The contact conductance of a one-dimensional wire partly embedded in a superconductor Raphael Matthews and Oded Agam arXiv:0709.0371v1 [cond-mat.str-el] 4 Sep 2007 The Racah Institute of Physics, The Hebrew University, Jerusalem, Israel The conductance through a semi-infinite one-dimensional wire, partly embedded in a superconducting bulk electrode, is studied. When the electron-electron interactions within the wire are strongly repulsive, the wire effectively decouples from the superconductor. If they are moderately or weakly repulsive, the proximity of the superconductor induces superconducting order in the segment of the wire embedded in it. In this case it is shown that the conductance exhibits a crossover from conductive to insulating behavior as the temperature is lowered down. The characteristic crossover temperature of this transition has a stretched exponential dependence on the length of the part of the wire embedded in the superconductor. The amount of this stretch is determined by the nature of the electron interactions within the wire. I. INTRODUCTION In recent years one dimensional interacting electron systems have attracted a large amount of attention. Part of the interest in these systems lies in the fact that electron-electron interactions in 1-D wires, even when weak, cannot be considered perturbatively. A question of practical importance, dealt with by a number of authors1 -11 , is that of the contact conductance of a one dimensional system connected to an external electrode. Most works on the subject picture the junction as a one dimensional wire connected to the electrode at a point. Here it has been found that electron-electron interactions strongly influence the conductance through the junction. In particular, repulsive interactions in the wire drive the system to be insulating at low enough temperatures, unless the contact is perfectly clean. Attractive interactions, on the other hand, mask obstructions at the interface between the two systems. These results do not qualitatively change whether the electrode is superconducting or metallic. In this paper we explore the behavior of a junction between a 1-D wire and a superconducting electrode of different geometry, specifically, the situation where the wire is embedded some distance into the electrode, as illustrated in Fig. 1. In this case one expects that superconducting order is induced in the part of the wire which is embedded in the superconductor, enhancing the conductance of the junction. On the other hand, for point contacts with even a small amount of normal reflection (and repulsive interactions in the wire), it has been shown that superconducting order suppresses the conductivity of the junction8,9 . The main goal of this work is to clarify how these two competing effects determine the behavior of the junction as a function of the temperature (or the applied voltage), and length of the embedded wire. As expected, provided interactions within the wire are not too strong, superconducting order is induced in the part of the wire embedded in the bulk, and an effective gap is formed in the wire whose value is determined by the tunneling rate between the superconductor and the wire, as well as the nature of the electron interactions. Figure 1: A schematic representation of a 1-D wire in contact with a superconductor: (a) A point contact. (b) A wire embedded a length L into the bulk Yet, in spite of this proximity effect, the junction conductance exhibits a crossover from a conductive state, governed by Andreev scattering, to an insulating behavior at low enough temperatures, characterized by a power law dependence on the temperature. This behavior results from the finite amount of normal backscattering within the wire. We found that the characteristic crossover temperature between the two regimes has a stretched exponential dependence on the length of the junction, and the amount of stretch is determined by the strength of the electron interactions in the wire. The article is organized as follows: In the next section the model whereby a 1-D wire is embedded infinitely deep into a bulk BCS superconductor is introduced. By integrating out the superconductor degrees of freedom the effective action of the embedded wire is obtained. In Sec. II and the Appendix, renormalization group (RG) techniques are employed in order to characterize the behavior of a 1-D wire embedded in a BCS superconductor. This RG flow allows one to deduce the low energy properties of the finite part of the wire embedded in the superconductor. In Sec. III the conductance of the wiresuperconductor junction is evaluated. Finally, the results are summarized and discussed in Sec. IV. II. MODEL We consider a single 1-D wire with interacting electrons (including backscattering), embedded inside a standard BSC superconductor (SC). The action of the system is a 2 sum of three contributions: ¯ ¯ ¯ S = Ssc (ϕ, ϕ) + SW ψ, ψ + St ψ, ψ; ϕ, ϕ ¯ (1) where Ssc is the action of the superconductor, SW is the action of the wire, and St describes the tunneling between the two systems. ψσ is the electronic field operator in the wire, ϕσ is the field operator in the SC, and σ denotes the spin index. The action of the SC, modeled by the standard BCS hamiltonian with a constant pairing amplitude ∆sc , has the form Ssc = ¯ d4 ξ Φ(ξ) ∂τ + H0 ∆sc ∆∗ ∂τ − H0 sc Φ(ξ) (2) where the vector ξ = (x, y, z, τ ) contains three space coordinates and an imaginary time, ΦT = (ϕ↑ , ϕ↓ ) is the SC ¯ h2 ¯ electron field in Nambu notation, and H0 = − 2m ∇2 − µ is the free Fermi gas Hamiltonian, with m and µ as the electron mass and chemical potential, respectively. The 1-D wire is modeled as a Bosonized Luttinger liquid. Since the literature on this subject is quite extensive Sc (θc ) = 1 2 β dτ uc Kc dx 0 describes the charge sector (after the dependence on the φc field has been integrated out13 ). A similar expression describes the spin sector, Ss (φs ), with the subscript c re−1 placed by s and Kc replaced by Ks . Here Kc , Ks , uc , and us are model specific parameters describing the interaction strength (K) and the mode velocity (u) of the charge and the spin fields. For the noninteracting case Kc = Ks = 1 and uc = us = vf , where vf is the Fermi velocity of the wire. For an interacting system, values of Kc > 1 and Ks < 1 correspond to attractive interactions, while Kc < 1, Ks > 1 correspond to repulsive ones. Generally in an interacting system the velocity of the two modes differ, uc = us . The third term of the wire’s action, Sbs , describes backscattering of two electrons with opposite spins. Namely a collision which effectively results in a spin flip between the right and left moving parts of the electronic † † field (∼ ψL,σ ψR,σ ψR,−σ ψL,−σ ). This term has the form Sbs = 2g (2πα)2 √ 8πφs (x, τ ) , dxdτ cos (6) where g is the backscattering coupling constant. Finally, the tunneling between the superconductor and (ref. [13] and therein), we will only point out the more relevant details to the topic at hand. We represent the electronic fields12 , √π 1 eirkf x e−i 2 [rφc (x)−θc (x)+σ(rφs (x)−θs (x))] , ψr,σ (x) = √ 2πα (3) in terms of boson charge and spin density fields: ∂x φν (x), and their conjugates, θν (x), where ν = c/s denotes the charge and spin sectors, respectively. In the above formula, kf is the Fermi wave number, α is the short distance cutoff, r is the chiral index representing the right (r = 1) and left (r = −1) moving part of the electronic field, and σ = 1 for ↑ spin while σ = −1 for ↓ spin. In terms of the boson fields, the action of the wire becomes a sum of three contributions: SW = Sc (θc ) + Ss (φs ) + Sbs (φs ), (4) where 1 2 2 (∂τ θc (x, τ )) + (∂x θc (x, τ )) u2 c (5) the wire is described by St = d4 ξ ′ ϕσ (ξ ′ )t(ξ ′ ; x, τ )ψσ (x, τ ) ¯ dxdτ σ=↑,↓ + H.C., (7) where ψσ = r ψr,σ , and t(ξ ′ ; x, τ ) is the tunneling matrix element. In the simplest case this tunneling is instantaneous, homogeneous in space, short distant and SU(2) spin symmetric. Under these assumptions it takes the form ˜ t(ξ ′ ; x, τ ) = t(x)δ(x′ − x)δ(y ′ )δ(z ′ )δ(τ ′ − τ ) (8) ˜ where t(x) has the characteristic function form: ˜ t(x) = t0 for 0 ≤ x ≤ L 0 otherwise (9) where L is the length of the part of the wire embedded in the superconductor (see fig. 1 (b)). The effective action of a wire embedded in a superconductor is obtained by tracing out the superconductor degrees of freedom, ¯ e−Sef f (ψ,ψ) = ¯ ¯ D[ϕ, ϕ]e−S (ϕ,ϕ;ψ,ψ) ¯ (10) 3 where S is given by (1). Since the action is quadratic in the superconductor field, this integration is straightforward. The result may be written as a sum of three terms, Sef f = SW + Sph + Spp , where Sph , and Spp represent contributions associated with Giever (particle-hole) and Andreev (particle-particle) tunneling, respectively. At energies above the superconductor gap ∆sc the particlehole term is dominant and it’s contribution, on integrating it out, will be to renormalize the chemical potential. On the other hand, in the low temperature regime (well below the superconductor gap ∆sc ) the particle-hole term vanishes and the main contribution comes from the Andreev tunneling term, Spp . In this limit, after averaging over the rapid spatial oscillations, the tunneling becomes local in space and time. Expressing it in terms of the bosonic fields we thus have Sef f = SW + Spp (11) where SW is given by (4), and Spp = 2∆ πα dxdτ cos III. THE RENORMALIZATION GROUP FLOW EQUATIONS The effective action (11) describes the physics of the junction for a temperature up to the order of the superconductor gap, ∆sc (which will subsequently provide the high energy cut-off of our system). In order to describe the behavior of the system at much lower energy scales, and take into account the electronelectron repulsive interactions, we shall employ a real space RG approach, following Giamarchi & Schulz13,14 . As usual in these cases the RG procedure manifest itself in a flow of the coupling constants of the problem as the ultraviolet cut off is reduced from 1/α to 1/α′ . In our problem these coupling constants are: The interaction strengths, Kν ; The mode velocities, uν ; The g dimensionless backscattering constant, y = πus ; And the α dimensionless pair tunneling strength ∆ = ∆ us . The flow equations of these coupling constants (see Appendix for the details of the derivation) are: √ √ 2πφs (x, τ ) cos 2πθc (x, τ ) , 2 us dKc ∆ = Xc dl uc y2 dKs 2 = −Ks Xs ∆2 + dl 2 (12) 2 t0 N0 πf p 2 where ∆ ≃ . Here N0 is the normal density of states of the SC at the Fermi level, and pf is the Fermi momentum. In terms of the original fermionic fields, the tunneling term has the form of the regular pairing term in the standard BCS theory: d∆ dl dy dl duc dl dus dl ¯ ¯ Spp = ∆ dxdτ ψ↑ (x, τ )ψ↓ (x, τ )+ψ↓ (x, τ )ψ↑ (x, τ ) In the absence of electron-electron interactions, this term, along with the free quadratic kinetic term of the model can be diagonalized by the standard Bogolubov transformation. The excitation spectrum of this system will be gapped with an energy of ∆. Xc(s) = Wc(s) = 1 2π 1 2π 2π dϕ cos2 (ϕ) + 0 2π 0 dϕ cos(2ϕ) × (13) (14) 1 −1 = ∆(2 − (Ks + Kc + y)) 2 (15) = y(2 − 2Ks ) − 2Xs ∆2 (16) −1 = uc Kc Wc us ∆ uc = us Ks Ws ∆2 2 (17) (18) where dl = d log α is the dimensionless change in the ultraviolet cutoff, us(c) uc(s) −gs(c) 2 2 cos2 (ϕ) + sin2 (ϕ) us(c) uc(s) (19) −gs(c) 2 2 sin2 (ϕ) (20) −1 and gc = Kc while gs = Ks . We shall restrict our analysis to the spin symmetric case (i.e. the situation where the interactions between electrons with parallel and opposite spins are identical). In this case14 Ks ≃ 1+ y , and equations (14) and (16) for 2 the dimensionless backscattering y and spin interaction strength Ks reduce to: dy = −y 2 − 2Xs ∆2 , dl thus maintaining spin invariance. (21) 4 A. Analysis of the Flow Equations 0.33 (0) (0) 4 − (Ks + Kc −1 + y (0) ) > 0, (22) since the flow equations can only drive Kc/s to be more ”attractive”. In Fig. 2 we depict the RG flow in the ∆-Kc plane, in the spin symmetric case, for several initial val0 ues of Kc in the range between 0.32-0.325. Fig. 3 shows the separatix between relevant and irrelevant tunneling for initial bare (repulsive) values of Kc and y. One can observe that Kc must be smaller than 1 for the tunneling 3 to be irrelevant. relevant tunneling SC regime 0.325 Kc The above RG equations imply that the charge and spin velocities, uc and us , renormalize towards each other. This is a consequence of the correlations between spin and charge excitations generated by the proximity effect. From Eqs. (17), (18), and (20) one can observe that the signs of Wc and Ws are determined by the ratio uc /us in such a way that the velocities approach each other. Another consequence of the RG equations,(13) and (14), is that Kc can only grow, while Ks can only be reduced. This flow stem from the finite value of the tunneling parameter ∆ (and y). Thus the RG behavior of ∆ controls the behavior of the system. The separatix between the regions where ∆ is relevant or irrelevant can be obtained numerically for a given set of bare parameters. It is clear that for initially repulsive (0) (0) interactions (Ks > 1, y (0) > 0, Kc < 1), ∆ will be relevant if: irrlevant tunneling 0.32 0.315 −0.2 0 0.2 0.4 0.6 0.8 y Figure 3: The curve describes the separatix between relevant tunneling, where the system flows to a superconducting state, and irrelevant tunneling, where the system flows to a decoupled wire, as a function of the bare values of Kc and y In the situation where ∆ is relevant, the system flows to a singlet superconducting state. The interaction parameters become attractive and spin-charge separation is no longer valid. Yet, before the interaction parameters obtain their asymptotic values (i.e Kc → ∞, Ks → 0 and y → −∞ which correspond to ”infinitely” attractive interactions), the RG equations (13- 18) will cease to be valid since the small parameters of our perturbation theory, ∆ and y, will flow to the strong coupling regime. B. Length Dependance of the Effective Gap 0.05 0.04 0.03 0.02 0.01 0.325 0.33 0.335 0.34 Kc Figure 2: Plots of the flow of ∆ as a function of Kc for initial (0) values of Kc between 0.32-0.325. The transition between irrelevant (∆ → 0) and relevant (∆ → ∞) tunneling occurs (0) at Kc ∼ 0.3223. The initial values of ∆(0) and y (0) are 0.05 and 0.1 respectively In the case where ∆ is irrelevant, the wire effectively decouples from the superconductor. The pair tunneling between the two systems is suppressed and the superconductor-wire junction becomes insulating. Notice however that this behavior takes place at very strong re1 pulsive interactions, Kc < 3 , where the system tends to 13 Wigner crystallize . In what follows we consider a weakly interacting Luttinger liquid, where the bare interaction parameters are close to unity. Moreover, in order to ensure that the perturbative RG equations (13- 18) remain valid we shall assume that L, the embedding length, is of order or smaller then the bare superconducting correlation length, vf /∆. Since L serves as an infrared cutoff for the RG flow, this condition ensures that the this flow is confined to the perturbative regime. We shall also assume that the temperature is much smaller than ∆, and approximate the velocities of the spin and charge sectors by their asymptotic renormalized values: uc = us = vf . In the limit of weak interactions the RG equations for Ks/c are of second order in the perturbation parameters ˜ (∆ ≪ 1, y ≪ 1), while the equation for ∆ (Eq. 15) is of first order (neglecting the y dependance, which is of second order as well). The equation for the coupling y (Eq. 16) is also second order in small parameters, at least if we consider the spin symmetric case . Thus one can assume Kc , Ks , and y to be approximately constants, and consider the simplified equation for ∆: d∆ = γ ∆, dl (23) 5 where 1 −1 γ = 2 − (Ks + Kc ). 2 (24) Integrating the above equations from l = 0 to l = ˜ log(L/α), and using the relation ∆ = ∆α/vf , we obtain the renormalized value of the gap of the wire embedded in the superconductor: ∆ef f = ∆ L α vf =ξ ∆ef f L α e where G0 S = 2G0 N = 4 2π¯ represents the conductance N N h of an ideal junction where only perfect Andreev reflections take place, and (26) THE WIRE-SUPERCONDUCTOR JUNCTION At this stage of the RG procedure (pursued up to the scale L) the junction between the wire and the superconductor may be assumed to be point-like. Thus if the temperature (or the applied voltage) is smaller than ¯ vf /L h one may continue to integrate out the high energy degrees of freedom in the part of the wire which is not embedded into the superconductor down to the relevant energy scale. This may be achieved following the procedure described in the literature9 . The important ingredient, now, is the magnitude of normal back scattering from the junction. The latter is of order of rN ≃ r0 e−2L/ξef f where r0 is system specific reflection amplitude in the absence of superconductivity, while ξef f is the correlation length (26) within the part of the wire embedded in the superconductor. This behavior of rN results form the fact that only the charge that is not converted to the condensate backscatters from the edge of the wire embedded in the superconductor. According to5 the normal current reduces exponentially with the distance on a length scale of ξ , which gives the above estimate for rN . Now, since ξef f depends on L, the magnitude of the normal reflection has a stretched exponential dependence: rN ≃ r0 e−aL γ 2 δG ∝ r0 e−2aL T −2(1−Kc ) . (25) 1−γ (27) where a = 2α1−γ /ξ, and the amount of stretch, γ, is dictated by the nature of the electron-electron interactions within the wire, as follows from (24). (28) 2 γ . is larger than the bare correlation length ξ = vf /∆. The above results are valid as long as the RG flow stays ˜ within the perturbative regime, namely ∆ < 1. This con1 dition implies that L should be shorter than ξ(ξ/α) γ −1 . IV. G = G0 S − δG, N γ−1 In particular, repulsive interactions (Kc < 1 and Ks > 1, and therefore γ − 1 < 0) reduce the effective gap in the wire. This renormalization of the gap implies that the effective correlation length, ξef f = The behavior of the backscattering, clearly, manifests itself in the conductance of the junction. In the limit of sufficiently high temperatures, one obtains8 : (29) The above formula holds for the range of temperature where δG ≪ G0 S , since δG cannot be larger than G0 S . N N Nevertheless, it has been shown6 -10 , that any scatterer, at a point contact between a wire with repulsive interactions and a SC, will eventually drive the conductance to zero as the temperature is lowered. Since any finite length junction will have some backscattering, the conductance should drop to zero for low enough temperatures, as illustrated in Fig. 4. The crossover temperature, T ∗ from the conductive and the insulating behavior of the junction depends on the length of the wire embedded in the superconductor and its scaling behavior may be deduced from (28) and (29): a T ∗ ∝ e− 1−Kc L γ (30) This result implies that the temperature scale at which the effects of backscattering becomes substantial reduce as a stretched exponent with the length of the junction, and the stretch is determined by the interactions, through the parameter γ. . V. SUMMARY AND CONCLUSIONS In this work we studied a junction of a 1-D wire embedded a certain length, L, into a bulk superconductor. We first characterized the nature of the contact between the 1-D wire and the superconductor using a real space RG scheme. We found that repulsive interactions in the wire compete against the superconducting order being imposed by the bulk superconductor. The system can flow to either of two phases, depending on the nature of the interactions. When the interactions are strongly repulsive the tunneling between the two systems becomes irrelevant and the wire essentially decouples from the bulk superconductor. For moderate repulsive interactions, and for attractive interactions, tunneling is relevant, and the bulk superconductor induces superconducting order in the wire. The gap opened in the wire depend on the tunneling strength, and electron-electron interactions modify its nominal value. The finite length of the part of the wire embedded in the superconductor, L, implies that the RG flow, in general, does not reach its asymptotic (non-perturbative) limit. Thus L introduces itself in the behavior of the effective gap, and the effective correlation length, in the 6 Dror Orgad. This work has been supported in part by the Israel Science Foundation (ISF) funded by the Israeli Academy of Science and Humanities, and by the USAIsrael Binational Science Foundation (BSF). G 0 2GNN VI. T ∗ ~e − a Lγ 1− K c APPENDIX The mathematical formulation used in this work follows closely the real space RG procedure used by Giamarchi & Schulz13,14 : In this procedure one evaluates a correlation function in the wire, of the form (the time ordering symbol is suppressed): T Rϕ (xa , τa ; xb , τb ) = eiγ Figure 4: A schematic graph of the conductance as a function of temperature for repulsive interactions in the wire. The decay of the conductance depends on the length of the junction as a stretch exponent. The longer the junction, the lower the temperature at which the decay sets in. In the limit of weak backscattering, represented by the solid curve, we can estimate the temperature dependance of the conductance through Eq. (29). In the zero temperature limit, represented by the dashed curve, any initial finite backscattering will eventually drive the conductance to zero. The crossover temperature, Eq. (30) between the two behaviors is estimated to scale as a stretched exponent in the length of the junction. Here ∆ = ∆α us ∞ α dr α is the dimensionless tunneling parameter, , (31) where ϕ can symbolize any of the boson fields and γ is some constant. For the following discussion it will suffice to examine only one of the sectors, for instance the spin sector. The same considerations can be carried on straightforwardly for the charge sector. The fact that the relevant boson fields in the spin sector is φs leads naturally to the evaluation of the correlation functions: Rφs (xa , τa ; xb , τb ) = eiγ √ 2π(φs (xa ,τa )−φs (xb ,τb )) . (32) Unfortunately this correlation function cannot be calculated exactly using the complete effective action (11). Though, if the tunneling and backscattering parameters, (t and g respectively), are small then it may be computed perturbatively. To second order in these parameters this function is found to be: wire. This, in turn, dictates a stretched exponential behavior of the normal reflection from the junction as function of L. For energy scales (temperature or voltages) beneath the effective gap, we described the qualitative picture of the conductance as a function of temperature and length of the embedded segment of wire, see Fig. 4. Several simplifications have been used for our analysis: One is that our model treats a semi-infinite wire with a single junction, while in practical situations a finite wire is usually connected to two reservoirs. This idealization holds as long as the segment of the wire outside the superconductor is long enough compared to ¯ vf /T , h where vf is the Fermi velocity and T is the temperature. Additional simplification is the assumption that the embedding of the wire into the bulk superconductor does not introduce inhomogeneities, i.e. the wire can be still considered to be clean, and that the tunneling to the superconductor is homogenous along the wire. This approximation holds when the transport mean free path in the wire is longer than L. In the opposite limit one expects a different behavior of the proximity effect which will change the conductance of the system. The authors gratefully acknowledge discussions with ef 2 Ks f = Ks − Ks ∆2 Xs √ 2π(ϕ(xa ,τa )−ϕ(xb ,τb )) Rs (ra,b ) = e ef ef −γ 2 Ks f Fs (ra,b )+Ds f sin2 (ϕra,b ,s ) , (33) where ra,b = ra − rb , and ϕr,s is the angle between the vector r = (x, us τ ) and the x axis. The function Fs is (at zero temperature)13 : Fs (x, τ ) = 1 ln 2 x2 + (us |τ | + α)2 α2 . (34) Apart from the term proportional to sin2 (ϕra,b ,s ), this functional form is identical to the free correlation function: (0) Rs (ra − rb ) = e−γ 2 Ks Fs (ra −rb ) , (35) ef but with an effective Luttinger interaction constant Ks f modified by the perturbations: r α −1 3−Ks −Kc y= g πus + y2 2 dr α r α 3−4Ks . (36) is the dimensionless spin backscattering param- 7 Applying the same renormalization scheme to Def f , will generate the flow of the parameter Ds : eter, and Xs is a geometrical term given by: Xs = 2π 1 2π dϕ cos2 (ϕ)+ 0 uc us −1 −Kc 2 2 sin2 (ϕ) Integrating out high energy degrees of freedom, near the ultraviolet cutoff, corresponds to integrating out a ”small ring” between α → α′ = α + dα, where α is the small distances parameter of the model. After integration and rescaling, an infinitesimal change is generated in the ef ef expressions for Ks f . In order to keep Ks f constant with the reduction of the cutoff, it is required that the bare parameters change. For instance, we find that: 2 Ks (α′ ) = Ks (α)−Ks (α) ∆2 (α)Xs (α)+ dDs 2 = Ks (l)Ws (l)∆2 (l). dl (37) y 2 (α) dα (38) 2 α which generates an (exact) differential equation for Ks : dKs y 2 (l) 2 = −Ks (l) Xs ∆2 (l) + dl 2 (39) It should be noted that Ds is initially zero but is generated under renormalization. The parameter Ds controls the renormalization of the velocity parameter. As long as the space and time directions are isotropic the velocity parameter does not flow under renormalization, but they should flow in the anisotropic case. Indeed, assuming that initially the correlation function is described by the function Ks Fs (ra,b ), then a small change of dus will generate the term: 2 Ks us sin (ϕra,b ,s ) · dus . Up to a factor, this is exactly the anisotropy term. Therefore, the renormalization of Ds is equivalent to that of the velocity us by the following u relation (e.g. Eq. 33): dus = Ks dDs . dl s dl The flow equations for the charge sector can be obtained by an identical procedure. The equations obtained are: (here dl = dα ). α In a similar fashion one obtains differential equations for the parameters ∆ and y: d∆2 −1 = ∆2 (4 − Ks − Kc ), dl dy 2 = y 2 (4 − 4Ks ). dl The sin2 (ϕra,b ,s ) contribution to the correlation function arises from the fact that the tunneling perturbation couples the spin and charge sectors, which were uncoupled without this term. Mathematically, this term characterizes the anisotropy between the space (x) and time (us τ ) directions. It’s pre-factor Def f is given by: ef 2 Ds f = Ds + Ks ∆2 Ws ∞ α dr α r α −1 3−Ks −Kc dKc = = Xc (l) dl , (42) 1 Ws = 2π 1 2 3 4 0 uc dϕ cos(2ϕ) cos2 (ϕ)+ us 2 sin2 (ϕ) 2 , us ∆(l) uc (44) 2 . 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