Study of the Skin Effect in Superconducting Materials Jeremie Szeftel, Nicolas Sandeau, Antoine Khater

Study of the skin effect in superconducting materials Jeremie Szeftel, Nicolas Sandeau, Antoine Khater

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Jeremie Szeftel, Nicolas Sandeau, Antoine Khater. Study of the skin effect in superconducting materials. Physics Letters A, Elsevier, 2017, 381 (17), pp.1525-1528. 10.1016/j.physleta.2017.02.051. hal-01504609

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The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. arXiv:1610.09501v1 [cond-mat.supr-con] 29 Oct 2016 materials o h rtcltmeaue,cnito esrn the measuring of consist temperature), critical the for ndwti hnlyro rqec eedn thick- dependent frequency of layer con- thin remain ness a to within field electromagnetic fined the causes quency, ue deo h conductor. the of edge outer edi qa to equal is Maxwell, field and Newton of entails those with combined equation, magnetic the by induced induction current, persistent the vacuum, ω < ω urn eemntosof determinations current effect where eypr easa o eprtr,a temperature, bound low at lower metals pure very eaelike behave nmlu kneet nnra odcos h real the constant conductors, dielectric normal the the In of of framework part effect. the skin within results anomalous his interpreted and lcrcfil n h esnreffect Meissner the and field electric properties e ffc a civdtak oLno’ equation London’s to thanks Meiss- achieved the was into effect insight bulk Early ner in field superconductor. magnetic a applied in an matter of decay rapid the lights n ocnrto fsprodcigeetos Thus electrons. of measurement superconducting the of concentration and where 10 hr sn te xeietlacs to access experimental other no is there lcrmgei nryasrto tafrequency a at absorption energy electromagnetic uecnutvt scaatrzdb w prominent two by characterized is Superconductivity Pippard GHz 13,14 δ p 1–5,7 ( µ ,m ρ m, e, ω where , 0 nsuperconducting in ,cle h kndph n oae tthe at located and depth, skin the called ), 15–20 ,λ j, , culy ntewk fPpadswr,all work, Pippard’s of wake the in Actually, . 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II. SKIN EFFECT of Ohm’s law for the superconducting material as ρe2τ Consider as in Fig.1 a superconducting material of j = σE , σ = . (3) cylindrical shape, characterized by its symmetry axis z m 2 and radius r0 in a cylindrical frame with coordinates Thus both Ohm’s law and σ are seen to display the same (r,θ,z). The material is taken to contain conduction form in normal and superconducting metals, as well. iωt electrons of charge e, effective mass m, and total concen- E induces a magnetic induction B(r, t) = Bz(r)e , tration ρ. It is subjected to an oscillating electric field parallel to the z axis. B is given by the Faraday-Maxwell iωt E(t, r)= Eθ(r)e , with t referring to time. As Eθ(r) is equation as normal to the unit vectors along the r and z coordinates, iωt ∂B E ∂E there is divE = 0. E induces a current j(t, r)= jθ(r)e = curlE = + . (4) along the field direction, as given by Newton’s law − ∂t r ∂r The displacement vector D, is parallel to E and is defined dj ρe2 j = E , (2) as dt m − τ D = ǫ0E + ρeu , (5) ρe2 j where m E and τ are respectively proportional to the driving force accelerating− the conduction electrons and a where ǫ0, u refer to the electric permittivity of vacuum generalized friction term, which is non zero in any super- and displacement coordinate of the conduction electron conducting material provided dj = 0. center of mass, parallel to E. The term ρeu represents the dt polarization of conduction electrons25. Because divE =0 The existence of a friction force6 in superconductors, entails that divD = 0, Poisson’s law warrants the lack of carrying an ac current, is known experimentally (see4 p.4, 2nd paragraph, line 9). For example, the mea- charge fluctuation around ρe. Thence since there is by definition j = ρe du , the displacement current reads sured ac conductivity for the superconducting phase of dt 18 rd BaFe2(As1 xPx)2 has been found (see p.1555, 3 col- ∂D ∂E nd − = j + ǫ . (6) umn, 2 paragraph, line 11) to be .03σn, where σn ∂t 0 ∂t stands for the normal conductivity21 measured≈ just above iωt the critical temperature Tc. However because the current Finally the magnetic field H(t, r) = Hz(r)e , parallel is carried by electrons, making up either a BCS state22 to the z axis, is given by the Ampère-Maxwell equation or a Fermi gas2 in a superconducting and normal metal, as respectively, the physical sense of τ in Eq.(2) for super- ∂H ∂D ∂E conductors may be different from that given by the Drude curlH = = j + =2j + ǫ0 . (7) − ∂r ∂t ∂t model2 for a normal metal. To understand this difference and to model the new τ, we shall next work out the equiv- Replacing E(t, r), j(t, r),B(t, r),H(t, r) in alent of Ohm’s law for a superconducting material when Eqs.(2,4,7) by their time-Fourier transforms submitted to an electric field. Eθ(ω, r), jθ(ω, r),Bz(ω, r),Hz(ω, r), while taking into The superconducting state, carrying no current in the account absence of external fields, is assumed to comprise two B (ω, r)= µ (ω) H (ω, r) , subsets of equal concentration ρ/2, moving in opposite di- z z rections with respective mass center velocity v, v, which − where µ (ω)= µ0 (1 + χs (ω)) (χs (ω) is the magnetic sus- ensures j = 0, p = 0, where p refers to the aver- ceptibility of superconducting electrons at frequency ω) age electron momentum. Under a driving field E, an yields ensemble δρ/2 of electrons is transferred from one sub- 1+iωτ set to the other, so as to give rise to a finite current Eθ (ω, r)= σ jθ (ω, r) j = δρev = eδp/m, where δp stands for the electron mo- Eθ(ω,r) ∂Eθ(ω,r) iωBz (ω, r)= r + ∂r (8) mentum variation. The generalized friction force is re- − ∂Bz (ω,r) = µ (ω) (2j (ω, r)+ iωǫ E (ω, r)) sponsible for the reverse mechanism, whereby electrons ∂r − θ 0 θ are transferred from the majority subset of concentration ρ+δρ ρ δρ Eliminating Eθ (ω, r) , jθ (ω, r) from Eqs.(8) gives 2 back to the minority one ( −2 ). It follows from flux 1–5,23 2 quantization and the Josephson’s effect that the el- ∂ Bz (ω, r) Bz (ω, r) ∂Bz (ω, r) 24 = . (9) ementary transfer process involves a pair rather than ∂r2 δ2(ω) − r∂r 1 a single electron. Hence if τ − is defined as the transfer probability per unit time of one electron pair, the net The skin depth δ and plasma frequency ωp are ρ+δρ (ρ δρ) δρ defined2,11,12 as electron transfer rate is equal to −2τ − = τ . By virtue of Newton’s law, the resulting generalized friction δ(ω)= λL , term is mvδρ/τ = δp/τ j/τ, which validates Eq.(2). 2iωτ ω2 (1+χs(ω)) 1+iωτ 2 ∝ − ωp Furthermore for ωτ << 1, the inertial term dj in s ∝ dt ρe2 Eq.(2) is negligible, so that we can write the equivalent ωp = ǫ m q 0 3

1 where both limits ωτs << 1, ωτs >> 1 are seen to correspond to the usual and anomalous skin eﬀect, re- 0.1 B (u) z spectively. Besides it is concluded that the conduction properties, at ω = 0 and T well below T , are assessed eu c 0.01 solely by the superconducting6 electrons, while the normal ones play hardly any role. 0.001

0.0001 IV. COMPARISON WITH EXPERIMENT

10-5 The experiments, performed at ω 10GHz, give ac- ≈ 10-6 cess to the imaginary part of the complex impedance of the resonant cavity, containing the superconducting sam- 0 5 10 15 ple, equal to µ0ωlp, where lp designates the penetration u = r / δ depth of the electromagnetic field, i.e. lp = δ. Moreover u at ω 10MHz, the cavity is to be replaced by a reso- FIG. 2. Semi-logarithmic plots of Bz(u),e . nant circuit,≈ combining a capacitor and a cylindrical coil of radius r0. The sample is inserted into the coil, which is flown through by an oscillating current I (ω)eiωt. The Note that the above formula of δ(ω) retrieves indeed 0 coil is made up of a wire of length l and radius r (see both, the usual11,12 expression δ = 1 , valid for c | | √2µ0σω Fig.1). Since the relationship between the penetration ωτ << 1, and the ωτ >> 1 limit δ = λL , typical of the a √2 depth lp and the observed self inductance of the coil L anomalous skin effect13,14 and widely used in the inter- is seemingly lacking, we first set out to work out an ex- pretation of the experimental work8–10,15–20. pression for L. As Eqs.(8) make up a system of 3 linear equations in Applying Ohm’s law yields terms of 3 unknowns jθ, Eθ,Bz, there is a single solution, embodied by Eq.(9). The solution of Eq.(9), which l (Ea(ω)+ Eθ(ω, r0)) = RI0(ω) − U(ω) RI0(ω) ⇒ , (11) has been integrated over r [0, r ] with the initial con- Eθ(ω, r0)= − ∈ 0 l dBz dition (r = 0) = 0, is a Bessel function, having the iωt iωt iωt iωt dr where Ea(ω)e , Eθ(ω, r)e ,Ue = lEa(ω)e , R r/δ(ω) − property Bz(r) e if r >> δ(ω) , as illustrated in are the applied and induced electric fields, both normal to Fig.2. ≈ | | the r, z axes, the voltage drop throughout the coil and its resistance, respectively (E (ω), E (ω, r),U(ω) C). Be- a θ ∈ sides Eθ(ω, r0) is obtained from Eq.(8) as III. THE TWO-FLUID MODEL E (ω, r )= iωδ(ω)B (ω, r ) . (12) θ 0 − z 0 − The total current jt reads as r r0 where E (ω, r r ) E (ω, r )e δ(ω) . Working θ → 0 ≈ θ 0 jt = jn + js = (σn + σs)E, out Bz(ω, r0) in Eq.(12) requires to solve the Ampère- iωt 2 2 Maxwell equation for Bz(ω, r)e inside a cross-section where j , σ = ρne τn (j , σ = ρse τs ) designate the n n mn s s ms of the coil wire normal (superconducting) current and conductivity, with ∂Bz ρn, τn,mn (ρs, τs,ms) being the concentration, decay = µ (2j + iǫ ωE ) , (13) ∂r − 0 c 0 c time of the kinetic energy, associated with jn (js), and effective mass of normal (superconducting) electrons. Re- I0(ω) where jc(ω)= 2 and Ec(ω) = Ea(ω)+ Eθ(ω, r0) are placing jθ in Eqs.(8) by the expression jt hereabove leads πrc to the following expression of the skin depth for the prac- both assumed to be r-independent. Moreover integrating Eq.(13) for r [r +2r , r ] with the boundary condition tical case ω <<ωp and χs << 1 0 c 0 B (ω, r +2r ∈)=0, t (see Fig.1), while taking advantage z 0 c ∀ 2 σn σs of Eq.(11), yields δ− =2iµ0ω + . (10) 1+ iωτn 1+ iωτs 2 iǫ0ωrcR Bz(ω, r0)=2µ0 I0(ω) . (14) The inequalities ρn << ρs, τn << τs, σn << σs can be πrc − l 15–20 inferred from observation to hold at T well below Tc, so that δ in Eq.(10) is recast finally as Finally it ensues from the definition of L and Eq.(14) that 1 1 λL δ ω <<τ − = = , s √2µ0σsω √2ωτ r0 − | | r r0 5 r0 1 ms λL LI0 =2πRe Bz(ω, r0)e δ rdr L 2 2 µ0 δ , δ ω >>τs− = 2µ ρ e2 = , 0 s √2 Z0 ⇒ ≈ | | rc q 4 where Re means real part and δ = λL/√2iωτ for ωτ << in the limit ωτs << 1. Therefore it is suggested to work 1. at higher frequencies, such that ω >> 1/τs,ω << ωp, Both imaginary parts of the coil and cavity impedance because δ(ω) = λL/√2 is independent from τs in that are thus found to be equal to C(ω)ω δ(ω) , where C(ω) range. Typical values τ 10 11s,ω 1016Hz would | | s − p is an unknown coefficient depending on the experimen- imply to measure light absorption≈ in the≈ IR range. Then tal conditions. As C(ω) is likely to be T independent for an incoming beam being shone at normal incidence at least inside the limited range 0 > V. CONCLUSION 1, ω2τs << 1, characterized by δ(ω1,T < Tc) = λL(T ) λL(T ) , δ(ω2,T

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