Anisotropic zero-resistance onset in organic superconductors
Vladislav D. Kochev, Kaushal K. Kesharpu, Pavel D. Grigoriev
AAnisotropic zero-resistance onset in organic superconductors
Vladislav D. Kochev, Kaushal K. Kesharpu, and Pavel D. Grigoriev
2, 1, 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 (Dated: July 29, 2020)We study the coexistence of superconductivity (SC) and density-wave state and reconcile variouspuzzling experimental data in organic superconductors (TMTSF) PF and (TMTSF) ClO . Theanisotropic resistance drop above T c is qualitatively described by nascent isolated SC islands withina bulk analytical model. However, the observed anisotropic SC onset is explained only when thefinite size and flat needle shape of samples is considered. Our results pave a way to estimate thevolume fraction and the typical size of SC islands in far from the sample surface, and apply to manyinhomogeneous superconductors, including high- T c cuprate or Fe-based ones. The interplay between various types of electronic or-dering is a subject of extensive research in condensedmatter physics. It is crucial for understanding the elec-tronic properties of various strongly correlated electronsystems. The coexistence of charge- or spin-density wave(CDW/SDW) and superconductivity (SC) is very com-mon [1–3] and especially important for high- T c super-conductors, both cuprate [4–6] and iron-based [7, 8], fortransition metal dichalcogenides [9, 10] and tetrachalco-genides [3], for organic superconductors [11–22]. In thesematerials the density wave (DW) is suppressed by someexternal parameter, such as pressure or doping. The SCtransition temperature T c is, usually, the highest in thecoexistence region near the quantum critical point whereDW disappears. The upper critical field H c is oftenseveral times higher in the coexistence region than in apure SC phase [13, 22], suggesting possible applicationsof SC/DW coexistence.The microscopic structure of SC and DW coexistenceis important for understanding the DW influence on SCproperties and SC transition temperature T c . The DWand SC phase separation may happen in the momentumor coordinate space. The first scenario assumes a spa-tially uniform structure, when the Fermi surface (FS) ispartially gapped by DW and the ungapped parts giveSC [3, 23]. The second scenario means that SC and DWphases are spatially separated on a microscopic or macro-scopic scale, depending on the ratio of SC domain size d and the SC coherence length ξ SC . An example of micro-scopic SC domains with size d < ξ SC is the soliton DWstructure, where SC emerges in the soliton walls [24–28].The SC upper critical field H c may theoretically increaseseveral times in both coexistence scenarios [23, 26].It is yet unknown or debated how SC and DW coexisteven in the relatively weakly correlated organic supercon-ductors, such as (TMTSF) PF [14–17], (TMTSF) ClO [20, 21] or α -(BEDT-TTF) KHg(SCN) [22]. Amongthese materials the most extensive and detailed exper-imental data are available for (TMTSF) PF [11–19].This compound attracts special attention because super-conductivity there appears on a spin-density wave back- FIG. 1. (color online)
Pressure-temperature phase di-agram of (TMTSF) PF recreated from resistivity data inRef. [15]. Filled (blank) symbols show the transition towardsSC (SDW) phase. The intensity of green (orange) color showsthe SC (SDW) volume fraction coexisting with SDW (metal)phase. ground, which violates the conservation of electron spinand, in the case of a microscopic SDW/SC coexistence,favors [23, 28] the unconventional spin-triplet SC. Thelatter is supported by the observed high in-plane uppercritical field [18], exceeding several times the expectedparamagnetic limit, and by the NMR Knight shift mea-surements [19]. However, an indisputable experimentalconfirmation of a triplet SC in (TMTSF) PF is stillmissing.At ambient pressure (TMTSF) PF undergoes a tran-sition from metallic to SDW insulating state at temper-ature T cSDW ≈
10 K. The SDW transition temperaturedecreases with the raise of pressure [11–16], and SDWbecomes finally suppressed at pressure[29] P c ≈ . P c
µ m [16].The most puzzling feature of SDW/SC coexistencein (TMTSF) PF , unexplained in any scenario, is theanisotropic SC onset [15, 16]: with the increase in pres-sure at P = P c ≈ . z -direction, then at P = P c ≈ . z - and y -directions, and only at P = P c ≈ . x -direction. This is opposite to a weak intrinsic interlayerJosephson coupling, typical in high- T c superconductors[30]. Other organic metals manifest similar anisotropicSC onset [20]. Note that the observed [15, 16] anisotropiczero-resistance T c contradicts the general rule that thepercolation threshold in large heterogeneous media mustbe isotropic [31], provided the high-conducting inclusionsare not thin filaments [15] connecting opposite edges of asample. However, such a filament scenario cannot be sub-stantiated microscopically in (TMTSF) PF and seemsto be absent in the metal/SDW coexistence region at T > T c . Below we resolve this paradox and reconcile rel-evant experimental data on SC onset in (TMTSF) PF .The proposed model and the results obtained are appli-cable to many other superconductors and can be used toestimate the volume fraction and the size of SC domains.A possible clue to explain the observed SC anisotropywithout invoking SC filaments may come from a similareffects in iron selinide FeSe, where the resistivity drop∆ ρ above T c was also observed to be very anisotropic,being much greater along the least conducting interlayerdirection [32, 33]. Its superconducting origin was con-firmed by the simultaneous measurements of a diamag-netic response and of the critical current [32]. ThisSC anisotropy was explained within a model of a het-erogeneous SC onset in the form of isolated SC islands[32, 33]. This effect originates from a strong conduc-tivity anisotropy η z = σ zz /σ xx (cid:28) z -axis. Isolated spherical SC islands increase con-ductivity in all directions similarly, but their relative ef-fect ∆ σ i /σ ii for the interlayer current is ∼ /η z (cid:29) η y = σ yy /σ xx < a i , can be obtained us-ing the Maxwell-Garnett approximation (MGA), valid inthe limit of small volume fraction φ (cid:28) ρ i = 1 /σ ii alongthe axis i ∈ { x, y, z } is given by [36]: ρ MGAi = ρ i (cid:20) A ∗ i (1 − φ ) A ∗ i + (1 − A ∗ i ) φ (cid:21) , (1)while in SCA we obtain[37]: ρ SCAi = ρ i (1 − φ/A ∗ i ) , (2)where the diagonal components of depolarization tensorare given by Eq. (17.25) of Ref. [35]: A ∗ i = 12 Y n =1 a ∗ n ∞ Z d t ( t + a ∗ i ) vuut Y n =1 ( t + a ∗ n ) − , (3)where a ∗ i = a i / √ η i , η i = σ ii /σ xx .Unfortunately, the SCA gives qualitatively incorrectresult in the limit of strong anisotropy η i (cid:28) σ SC /σ = ∞ . This is illustrated by our numerical calcu-lations in 2D case shown in Fig. S1[37]. From Eq. (1) onecan also solve an inverse problem to express the volumefraction φ through the conductivity with and without SCinclusions: φ MGA = A y (cid:0) σ yy − σ yy (cid:1) σ yy + A y (cid:0) σ yy − σ yy (cid:1) . (4)We apply Eqs. (1)–(4) to fit the observed resistivityanisotropy ρ i ( T ) in (TMTSF) PF [15, 16] at T > T c (see Fig. 2). The required temperature dependence φ ( T ) is extracted using Eq. (4) from the resistivity data[15] without and with magnetic field destroying SC (seeFig. 2d). From Fig. 2 one sees that the observed veryanisotropic temperature dependence of resistivity ρ i ( T )is qualitatively described by isolated SC islands withinMGA. The effect of SC inclusions on resistivity in MGAis clearly seen from the difference between the solid blueand dotted green curves in Fig. 2, showing ρ i ( T ) withand without SC islands. However, the MGA cannot ex-plain the anisotropic zero-resistance onset observed in(TMTSF) PF [15, 16] and (TMTSF) ClO [20], i.e. theanisotropy of SC transition temperature T c where the ob-served resistivity drops by several orders of magnitude.Moreover, such a T c anisotropy seems to contradict thepercolation theory [31].To resolve this puzzle we note that the percolationthreshold is isotropic only in infinite heterogeneous me-dia [31], i.e. when the sample dimensions are much larger MGAnon - SC fitexperiment
FIG. 2. (color online)
Temperature dependence of resis-tivity ρ along (a) x , (b) y and (c) z axes. Used experimentaldata for (TMTSF) PF at P = 8 . ρ y in magnetic field (inset)at B = 0 .
22 T and B = 0 T. than the size d of SC islands. Usually, the single crys-tals of organic metals are flat whiskers elongated in themost conducting x -direction with a tiny thickness alongthe interlayer z -axis. The (TMTSF) PF samples in theexperiments of Refs. [14, 15] were 3 × . × . .The typical dimensions of (TMTSF) ClO single crys-tals are similar: 3 × . × .
03 mm in Ref. [20], or2 . × . × . in Ref. [21]. The observation ofAMRO and FISDW in (TMTSF) PF at field B ≈ d min of SC islands to d min > µ m [16]. On the other hand, the observed [13, 22] in-crease in H c restricts the maximal SC size to d max < λ ,where the penetration depth λ ( T = 0 .
19 K) ≈ µ m in (TMTSF) ClO [38], and a close λ is expectedin (TMTSF) PF [39]. Similar H c enhancement andAMRO were also observed in (TMTSF) ClO [20, 40].These experimental data suggest that the typical size d of SC islands in (TMTSF) PF and (TMTSF) ClO getsinto the interval 1 µ m < d . µ m, being comparableto the sample thickness L z ∼ µ m. Thus we need toanalyze the effect of finite sample size.For this end we calculated percolation thresholds φ c numerically for randomly distributed spherical SC inclu-sions of various diameter d in a sample of dimensions3 × . × . , as in the experiment [14, 15]. For d > µ m φ c strongly depends on the distribution pat-tern of SC islands, hence, the percolation probabilities p ( φ c ) in Fig. 3 obtained by averaging over the large num-ber of distribution patterns[37]. In Fig. 3 we see that p is the largest along the shortest sample dimension in allcases. With the increase in SC volume fraction φ the SCtransition, i.e. the supercurrent percolation, first appears FIG. 3. (color online)
Percolation probability p along x (solid blue), y (dotted green) and z (dashed red) axes as afunction of SC volume fraction φ . Spherical SC inclusionshave diameter (a) d = 40 µ m, (b) d = 15 µ m, and (c) d =40 ± µ m with standard deviation of 20 µ m. (d) Dependenceof p on pressure P along main axes for spherical SC inclusionsof d = 15 µ m, calculated from Fig. 3b using the experimentaldata φ ( P ) (inset) extracted from Tab. 1 of Ref. [14]. along z , then along y , and only at much larger φ alongthe most conducting x -axis. Since φ increases with pres-sure P (see Fig. 3d inset), it explains the anisotropic SCtransition observed in Refs. [15, 16, 20]. Notably, we donot need a questionable filamentary z -elongated shapeof SC islands to describe these experiments: the effectemerges even for their opposite flattened shape. Thus,our scenario reconciles the relevant experimental facts onSC onset in (TMTSF) PF and (TMTSF) ClO : (i) theanisotropy of SC onset [15, 16, 20, 40], (ii) the observationof AMRO [16, 40], and (iii) the strong H c enhancementin the DW/SC coexistence region [13, 20, 40].Our numerical result of anisotropic percolation thresh-old can be easily understood. In thin elongated sam-ples with L x (cid:29) L z the probability to find a chain of n ≈ L z /d ∼ N ≈ L x /d (cid:29) L x andthickness L y at other parameters d, L z , φ fixed, the per-colation probability p z along the sample thickness grows.At small p z (cid:28) p z ∝ L x × L y . The anisotropy of SCpercolation transition also depends on the ratio of SCgrain size d and of the sample thickness L z (see Figs.3). This dependence is important because it allows anexperimental study of the typical size d of SC islandsin various materials and far from the sample boundaryusing resistivity measurements.To investigate the main features of this dependence,we calculated percolation probabilities p x and p y as afunction of diameter d of SC islands in a 2D rectangular (a) (b) (c) fit0.5 5 50 5000.050.10.5 FIG. 4. (color online)
Dependence of percolation thresh-old φ c on sample shape and size in 2D. (a) Circular SC islands(blue) with diameter d = 0 . ×
1, forming SCchannels between contact electrodes (green). It shows thatthe percolation along sample thickness is much easier thanalong sample length. (b,c) φ cx , φ cy as a functions of samplelength to sample thickness ratio L x /L y and of sample thick-ness to SC island diameter ratio L y /d . (d) Dependence of φ cx and φ cy on L x /L y for d = L y /
5. Same plot in logarithmicscale (inset) shows that φ cy ∝ ( L x /L y ) − . . (e) Dependenceof φ cx and φ cy on L y /d for L x /L y = 20. sample of dimension L x × L y . The results are shown inFigs. 4b-e with φ ci plotted instead of p i , where i ∈ { x, y } . φ ci is found by solving the equation p ci ( φ ) = 1 /
2. Wefound that φ cx depends weakly on L x /L y (Figs. 4b,d),but strongly on L y /d (Figs. 4b,e). It means that thepercolation threshold along the sample length is moresensitive to the size of SC islands than to the samplelength. Comparison of Figs. 3a and 3b shows a similardependence of p x ( φ ) on d in 3D case. On the contrary, as φ cy depends strongly on L x /L y (Figs. 4c,d) and weaklyon L y /d (Figs. 4c,e), percolation threshold along samplethickness is more sensitive to the length of the samplethan to the size of SC islands. When d ≈ L y we seea cusp in φ cy , representing percolation due to a singleSC inclusion. Numerical fitting shows φ cy ∝ ( L x /L y ) − α ,where α increases with d : α ≈ .
34 at d = L y / α ≈ .
41 at d = L y /
3. Difference in φ cx − φ cy growswith the increase in L x /L y (Fig. 4c) and d (Fig. 4d).Similar effect is also observed in Figs. 3a,b for 3D case.Hence, the anisotropy of SC onset grows when the samplebecomes thinner and longer, and when the SC grain size d increases. This shows the importance of finite-size effectsfor the anisotropy of SC onset.Superconductivity onsets heterogeneously in all known high-temperature superconductors, as confirmed by nu-merous scanning tunneling microscopy and spectroscopymeasurements [41–48]. However, these and other elabo-rated experimental techniques provide detailed informa-tion about the electronic structure at the surface, whichmay differ from the structure deep in the bulk. The pro-posed effect allows one to estimate the typical size ofSC islands far from the surface by measuring the tem-perature dependence of resistivity along three main axesin the samples or artificial bridges of thickness compa-rable to or 1-2 orders less than the expected size of SCgrains. It is helpful for understanding the properties andthe electronic structure across the phase diagram of var-ious high- T c superconductors.This article is partly supported by the Ministry of Sci-ence and Higher Education of the Russian Federation inthe framework of Increase Competitiveness Program ofMISiS, and by the “Basis” Foundation for developmentof theoretical physics and mathematics. V. D. K. ac-knowledges the project No. K2-2020-001, and K. K. K.the MISiS support project for young research engineersand RFBR Nos. 19-32-90241 & 19-31-27001. P. D. G.acknowledges the State Assignment No. 0033-2019-0001and RFBR grants Nos. 19-02-01000 & 18-02-00280. ∗ [email protected][1] A. M. Gabovich, A. I. Voitenko, J. F. Annett, andM. Ausloos, Supercond. Sci. Technol. , R1 (2001).[2] A. M. Gabovich, A. I. Voitenko, and M. Ausloos, Phys.Rep. , 583 (2002).[3] P. Monceau, Adv. Phys. , 325 (2012).[4] J. Chang, E. Blackburn, A. Holmes, N. Christensen,J. Larsen, J. Mesot, R. Liang, D. Bonn, W. Hardy,A. Watenphul, M. Zimmermann, E. Forgan, and S. Hay-den, Nature Phys , 871 (2012).[5] S. Blanco-Canosa, A. Frano, T. Loew, Y. Lu, J. Porras,G. Ghiringhelli, M. Minola, C. Mazzoli, L. Braicovich,E. Schierle, E. Weschke, M. Le Tacon, and B. Keimer,Phys. Rev. Lett. , 187001 (2013).[6] W. Tabis, B. Yu, I. Bialo, M. Bluschke, T. Kolodziej,A. Kozlowski, E. Blackburn, K. Sen, E. Forgan, M. Zim-mermann, Y. Tang, E. Weschke, B. Vignolle, M. Hepting,H. Gretarsson, R. Sutarto, F. He, M. Le Tacon, N. Bar-išić, G. Yu, and M. Greven, Phys. Rev. B , 134510(2017).[7] Q. Si, R. Yu, and E. Abrahams, Nat Rev Mater , 16017(2016).[8] X. Liu, L. Zhao, S. He, J. He, D. Liu, D. Mou, B. Shen,Y. Hu, J. Huang, and X. J. Zhou, J. Phys.: Condens.Matter , 183201 (2015).[9] C.-S. Lian, C. Si, and W. Duan, Nano Lett. , 2924(2018).[10] K. Cho, M. Kończykowski, S. Teknowijoyo, M. Tanatar,J. Guss, P. Gartin, J. Wilde, A. Kreyssig, R. McQueeney,A. Goldman, V. Mishra, P. Hirschfeld, and R. Prozorov,Nat Commun , 2796 (2018).[11] T. Ishiguro, K. Yamaji, and G. Saito, Organic Supercon- ductors (Springer Berlin Heidelberg, 1998).[12] A. Lebed, ed.,
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2, 1, 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
DERIVATION OF RESISTIVITY OF HETEROGENEOUS ANISOTROPIC COMPOUND IN THESELF-CONSISTENT APPROXIMATIONIsotropic case with ellipsoidal inclusions
In the self-consistent approximation (SCA), the effect of all the material outside any inclusion is to produce ahomogeneous medium whose effective conductivity σ ∗ ii is the unknown to be calculated [S1]. The diagonal componentsof the effective conductivity tensor σ ∗ ii along the axis i ∈ { x, y, z } of a heterogeneous media with unidirectionally aligned isotropic ellipsoidal inclusions in SCA can be calculated from Eqs. (18.18) and (18.19) of Ref. [S1]: N X j φ j (cid:0) σ j − σ ∗ ii (cid:1) σ ∗ ii σ ∗ ii + A ∗ i ( σ j − σ ∗ ii ) = 0 , (S1)where j numerates the phase, φ j is its volume fraction, σ j is its conductivity, which is assumed to be isotropic , andthe diagonal components A ∗ i of depolarization tensor for ellipsoidal inclusions with semiaxes a ∗ i are given by Eq. (3)of the main text: A ∗ i = Q n =1 a ∗ n ∞ Z d t ( t + a ∗ i ) s Q n =1 ( t + a ∗ n ) . (S2)In the next subsection we generalize these results for the case of anisotropic conductivities σ jii of constituent phases.For only two different phases, m and s , with isotropic conductivities σ m and σ s , according to Eq. (S1), the effectiveconductivity σ ∗ ii along the axis i of such a heterogeneous media satisfies the equation:(1 − φ ) ( σ m − σ ∗ ii ) σ ∗ ii σ ∗ ii + A ∗ i ( σ m − σ ∗ ii ) + φ ( σ s − σ ∗ ii ) σ ∗ ii σ ∗ ii + A ∗ i ( σ s − σ ∗ ii ) = 0 , (S3)where φ is the volume fraction of phase s , which in our case is superconducting (SC). The conductivity of SC inclusions σ s → ∞ . Then from Eq. (S3) we obtain a simple formula for the effective conductivity: σ ∗ ii = σ m A ∗ i A ∗ i − φ . (S4) Anisotropic case with ellipsoidal inclusions
The generalization of Eq. (S4) to the case of anisotropic conductivity σ m of the parent media is performed by themapping of the initial anisotropic problem to an isotropic one in a similar way as used in Ref. [S2] for the derivation ofeffective conductivity in the Maxwell-Garnett approximation (MGA), given by Eqs. (1) and (3) of the main text. Let J and V be the current density and the electric potential respectively in the real space, and σ mii be the conductivitycomponents of the parent phase. The electrostatic continuity equation in real space is written as: −∇ · J = X i ∂∂r i (cid:18) σ mii ∂V∂r i (cid:19) = 0 , (S5) a r X i v : . [ c ond - m a t . s up r- c on ] J u l where i ∈ { x, y, z } . After the mapping, i.e. the change of coordinates r i as: r i = r ∗ i √ η i , η i = σ mii /σ mxx , (S6)with the simultaneous change of conductivity to σ m = σ mxx , Eq. (S5) transforms to the electrostatic continuityequation for an isotropic media: −∇ · J = X i ∂∂r ∗ i (cid:18) σ m ∂V∂r ∗ i (cid:19) = 0 . (S7)Coordinate dependence of the electrostatic potential V ( x, y, z ) in an inhomogeneous medium, given by solutions of theequations (S5) or (S7) with proper boundary conditions, determines the effective conductivity of this inhomogeneousmedium. Consequently, the initial problem of conductivity in anisotropic media with some boundary conditions canbe mapped to the conductivity problem in isotropic media with new boundary conditions, obtained from the initialboundary conditions by anisotropic dilatation given in Eq. (S6). These boundary conditions are determined bothby the sample boundaries and by the inclusions of second phase. If these inclusions have ellipsoidal shape with theprincipal semiaxes a i , then after the mapping to the isotropic media these inclusions keep an ellipsoidal shape butchange the principal semiaxes to: a ∗ i = a i / √ η i . (S8)Eqs. (S4) and (S2) with semiaxes a ∗ i give the effective conductivity in the mapped space. Making the reverse mappingto the real space, we obtain the effective conductivity of initial heterogeneous media in real space in SCA: σ ii = σ mii A ∗ i A ∗ i − φ , (S9)which gives Eq. (2).Note that in the final formula (S9) the effective conductivity σ ii in the real space depends on the parameters A ∗ i and a ∗ i in the mapped space. This is because the coordinate dependence of electrostatic potential V ( r i ) in the real space isobtained from the electrostatic potential V ∗ ( r ∗ i ) in the mapped space (with semiaxes a ∗ i ) via the simple substitutionof Eq. (S6): V ( r i ) = V ∗ ( r ∗ i ). The dilatation r i → r i / √ η i changes √ η i times the electric field E i ( r i ) = −∇ i V ( r i ),while the electric current J i = σ ii E i changes 1 / √ η i times, because the local conductivity σ mii changes 1 /η i times. Theeffective conductivity σ ii also changes 1 /η i times: σ ii = J i ¯ E i , where the averaged (over sample size L i ) electric field¯ E i = L i / [ V ( r i = 0) − V ( r i = L i )] changes 1 / √ η i times due to the dilatation. COMPARISON OF THE RESULTS OF BULK ANALYTICAL MODELS AND NUMERICALCALCULATIONS
In this section we compare the results, given by analytical formulas (1)-(3) obtained in the Maxwell-Garnett(MGA) and self-consistent (SCA) approximations, with the numerical calculations in 2D case (see Fig. S1). Thisallows to estimate the applicability of these two bulk analytical models to describe real experiments on conductivityin heterogeneous superconductors. The calculated conductivity along two axes, x and y , for a square heterogeneousmedia of conductivity σ xx = 1 and σ yy = η = 1 /
400 with circular superconducting islands as a function of their volumefraction φ is shown in Fig. S1. For numerical calculations three different distributions of SC islands are considered:random, rectangular and chess order. For rectangle order our numerical calculations give the largest conductivity σ xx ( φ ) along the most conducting axis and the smallest conductivity σ yy ( φ ) along the most conducting direction. Forconductivity σ xx ( φ ) all approximations, both numerical and analytical, give similar results (see Fig. S1a). However, σ yy ( φ ) in various approximations differ much, as shown in Fig. S1b. The numerical calculations of σ yy ( φ ) for all threedistributions of SC islands give rather close results, but the analytical models MGA and SCA differ very strongly.The MGA approximation for σ yy ( φ ) is much closer to the numerical results than SCA: the conductivity σ yy in SCAdeviates crucially and diverges at φ ∼ η (cid:28)
1. This calculation illustrates the known fact [S1] that SCA, usually, givesqualitatively incorrect results in the limit of strong contrast between the conductivities of two phases in heterogeneousmedia, especially in the limit of strong anisotropy. randomchessrectangleMGASCA (a) (b)FIG. S1. The conductivity of an anisotropic (square 1 ×
1) heterogeneous media with superconducting inclusions calculatedusing analytical models, Eq. (1) for MGA and Eq. (2) or (S9) for SCA, and numerically for three different distributions ofSC islands: random, rectangular and chess order. In SCA σ yy ( φ ) goes up sharply and even diverges at φ ∼ η (cid:28)
1, whichdrastically contradicts the numerical results.
DETAILS OF FITS AND CALCULATIONS
In plotting Fig. 2d we assume that the magnetic field B z = 0 .
22 T is strong enough to suppress superconductivity.In fact, such a field at P = 8 . T c ( B z = 0) ≈ . T c ( B z = 0 . ≈ . φ ( T > . B z = 0 .
22 T also leadsto small metallic magnetoresistance ρ b ( B ), which is almost temperature independent at T c < T < . φ ( T ) by the offset ρ y ( T ) = ρ y ( T, B z = 0 . − [ ρ y ( T = 1 .
15 K , B z = 0 .
22 T) − ρ y ( T = 1 .
15 K , B z = 0)].The percolation probability in Figs. 3, 4 was calculated numerically using Monte-Carlo simulation. For eachdistribution of diameters d = µ ± σ , which is taken Gaussian with a half-width σ , a random state with proper numberof spherical inclusions in a box with given dimensions ( L x × L y × L z = 3 × . × . in Fig. 3 and various L x × L y in Fig. 4) was generated. The number of SC inclusions is determined by the fixed volume fraction φ of SC phase. Eachstate is associated with a graph whose vertices are SC islands. The vertices of the graph are connected by edges ifthe corresponding inclusions overlap. Thus, the problem of detecting the presence of percolation is reduced to findingthe connected components of the graph, which contain the vertices corresponding to SC inclusions on the oppositesample edges. For each state along each axis the percolation, i.e. the existence of a continuous path via intersectinginclusions, was checked, and the averaging over random realizations was made. Depending on the parameters, from10 to 10 generated realizations were enough to estimate the average probability of percolation in our calculations.The conductivity of an anisotropic media (in Fig. S1) was calculated numerically by solving the electrostaticcontinuity equation (S5) for the heterogeneous medium using the finite element method.A quantitative comparison with experiment requires the exact functions φ ( P ) and φ ( T ), which are known onlyapproximately. Fig. 2d, based on resistivity in MGA, overestimates φ ( T ), because MGA gives a lower bound ofconductivity in heterogeneous media [S1]. On contrary, Fig. 3c inset, based on the resistivity fit above T c in themetal/SDW phase [S4], underestimates φ ( P ), because the volume fraction of SC phase at T < T c should be largerthan the volume fraction of metal phase at T c < T (cid:28) T cSDW for two reasons: (i) superconducting phase has lowerenergy than metallic phase, and (ii) the SC proximity effect increases the effective SC volume fraction. ∗ [email protected][S1] S. Torquato, Random Heterogeneous Materials (Springer New York, 2002).[S2] S. S. Seidov, K. K. Kesharpu, P. I. Karpov, and P. D. Grigoriev, Phys. Rev. B , 014515 (2018).[S3] N. Kang, B. Salameh, P. Auban-Senzier, D. Jerome, C. R. Pasquier, and S. Brazovskii, Phys. Rev. B , 100509(R)(2010).[S4] T. Vuletić, P. Auban-Senzier, C. Pasquier, S. Tomić, D. Jérome, M. Héritier, and K. Bechgaard, Eur. Phys. J. B25