size. This knowledge may help to better understand the microscopic structure and electronic properties ofthe SDW/SC coexistence phase in organic superconductors.In Sec. II we briefly describe the important properties of (TMTSF) ClO and argue that MGA can be applied toestimate temperature dependence of SC volume ratio ( φ ) in the presence of all 3 phases, i.e. metallic, SDW and SC.In Sec. III we describe the theoretical model in MGA. Further, in Sec. IV we apply our theoretical model to analyzethe experimental data on (TMTSF) ClO . Using the experimental data from Ref. [8] we find out how cooling rateeffects φ . Similarly, using the experiments of Ref. [7] we analyze the evolution of aspect ratio of SC inclusions withsample disorder. Finally, in Sec. V we discuss the main results of our investigation and their consequences. II. MATERIAL AND METHODA. Material (TMTSF) ClO is the only member of Beechgard salts which becomes superconducting at ambient pressure[36, 37].It is a quasi-1D superconductor in which cooper-pair ordering (SC) coexists with insulating Pierels ordering (SDW) .This very competition between SC and SDW is the key to understand the unconventional superconductivity[38–41].Among quasi-1D superconductors (TMTSF) ClO is the only compound in which superconductivity can be controlledby the cooling rate of samples, which affects the disorderliness of ClO anions. Slowly cooled (TMTSF) ClO samples(relaxed state) undergo an SC transition at T c ≈ . K [42, 43]. However, when cooled very fast (quenched state)(TMTSF) ClO has an insulating SDW transition at T SDW ≈ − K [44–46]. This behavior can be ascribed tostructural change that occurs at the anion ordering temperature[47] T AO ≈ . K . For T > T AO the noncentrosym-metric tetrahedral ClO anions, which are located at the inversion centers [48, 49], preserve the inversion symmetrydue to thermal motion of ClO anions. ClO anions randomly occupies one or other orientations, hence, on averagethe inversion symmetry is preserved [48]. For T < T AO , if the sample is cooled fast enough then the randomness oforientation of ClO anions is preserved [45]. However, if the sample is cooled slowly through T AO , ClO anions alonga,c-axes are ordered uniformly; and along b-axis ordered alternatively [47, 50, 51]. The anion ordering introducesthe new wave vector and the Fermi-surface folding. It disturbs the Fermi-surface nesting, preventing the SDW andfavoring SC. The analysis below equally applies for a charge-density wave (CDW) or SDW Pierels ordering. Our current study is mainly devoted toSDW/SC coexistence in (TMTSF) ClO , therefore we keep SDW notation for the insulating phase. However, it may equally be appliedto other compounds with CDW/SC mixed phase. Recent experiments strongly support the presence of SC inclusions embedded in the background of metallic/SDWphase when (TMTSF) ClO is cooled at intermediate rate [7, 8]. This means granular superconductivity for partiallyordered samples. From the crystallographic point of view, in this coexistence phase the domains of ordered ClO anions are embedded inside disordered background. In these domains the anions have alternating ordering pattern,being disordered outside the domains [7, 8, 47]. The electronic state inside these domains is assumed to remainmetallic even at T < T
SDW , and at
T < T ∗ these ClO -ordered grains becomes superconducting [8]. The volumefraction φ of superconducting phase increases with the decrease of temperature. For slow or intermediate cooling rateat T = T c the phase coherence of these SC islands establishes in the entire sample, leading to its zero resistance.In the temperature interval T c < T < T ∗ the effective medium model for highly anisotropic heterogeneous layeredcompounds [35], developed by the authors in Refs. [32–34, 52], can be used to estimate the volume fraction and theshape of SC inclusions inside the samples. B. Method
To analyze the temperature dependence of resistivity at T c < T < T ∗ we use the Maxwell-Garnett approximation(MGA) [35], valid when the volume fraction of SC phase φ ≪ . One can show that in 3D anisotropic samples eventhe SC percolation threshold φ c is considerably smaller than unity[1], so that the MGA applicability region is ratherwide. In recent works [32–34, 52] using MGA the superconducting volume ratio φ was found when SC inclusionswere embedded inside metallic background. Similarly, here we also assume SC inclusions are embedded inside abackground phase. However, here the background phase consists of SDW and metallic phases. This is permissible inMGA approximation as long as we know the effective conductivity of this mixed background phase. We make twomore assumptions: (i) the SC inclusions are of ellipsoidal shape, which simplifies the calculations and allows derivinganalytical formulas for conductivity; (ii) the size d and distance between SC inclusions l are much greater than the SCcoherence length ξ , so that the SC proximity effects and the Josephson coupling between the SC grains do not changethe results considerably. Both in (TMTSF) PF and (TMTSF) ClO the size d & µm of metal/SC inclusions isindeed much larger than the SC coherence length [8, 53] ξ = 70 , , and nm along the a, b , and c axes, respectively.Such a large metal/SC grain size is evidenced by the observation [16, 31] of angular magnetoresistance oscillations(AMRO) in the mixed phase of these compounds. If φ ≪ , one can also take l ≫ ξ .Due to the proximity effect a shell of SC condensates around SC inclusions with thickness ∼ ξ gets created,which changes the effective size and shape of SC islands. Another quantum mechanical effect is the Josephsoncoupling between SC grains. The Josephson coupling energy E J depends directly on Josephson junction current I c , E J ≡ ~ I c / , which exponentially decreases with the increase of distance l between neighbouring SC grains: I c = I exp( − l/ξ ) . If E J ≪ T , the Josephson coupling can be disregarded [54]. As we assume l, d ≫ ξ , both theseeffects can be neglected.The conductance through N-S boundaries of a normal metal and SC may increase, maximum two times, due tothe Andreev reflection . However, this increase of the interface conductance should not considerably change theeffective conductivity of the whole sample because the main voltage drop (at a given current) comes not from the N-Sinterfaces, but from the large non-SC parts of the sample. Even if we take an infinite N-S interface conductance, thisdoes not much affect the total sample resistance.Considering the above facts we neglect these small quantum mechanical effects, and treat conductivity due toembedded SC inclusions using the classical MGA approximation and percolation. III. THEORY
We use the Maxwell-Garnett approximation (MGA) [55, 56] to find the SC volume ratio φ . We derive a generalformula for effective conductivity in anisotropic heterogeneous media, where unidirectional ellipsoidal SC (SDW) in-clusions are embedded inside the background SDW/metallic (metallic) phase with anisotropic resistivity. We denotethe diagonal effective conductivity tensor of this materials as diag ( σ xx , σ yy , σ zz ) . Similarly, the background conduc-tivity tensor of this material is denoted as diag ( σ bxx , σ byy , σ bzz ) . Background phase consists of both metallic and SDWphase, or only of the metallic phase. Since the MGA in its standard form[35] can be applied only to an isotropicmedium, we use the coordinate-space dilation to transform the initial problem with anisotropic conductivity intoisotropic one [32–34] (see appendix A for the details of this mapping and Eqs. (A4) and (A5) for the definition ofdilation coefficients µ and η ). See Sec. 11.5.1 of Ref. [54] for the basic description of Andreev reflection
Before explaining the idea of MGA [35, 55, 56] we note that the stationary-current equation for electrostatic potential V , coming from the continuity equation for the electric current J i in heterogeneous media with coordinate-dependentdiagonal conductivity σ ( r ) , − ∇ i J i = ∇ i [ σ ij ( r ) ∇ j V ( r )] = 0 , (1)is completely equivalent to the electrostatic equation in heterogeneous media with coordinate-dependent dielectricconstant ε ( r ) , as comes from the Maxwell’s equations: ∇ i D i ≡ ∇ i ( E j ε ij ) = −∇ i [ ε ij ( r ) ∇ j V ( r )] = 0 . (2)Hence, the problem of an effective dielectric constant of such a medium with heterogeneous dielectric function ε ( r ) isequivalent to the problem of effective conductivity of a heterogeneous medium with the same coordinate dependenceof σ ( r ) . P Rr E
11 111 122 1: Inclusion phase2: Background phase
FIG. 1. Schematic representation of Maxwell-Garnett approximation. The distance of point P from the center of sphere is verylarge compared to the sphere size, i.e. r ≪ R . E is the applied electric field. To explain the idea of MGA [35, 55, 56] of calculating the effective dielectric constant of heterogeneous isotropicmedia with a small volume fraction φ of inclusions of the second phase, we refer to Fig. 1. The background phase isrepresented as “2”, and the inclusion phase as “1” in the figure. We take a sphere of our heterogeneous media. Insidethis sphere the inclusions are embedded as shown in Fig. 1. Let this sphere be embedded in an infinite medium ofbackground phase. An electric field E is applied to the infinite medium. Let E ′ be the electric field at a distant pointP. E ′ includes polarization due to individual inclusions. We denote the inclusion volume ratio as φ and the backgroundphase volume ratio as ( − φ ). We assume that the heterogeneous sphere with both inclusion and background phasecan be substituted by a sphere of homogeneous phase with an effective dielectric constant ε eff . Then the electricfield at point P can be found in two ways: (i) By summing the polarization effect due to individual inclusions; thecorresponding electric field at point P we denote as E ′ . (ii) By taking the polarization effect at point P of homogeneoussphere with effective dielectric constant ε eff ; the corresponding electric field at point P we denote as E ′′ . The MGAassumes these fields to be the same, i.e. E ′ = E ′′ , which gives an equation on ε eff . Using the well-known formulafor the polarization of dielectric ellipsoid in isotropic medium, and replacing ε ii by σ ii , we find the following equation(see Eqs. 18.9 and 18.10 of Ref. [35]) for the effective conductivity σ ∗ i along main axis i in the mapped space: (1 − φ )( σ ∗ i − σ b ) + φ " σ b (cid:0) σ ∗ i − σ isl (cid:1) σ b + A i ( σ isl − σ b ) = 0 . (3) In Eqs. (1) and (2) the summation is assume over the repeated coordinate indices i and j . Originally, MGA was formulated in 1904 only for spherical inclusions in isotropic medium [57]. See sec-18.1.1 of Ref. [35] for complete discussion on MGA Eqs. (3),(5) and (8) assume that in the mapped space the conductivities of both phases, σ isl and σ b , are isotropic. For a generalizationto anisotropic σ isl or σ b in the mapped space see Refs. [58, 59]. In the case of superconducting inclusions, applied in Sec. IV to analyzethe experimental data, σ isl = ∞ is naturally isotropic. Here A i is the depolarization factor of a dielectric ellipsoid in the mapped space: A i = Y n =1 a n Z ∞ dt , t + a i ) vuut Y n =1 ( t + a n ) . (4)The analytical solution of this integral can be found in terms of incomplete elliptic integrals of first and second kind .If the inclusions have finite conductivity, then solving Eq. (3) for σ ∗ i we obtain σ ∗ i = σ b (cid:20) ( A i + (1 − A i ) φ )( σ isl − σ b ) + σ b A i (1 − φ )( σ isl − σ b ) + σ b (cid:21) . (5)If the inclusions are superconducting, we take σ isl ≈ ∞ . Eq. (3) in this case simplifies to (1 − φ )( σ ∗ i − σ bi ) − φ σ b A i = 0 (6)Solving Eq. (6) for σ ∗ i we obtain σ ∗ i = σ b (cid:20) A i + (1 − A i ) φA i (1 − φ ) (cid:21) (7)Both Eqs. (5) and (7) gives the effective conductivity σ ∗ in the mapped space. Although the background conductivityin the mapped space is isotropic, the effective conductivity σ ∗ is anisotropic because of the anisotropic ellipsoidalshape of inclusions. The effective conductivity of original anisotropic material in real space is found by the reversemapping. It is done via multiplying the effective conductivity matrix in the mapped space σ ∗ = diag ( σ ∗ xx , σ ∗ yy , σ ∗ zz ) bythe inverse mapping coefficient matrix diag (1 , µ, η ) . Hence, multiplying Eq. (5) by diag (1 , µ, η ) , we find the effectiveconductivity in real space: σ xx σ bxx ≡ σ xx σ b = ( A x + (1 − A x ) φ )( σ isl − σ b ) + σ b A x (1 − φ )( σ isl − σ b ) + σ b ,σ yy σ byy ≡ σ yy µσ b = ( A y + (1 − A y ) φ )( σ isl − σ b ) + σ b A y (1 − φ )( σ isl − σ b ) + σ b ,σ zz σ bzz ≡ σ zz ησ b = ( A z + (1 − A z ) φ )( σ isl − σ b ) + σ b A z (1 − φ )( σ isl − σ b ) + σ b . (8)Here σ bii is the background conductivity along the axis i in real space. Similarly, multiplying Eq. (7) by diag (1 , µ, η ) ,we find the effective conductivity of original inhomogeneous material with SC inclusions of volume fraction φ : σ ii σ bii = A i + (1 − A i ) φA i (1 − φ ) . (9)From Eq. (9) one can express the volume fraction φ via the effective σ ii and background σ bii conductivities anddepolarization factor A i along the same axis: φ = A i (1 − σ bii /σ ii ) A i + (1 − A i ) σ bii /σ ii . (10)Eq. (8) is helpful when the conductivities of background and inclusion phases are both finite. Hence, it can beused to find the effective conductivity of heterogeneous material, when, e.g., SDW domains are embedded inside ametallic background, or vice versa. Eq. (9) can be used when superconducting inclusions are embedded inside abackground phase of finite conductivity. Below we use Eqs. (9) and (10) to analyze experimental resistivity data in(TMTSF) ClO in the mixed SC/SDW state. Here the background phase is made up of metallic as well as SDWphases. Resistivity along the corresponding axis is found by taking the inverse of Eq. (8) and Eq. (9). See appendix B of Ref. [34]
IV. ANALYSIS OF EXPERIMENTAL DATA IN (TMTSF) ClO We consider partially ordered (TMTSF) ClO samples. We denote T ∗ as superconducting onset temperature. Inthese compounds for T > T ∗ there is no superconductivity. However, for T < T ∗ the domains containing orderedClO anions partially transform to superconducting inclusions [8]. These ordered domains are embedded inside thephase of unordered ClO anions, where SDW prevails but may coexist with metallic phase. Further cooling resultsin the increase of SC volume fraction φ and in the formation of coherent clusters of SC inclusions. At T = T c < T ∗ acomplete SC channel gets opened [7, 8], i.e. the SC phase coherence establishes in the whole sample. In Sec. IV A, weuse our theory to calculate the SC volume ratio φ . In Sec. IV B, we study the influence of cooling rate on SC volumeratio. In the end, in Sec. IV C, we find the approximate shape of SC inclusions in various disordered samples. A. Application of MGA theory to describe resistivity and to find superconducting volume ratio to describe resistivity To find a typical temperature dependence of SC volume ratio φ ( T ) in (TMTSF) ClO at f T, K r xx , m W . c m T, KBackgroundEffective
FIG. 2. Dependence of SC volume ratio φ on temperature. φ (red,circle) is calculated using Eq. (10) for i = x and theexperimental data on resistivity along the x-axis, taken from Fig. 2c of Ref. [7] and shown in the inset. (inset) The effectivemedium resistivity ρ xx containing SC, metallic and SDW phases (green squares), and the background resistivity ρ bxx in theabsence of SC islands (blue triangles), taken from the data in Fig. 2c of Ref. [7] in magnetic field. cooling rate − dT /dt ≤ K/min we choose the sample due to the availability of experimentalresistivity data on ρ xx ( T, H = 2 T ) in a magnetic field H = H z , shown in Fig. 2(c) of Ref. [7]. The magneticfield destroys superconductivity, and we can use these data to find the conductivity of background phase σ bxx ( T ) =1 / ( ρ xx ( T, H = 2 T ) − ∆ ρ xx ) (see the inset in Fig. 3), where the offset ∆ ρ xx = ρ xx ( T ∗ , H = 2 T ) − ρ xx ( T ∗ , H = 0) accounts for magnetoresistance of metallic phase at H = 2 T . A magnetic field H '
500 Oe is usually enough todestroy superconductivity in (TMTSF) ClO [60, 61], but we take the data at H = 2 T where the SC effects canbe safely ignored. Since the experimental data on ρ zz ( T ) under magnetic field for the same samples are absent, thebackground-phase conductivity σ bzz along the z-axis is found by extrapolating the metallic ρ zz ( T ) resistivity to lowtemperature by a second-order polynomial, similar to Ref. [61]. Here the second-order term comes from the electron-electron scattering at low temperature [61, 62]. We take the x-axis as the reference axis for mapping to isotropicmedium, i.e. σ bxx is taken as the background isotropic conductivity in the mapped space: σ b = σ bxx . According to Eq.(A4), the mapping coefficient along the z-axis is defined as η = σ bzz /σ bxx .The SC volume ratio φ is found from Eq. (10) for i = x . The calculated volume ratio as a function of temperature isplotted in Fig. 2. Substituting φ found from Eq. (10) for i = x to Eq. (9) for i = z , we predict the effective resistivityalong the z-axis, σ zz ( T ) . Its comparison with the experimental data from Fig. 2b of Ref. [7] is shown in Fig. 3. Wesee a rather good agreement. In this calculation we have one fitting parameter – the unknown ratio of the semiaxes a x and a z of ellipsoidal SC inclusions. We found that at high temperature T ≈ . K, typical inclusions have the aspectratio a z /a x ≈ . . At low temperature T ≈ . K, the inclusions have aspect ratio a z /a x ≈ . . It means thatwith a decrease in temperature the SC inclusion becomes more isotropic along x and z-axes, i.e. a z /a x → . It mayindicate the formation of large and almost isotropic clusters of small SC inclusions. Note that at any temperature This sample − dT/dt = 100 K/min and then annealed for some time at varying temperature between 15 and 23K.Therefore, its disorder, presumably, corresponds to a slower cooling rate. the found aspect ratio a z /a x is much larger than the ratio of coherence lengthes in (TMTSF) ClO , ξ z /ξ x ≈ . .This supports the fact that in (TMTSF) ClO the heterogeneity and SC islands originate from disorder and anionordering rather than from usual SC fluctuations, because for SC fluctuations a z /a x ∼ ξ z /ξ x .Fig. 3 and the inset in Fig. 2 show the temperature dependence of resistivity for the same sample along the z and x axes correspondingly. From their comparison one observes that the resistivity drop near T c for ρ zz is muchstronger than along for ρ xx . This feature originates from the strong anisotropy of background-phase resistivity ρ bii andis naturally described in generalized MGA theory [32–34]. The qualitative interpretation of this anisotropic resistivitydrop due to SC onset is illustrated in Fig. 1 of Refs. [32] or [33]. Because of high resistivity ρ bzz , the interlayer currentmainly flows via the SC islands serving as shortcuts for this current direction. The effective resistivity ρ zz is thendetermined by the much smaller intralayer resistivity and by the typical length of in-plane path between two close SCdomains, which is inversely proportional to the SC volume fraction φ . r zz , W . c m T, K
ExperimentBackgroudTheory
FIG. 3. Temperature dependence of resistivity along z-axis. Experimental values (blue circles) are taken for sample ρ bzz = 8 .
17 + 4 . T + 1 . T Ω .cm . Theoretical values (red triangles) are found from z-axis resistivity in Eq. (9). B. Effect of cooling rate on superconducting volume ratio
The cooling rate of (TMTSF) ClO samples controls the fraction of ClO -ordered domains. At slow cooling theClO anions have enough time to relax into ordered state. At fast cooling the thermal disorder remains in the samples,so that both ordered and disordered domains coexist. It was corroborated by resistivity [63], specific heat[64] andx-ray scattering [65] experiments.The volume fraction φ o of anion-ordered domains as a function of cooling rate has been studied using the x-rayscattering [65, 66] and, recently, by resistivity and magnetic susceptibility [8] measurements. The correspondingresults are compared in Fig. 4 of Ref. [8] in the range of cooling rate < − dT /dt < K/min. Several assumptionsare made in extracting φ o from these experiments[8, 65]. First, (i) all these results assume that at the slowest coolingrate − dT /dt | min ∼ K/min all ClO are ordered at low temperature. It is not evident, as some degree of aniondisorder may remain. Second, (ii) in the estimate of volume fraction φ o from the resistivity measurements in themixed SDW/metal phase in Ref. [8] the following equation (see Eq. 1 of Ref. [8]) for the effective conductivity σ zz along z -axis has been used: σ zz = h φσ / min + (1 − φ ) σ / max i , (11)where ρ min = 1 /σ min = 0 . cm is taken as a residual resistance of the sample with lowest cooling rate and ρ max = 1 /σ max = ρ min + ∆ ρ c ∗ = 0 . cm is determined assuming that the difference ∆ ρ c ∗ = ρ max − ρ min = 0 . cm is equal to the jump of resistivity at anion-ordering temperature T AO = 24 . K due to the scattering by anion disorder.In fact, at low temperature the anion disorder has much stronger effect on conductivity than just the electron scatteringby this disorder itself, because it also favors the formation of insulating SDW state. Even if the fraction of insulatingSDW domains is about one half, as in Fig. 4 of Ref. [8], it may considerably affect the electron conductivity. Inaddition, (iii) Eq. (11) does not take into account the conductivity anisotropy of (TMTSF) ClO , which stronglyenhances the effect of metal/SC domains on resistivity along the least-conducting axis [32–34], as given by Eq. (8).(iv) The extraction of φ o from the magnetic susceptibility χ ( T ) data, especially at rapid cooling rate when the SCvolume fraction φ ≪ , depends strongly on the size and shape of SC domains [34, 54]; therefore the assumption[8]that φ o = [ χ ( T → − χ ( T c )] / [ χ ( T → − χ ( T c )] dT/dt =0 . K/min is not valid when the size of SC domains is smallerthan the London penetration depth.In this subsection we estimate the volume fraction φ of SC phase in (TMTSF) ClO using the resistivity data fromRef. [8] and applying Eqs. (9) and (10) derived in MGA. Note that the volume fractions φ of SC phase and φ o ofanion-ordered phase may differ, e.g., because the former depends on temperature. Since the MGA approximation isvalid only at φ ≪ , it can only give φ ( T ) at T > T c . Thus, our method of estimating φ works better for highercooling rate when T c is lower, therefore our results are rather complimentary to those in Ref. [8]. However, thecooling-rate dependence of φ ( T > T c ) also gives the general tendency. Note that at cooling rate dT /dt = 100 K/minand some annealing, by extrapolating the φ ( T ) curve in Fig. 3 to T = 0 we obtain φ ( T → ≈ . , which is in agood agreement with other data in Fig. 4 of Ref. [8].To observe the influence of cooling rate on SC volume ratio φ ( T ) , we use resistivity data from the inset of Fig.1(d) of Ref. [8]. These experimental values are taken as the effective resistivity ρ zz along z-axis for different coolingrates. For background-phase resistivity ρ bzz along the z-axis we use the 2nd order polynomial fit of metallic resistivityat T > T ∗ . Substituting these ρ zz ( T ) and ρ bzz in Eq. (10) we obtain φ ( T ) for various cooling rates. Unfortunately,in Ref. [8] there is no resistivity data along other two axes which would allow us to find the ellipsoid aspect ratio.Therefore, in Fig. 4 we take the depolarization factor A i = 1 / , i.e. the spherical inclusions in the mapped spaceinstead of ellipsoidal. This choice corresponds to ellipsoid semiaxes a i ∝ p σ ii ( T ∗ ) ∝ ξ i in real space, as one expectsfor SC fluctuations. The obtained φ ( T ) at cooling rates 0.02K/min, 0.052K/min, 2.5K/min, 7.6K/min and 18K/minare shown in Fig. 4. The curves in Fig. 4 are similar to those in Fig. 4 of Ref. [8], but the values of SC volumefraction φ are expectedly smaller than φ o in Ref. [8], because at T > T c only a fraction of ClO -ordered domainsbecomes superconducting. But φ increases with decreasing temperature and, probably, reaches φ o at T → . f Colling Rate, K/Min.
FIG. 4. Dependence of SC volume ratio φ on cooling rate calculated using Eq. (10) for different temperatures. For thiscalculation the experimental data from the inset of Fig. 1(d) in Ref. [8] are used. C. Effect of disorder on the shape of superconducting inclusions
In Fig. 4 we investigated the effect of cooling rate on φ . However, along with φ the shape of SC domains alsoplays an important role. Recent work [1] has shown that the probability of percolation along the shortest sampledimension, i.e. along sample thickness, is higher than along other directions. It was corroborated by the experimenton FeSe [33, 52], where by reducing the z-axis thickness of sample from 300 nm to ∼ nm, one raised T c from K to K [52]. The (TMTSF) ClO samples are also usually flat. This effect of anisotropic superconductivity onsetalso depends on the shape of SC inclusions [1]. The knowledge of the shape of SC domains in (TMTSF) ClO is alsohelpful to better understand the mechanism of their formation. In Sec. IV A we found a z /a x for sample K/min. Below we find a z /a y and a y /a x for the samples cooled at rate K/min with varioustimes of subsequent annealing. This gives the effect of disorder on the shape of SC inclusions.To study the evolution of aspect ratios a z : a y : a x with disorder we use the experimental data from Figs. 3 and 4of Ref. [7]. Unfortunately, the curves with equal numbers in these two figures correspond to different samples. Thus,we do not have the data on resistivity along all three axes for the same sample and parameters, required to determine The metallic conductivity σ ii ∝ v i ∝ ξ i , where v i is the Fermi velocity along axis i , and the BCS coherence length ξ i = ~ v i /π ∆ . the full shape of ellipsoidal inclusions. However, we use the fact that the depolarization factors A i in Eq. (4) dependmost strongly on the semiaxis a i along the same direction, which allows us to vary only one parameter for each fit.First we find the evolution of a z /a y with disorder. From the resistivity data along the z-axis, given in Fig. 4a ofRef. [7], using Eq. (10) we find φ ( T ) for different degrees of disorder. Using these φ ( T ) in Eq. (9) we predictedresistivity along y-axis. From best fit values of predicted and experimental resistivity along y-axis, given in Fig. 4bof Ref. [7], we find the ratio a z /a y for various degrees of disorder. The results are shown in Fig. 5a and reveal thatat low disorder, up to sample a z /a y almost remains same. However, at higher disorder the ratio a z /a y increases.Similarly, to find the evolution of a y /a x with disorder we use the data from Fig. 3 of Ref. [7]. As before, from theresistivity data along y-axis using Eq. (10) we find φ ( T ) for different degrees of disorder. Using this φ ( T ) we predictresistivity along x-axis. We change the semiaxes of ellipsoids along y- and x-direction, so that the theoretical andexperimental values of resistivity agree. Thus obtained ratio of a y /a x is shown in Fig. 5b. a a z / a y DisorderT=0.8 KT=0.6 K b a y / a x DisorderT=0.8 KT=0.6 K
FIG. 5. The dependence of aspect ratios a z /a y (Fig. a) and a y /a x (Fig. b) of superconducting domains in (TMTSF) ClO ondisorder at two temperatures T = 0 . K and . K, calculated using Eqs. (4,9,10) and resistivity data from Figs. 4 and 3 of Ref.[7], taken at cooling rate 600K/min and various annealing times. At longer annealing time, i.e. at weaker disorder, the shapeof SC domains is more anisotropic.
V. DISCUSSION
In this paper we propose a method based on MGA to investigate the microscopic parameters of heterogeneoussuperconductors from resistivity data. We apply this method to the organic superconductor (TMTSF) ClO , whereSC coexists with SDW in the form of isolated domains. Using our method we study the SC volume fraction φ andthe shape of SC islands as a function of external parameters, such as temperature and ClO anion disorder, whichcan be experimentally controlled by the cooling rate through the anion-ordering transition at T AO ≈ . K [8, 65] orby the annealing of (TMTSF) ClO samples [7].For the best use of proposed method, one needs the following experimental data: (i) Temperature dependenceof resistivity ρ ii ( T ) along each of non-equivalent main crystal axes ; (ii) ρ ii ( T, H ) in a magnetic field H > H c destroying SC, to get the resistivity ρ bii ( T ) of the background homogeneous phase. In the absence of ρ ii ( T, H ) orin the case of non-SC inclusions, one needs to make an extrapolation of ρ ii ( T ) from T > T ∗ to lower T in orderto get ρ bii ( T ) , which is less accurate. In the case of SDW/CDW inclusions one can also apply an external pressuredestroying SDW/CDW to get ρ bii ( T ) . If also (iii) magnetic susceptibility data are present, especially for all non-equivalent magnetic-field orientations, they help to independently check the obtained microscopic parameters andallow the estimate of the average size of SC inclusions as compared to the SC penetration depth [33, 34]. Ideally, allthese data are available for several values of external parameters that one is interested in, for example, at each studiedcooling rate of (TMTSF) ClO . Unfortunately, in spite of an active experimental investigation of (TMTSF) ClO byresistivity measurements, e.g. performed recently in Refs. [7, 8], this full set of data is absent. Nevertheless, we haveanalyzed the available data from Refs. [7, 8] to make some physical predictions concerning the mixed SC/SDW phasein (TMTSF) ClO . If the crystal has orthorhombic or lower symmetry, one needs the data along all three axes. If two or three crystal main axes areequivalent by symmetry, one only needs the data along two or one axes correspondingly. φ for various cooling rates and temperatures T c < T < T ∗ .Fig. 3 illustrates how well the MGA model typically fits the experimental data. Figs. 5 and 6 show the evolution ofthe shape of SC grains with the change of disorder by the annealing of rapidly cooled samples.From Fig. 5a we observe that, for partially ordered samples a z /a y ≈ . of SCdomains at temperature T ≈ . − . K depends weakly on disorder. The corresponding SC volume ratio φ ≈ . also weakly depends on disorder. At shorter annealing time, i.e. at larger disorder, the SC volume ratio decreases to φ ≈ . , while the aspect ratio increases to a z /a y ≈ . . Another aspect ratio a y /a x ≈ . depends much weakeron disorder, as shown in Fig. 5b. Note that the obtained ratios a x : a y : a z at cooling rate 600K/min are close tothose of SC coherence lengthes. In Fig. 2 we found that for partially ordered sample a z /a x ≈ . at temperature T ≈ . K, which differs considerably from what weget at 600K/min even for long annealing time. This means that the decrease of cooling rate, as in Refs. [8, 65] isnot completely equivalent to the increase of annealing time at
T < T AO used in Ref. [7]: they have similar effecton SC volume fraction φ but different effect on the shape of SC domains. Therefore, it would be very interestingto study their effect on the SC domain size, which can be extracted from the simultaneous magnetic susceptibilitymeasurements as done for other compounds [33, 34].In organic superconductors (TMTSF) ClO [7] and (TMTSF) PF [15, 16] superconductivity onsets anisotropically,i.e. first along the highest conducting z-axis and only in the end along the lowest-conducting x-axis. This behaviorwas first explained by assuming filamentary SC inclusions elongated along z-axis [15], but this hypothesis receivedneither theoretical nor experimental proof till now. Our analysis also shows that the SC domains are not elongatedalong z-axis but, on contrary, are oblate. Nevertheless, we predict much stronger decrease of resistivity along the leastconducting direction (compare Fig. 3 and the inset in Fig. 2), similar to experimental observations in (TMTSF) ClO [7] and in many other heterogeneous anisotropic superconductors [32–34]. In our recent work [1] we proposed a simplemodel to explain the anisotropic zero-resistance onset also. We have shown [1] that the percolation probability alongSC islands in needle or flat shaped samples is the highest along the shortest direction. A schematic illustration ofthis idea is given in Fig. 6. The same idea can be applied to (TMTSF) ClO to explain the anisotropic onset ofsuperconductivity. Usually, the (TMTSF) ClO samples are much shorter along the interlayer z-axis than along othertwo, e.g., the dimensions of samples in Ref. [7] are × . × . mm . Hence, the probability of percolation for theellipsoidal inclusion, even with the obtained anisotropic aspect ratios a z /a x and a z /a y , will be the highest along thez-axis. Hence, without invoking a filamentary SC one can easily explain the anisotropic onset of superconductivity inorganic metals. xz FIG. 6. Schematic illustration of superconducting ellipsoid inclusions embedded inside a long thin conductor with dimensions L z ≪ L x . It shows that the percolation probability along z is higher than along x -axis if a z /a x > L z /L x . For all our calculation in Sec. 4 we have used Eqs. (9) and (10) because the inclusions are superconducting.However, the similar approach and Eq. (8) can be used when the conductivity of inclusions is finite, e.g., whenSDW inclusions are embedded inside a metallic background or vice versa. This is a usual occurrence in organicsuperconductors [5]. Thus, instead of using Eq. (11) or similar phenomenological formulas to analyze the resistivitydata in mixed metal/SDW or metal/CDW phases in organic metals, we recommend to use MGA formulas (8) and(4), which take into account the strong anisotropy of layered organic metals and the actual shape of domains. Apartfrom that, in several high-Tc superconductors as Bi Sr x Ca − x Cu O [67], La − x Sr x CuO [68], Ba(Fe − x Co x ) As [69–71], and in many other materials, e.g., HfTe [72], there is also a spatially-separated coexistence of metallic andSC or SDW/CDW phases. Hence in all these heterogeneous materials, provided the conductivity of both metallic andSDW/CDW phases is known and the domain size exceeds the coherence length, one can estimate the second-phasevolume ratio and the domain shape from resistivity data using the above method. VI. CONCLUSIONS
We develop a method to estimate the volume fraction and the shape of superconducting domains from resistivitymeasurements. We apply this method to investigate the heterogeneous electronic structure in organic superconductor1(TMTSF) ClO , where superconductivity coexists with the spin-density wave in the form of isolated domains. Thismaterial is especially important to study such coexistence because it appears even at ambient pressure and can beeasily controlled by changing the cooling rate or annealing time of the samples. From available resistivity data westudy the evolution of the volume fraction and of the shape of superconducting domains in (TMTSF) ClO withdisorder and temperature. Our method applies not only to superconductors, but also when the density-wave orother-type domains are embedded inside a metallic background, or vice versa. ACKNOWLEDGMENTS
P.G. conceptualized and developed the Methodology. Formal analysis, validation, manuscript writing and reviewwas done by K.K, P.G and V.K. All authors have read and agreed to the published version of the manuscript.This article is partly supported by the Ministry of Science and Higher Education of the Russian Federation inthe framework of Increase Competitiveness Program of MISiS, and by the “Basis” Foundation for development oftheoretical physics and mathematics. V. D. K. acknowledges the project No. K2-2020-001, and K. K. K. the MISiSsupport project for young research engineers and RFBR grants Nos. 19-32-90241 & 19-31-27001. P. D. G. acknowledgesthe State Assignment No. 0033-2019-0001 and RFBR grants Nos. 19-02-01000 & 18-02-00280.The authors declare no conflict of interest.
Appendix A: Anisotropic dilation of the problem of static current distribution
Anisotropic medium is converted to isotropic one by mapping the real space to a mapped space, where the solutionis simpler. The mapping should satisfy the following conditions: (i) The conductivity of background phase in themapped space should be isotropic, and (ii) the electrostatic continuity equation should be satisfied in the mapped spacewith the same solution.
In our notations, σ bxx , σ byy , σ bzz are the constant conductivity components of background phase in the original het-erogeneous medium. Let J and V be the current density and the applied potential respectively. The electrostaticcontinuity equation in the background phase is then written as − ∇ J = σ bxx ∂ V∂x + σ byy ∂ V∂y + σ bzz ∂ V∂z = 0 . (A1)Heterogeneity is hidden in the boundary conditions on the surface of each grain and of the sample.Let x ′ , y ′ and z ′ be the axes in mapped space, where conductivity should be isotropic: σ bx ′ x ′ = σ by ′ y ′ = σ bz ′ z ′ = σ b . If J ′ and V ′ are the current density and electrostatic potential respectively in the mapped space, the continuity equationin the mapped space is written as − ∇ ′ J ′ = σ b (cid:18) ∂ V ′ ∂x ′ + ∂ V ′ ∂y ′ + ∂ V ′ ∂z ′ (cid:19) = 0 . (A2)The condition (ii) for our mapping means that the solution V ( x, y, z ) = V ′ ( x ′ , y ′ , z ′ ) (A3)of Eqs. (A1),(A2) is the same. This solution completely determines the effective conductivity of heterogeneousmedium, as it also gives the current density via J i = σ bii ∇ i V = J ′ i = σ b ∇ ′ i V ′ . Eqs. (A1),(A2) and (A3) are consistentif the mapping is the anisotropic scaling, e.g., x = x ′ , y = √ µy ′ , z = √ ηz ′ , (A4)with constant mapping coefficients µ and η determined by conductivity anisotropy in real space: µ = σ byy σ bxx , η = σ bzz σ bxx . (A5)Instead of (A4) one could choose a product of the mapping (A4) and of any isotropic scaling by a factor α with thesimultaneous change of σ b → α σ b . We have chosen σ b = σ bxx , so that µ, ν < .The current components in the real space r = ( x, y, z ) and in the mapped space r ′ = ( x ′ , y ′ , z ′ ) are related as J x ( r ) = J ′ x ′ ( r ′ ) , J y ( r ) = √ µJ ′ y ′ ( r ′ ) , J z ( r ) = √ ηJ ′ z ′ ( r ′ ) . (A6)2The shapes of inclusions are not preserved during this mapping procedure. For example, a sphere with radius a x described by the equation x /a x + y /a x + z /a x = 1 in non-homogeneous medium will transform to an ellipsoiddescribed by the equation x ′ /a x ′ + y ′ /a y ′ + z ′ /a z ′ = 1 in mapped space, where the semiaxes are given by a x ′ = a x , a y ′ = a x / √ µ, a z ′ = a x / √ η. (A7)If z is the lowest conducting axis, and the highest conducting axis is x , then a sphere in real space transforms to anellipsoid elongated along z-axis. Due to the temperature dependence of conductivity anisotropy, the coefficients µ, η and the shape of inclusions change with temperature either. If in real space the inclusion is ellipsoid with semiaxes a = a x , b = βa x and c = γa x , then it transforms to ellipsoid in mapped space with semiaxes a x ′ = a x , a y ′ = a x β/ √ µ, a z ′ = a x γ/ √ η. (A8)In MGA we take ellipsoidal inclusions with fixed aspect ratios β and γ , but varing size. [1] V. D. Kochev, K. K. Kesharpu, and P. D. Grigoriev, (2020), arXiv:2007.14388.[2] K. M. Shen and J. S. Davis, Materials Today , 14 (2008).[3] M. R. Norman, Journal of Superconductivity and Novel Magnetism , 2131 (2012).[4] H. Hosono and K. Kuroki, Physica C: Superconductivity and its Applications , 399 (2015).[5] D. Jerome, Chemical Reviews , 5565 (2004).[6] G. R. Stewart, Advances in Physics , 75 (2017).[7] Y. A. Gerasimenko, S. V. Sanduleanu, V. A. Prudkoglyad, A. V. Kornilov, J. Yamada, J. S. Qualls, and V. M. Pudalov,Physical Review B , 054518 (2014).[8] S. Yonezawa, C. A. Marrache-Kikuchi, K. Bechgaard, and D. Jérome, Physical Review B , 014521 (2018).[9] D. Jerome, Molecular Crystals and Liquid Crystals , 511 (1982).[10] T. Ishiguro, K. Yamaji, and G. Saito, Organic Superconductors , edited by M. Cardona, P. Fulde, K. von Klitzing, H.-J. Queisser, and H. K. V. Lotsch, Springer Series in Solid-State Sciences, Vol. 88 (Springer Berlin Heidelberg, Berlin,Heidelberg, 1998).[11] A. Lebed, R. Hull, R. M. Osgood, J. Parisi, and H. Warlimon, eds.,
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