Anisotropic time-dependent London approach
AAnisotropic time dependent London approach
V. G. Kogan ∗ Ames Laboratory-DOE, Ames, Iowa 50011
R. Prozorov † Ames Laboratory–DOE, Ames, IA 50011, USA andDepartment of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA (Dated: September 16, 2020)The anisotropic London equations taking into account the normal currents are derived and ap-plied to the problem of the surface impedance in the Meisner state of anisotropic materials. It isshown that the complex susceptibility of anisotropic slab depends on the orientation of the appliedmicrowave field relative to the crystal axes. In particular, the anisotropic sample in the microwavefield is a subject to a torque, unless the field is directed along one one of the crystal principle axes.
I. INTRODUCTION
Its shortcomings notwithstanding, the approach basedon London equations played - and still does - a majorrole in describing magnetic properties of superconduc-tors away of the critical temperature T c where it is, infact, the only available and sufficiently simple techniquefor many practical applications. The physical reason forthis success is in its ability to describe the Meissner effect,the major feature of superconductors at all temperatures.The anisotropic version of this approach [1, 2] has provenuseful when strongly anisotropic high- T c materials cameto the forth. It was also realized that in time depen-dent phenomena the normal dissipative currents due tonormal excitations should be taken into account alongthe persistent currents [3, 4]. In particular, normal cur-rents influence superconductors behavior in microwavesabsorption [3] and perturb the field distribution of mov-ing vortices [5, 6]. In this work, the anisotropic version oftime dependent London equation is derived and appliedto problems of surface impedance and magnetic suscep-tibility in a simple geometry.Within the London approach, the current density con-sists, in general, of normal and superconducting parts: J = σ E − c πλ (cid:18) A + φ π ∇ θ (cid:19) , (1)where E is the electric field, λ is the penetration depth, A is the vector potential, θ is the phase, and φ is theflux quantum. The conductivity σ for the quasiparticlesflow is in general frequency dependent. If however thefrequencies ω are bound by inequality ωτ n (cid:28) τ n being the scattering time for the normal excitations, onecan consider σ as a real ω -independent quantity. As al-ways within the London approach, the order parameteris assumed constant in space. ∗ [email protected] † [email protected] In the absence of vortices, we have by applying curl:curl curl H + 1 λ H = − πσc ∂ H ∂t . (2)These are in fact London equations corrected by the timedependent right-hand side [5].
1. Surface impedance of the half-space isotropic sample
The surface impedance in isotropic superconductorshas been considered, e.g., by Clem and Coffey [3]. Eq. (2)provides a simple and direct approach to this problem.Let a weak magnetic field H = H ˆ x e − iωt be at the sur-face z = 0 of a superconducting half-space z >
0. Sincethe field is assumed weak, the order parameter f is un-perturbed and we can use the London Eq. (2). The fieldis uniform in plane ( x, y ) and depends only on z . Welook for solutions of − ∂ H x ∂z + 1 λ H x = − πσc ∂H x ∂t . (3)in the form H x ( z ) e − iωt and obtain: H x = H e − kz − iωt , k = 1 λ − iδ , (4)where δ = c/ √ πσω is the quasiparticles related skin-depth.The electric field is found from the Maxwell equationcurl E = − ∂ t H /c : E y = ( iω/ck ) H e − kz − iωt , so that thesurface impedance, see e.g. [7]: ζ = − E y H x (cid:12)(cid:12)(cid:12) z =0 = − iωck . (5)If δ (cid:29) λ , k ≈ λ (cid:18) − i λ δ (cid:19) (6)and ζ ≈ ωλ cδ − i ωλc . (7) a r X i v : . [ c ond - m a t . s up r- c on ] S e p Thus, the dissipative part of impedance is given byRe ζ ≈ πc ω σλ . (8)The imaginary part of the impedance is not affected byquasiparticles part of the current, see e.g. Ref. 7, i.e.it depends only on λ . It is worth noting that Eqs. (6)and (7) do not hold in immediate vicinity of T c , where λ diverges.
2. Susceptibility of a slab
It is instructive to consider Eq. (2) for a supercon-ducting slab of the thickness d in the applied ac field H x = H e − iωt parallel to the slab faces. The solution is H x = H cosh( kz )cosh( kd/ e − iωt . (9)with k of Eq. (4) and z counted from the slab middle.The electric field is E y = i ωH ck sinh( kz )cosh( kd/ e − iωt , (10)and the surface impedance ζ = − E y H x (cid:12)(cid:12)(cid:12) z = d/ = − iωck tanh kd . (11)Commonly measured quantity is the susceptibility de-fined as ratio of the average magnetization µ of the slabto the applied field: χ = µ x H = 14 πd (cid:90) d/ − d/ H x ( z ) − H H dz = − π + 12 πdk tanh kd . (12)Hence, we have a simple relation between the surfaceimpedance and the slab susceptibility: χ + 14 π = ic πdω ζ , (13)i.e. the surface impedance is proportional to the devia-tion of susceptibility from the Meissner value − / π .Hence, for λ (cid:28) δ one obtains with the help of Eq. (7): χ + 14 π = λ πd + i λ πdδ . (14) II. ANISOTROPIC MATERIALS
In the absence of vortices, the order parameter can betaken as real so that the current equation becomes J k = σ kl E l − c π (cid:0) λ − (cid:1) kl A l , (15) where σ kl and (cid:0) λ − (cid:1) kl are tensors of the conductivitydue to normal excitations and of the inverse square ofthe penetration depth. As usual, summation is impliedover double indices. Being interested in problems withno conversion of normal currents to super-currents, weimpose the conditionsdiv J n = σ kl ∂E l ∂x k = 0 , div J s = λ − kl ∂A l ∂x k = 0 , (16)i.e. the densities of normal excitations and of Cooperpairs are separately conserved. In particular, this impliesa certain gauge for the vector potential.In order to obtain an equation for magnetic field exclu-sively, one has to isolate E and apply the Maxwell equa-tion curl E = − ∂ t H /c . To this end, multiply Eq. (15) by σ − sk = ρ sk with ρ sk being the resistivity tensor and sumup over k : ρ sk J k = E s − c π ρ sk λ − kl A l . (17)In the following it is convenient to use the notationcurl u E = (cid:15) uvs ∂E s /∂x v where (cid:15) uvs is Levi-Chivita unitantisymmetric tensor: all components with even num-ber of transpositions from ( xyz ) are +1, − (cid:15) uvs ∂/∂x v toEq. (17), one obtains anisotropic London equations forthe magnetic field, the main result of this paper: c π ρ sk (cid:15) uvs (cid:15) kmn ∂ H n ∂x v ∂x m + ∂H u c ∂t = − c π ρ sk λ − kl (cid:15) uvs ∂A l ∂x v . (18)One can check that in the isotropic case this equa-tion reduces to the time-dependent London equation (2).Another limit to check is the static anisotropic Londonequations [1]. In this case we have ρ sk (cid:15) uvs (cid:18) (cid:15) kmn ∂ H n ∂x v ∂x m + λ − kl ∂A l ∂x v (cid:19) = 0 . (19)Clearly, this equation is satisfied if (cid:15) kmn ∂H n ∂x m + λ − kl A l = 0 . (20)We now introduce a tensor (cid:0) λ (cid:1) kl inverse to (cid:0) λ − (cid:1) kl , mul-tiply the last equation by (cid:0) λ (cid:1) kµ , and sum up over k : λ kµ (cid:15) kmn ∂H n ∂x m + A µ = 0 . (21)Finally, apply to this (cid:15) uvµ ∂/∂x v to replace curl A with H and obtain static anisotropic London equations [1]. A. Orthorhombic slab with plane faces ab Cumbersome Eqs. (18) are applicable in coordinatesystem ( x, y, z ) oriented arbitrarily relative to theanisotropic sample. One, of course, can choose ( x, y, z )as the crystal frame ( a, b, c ) where ρ sk and λ − kl are diag-onal. Consider a slab of a thickness d of orthorhombicmaterial with a, b (or x, y ) plane faces; z is counted fromthe slab middle. Let the ac applied field H be parallelto x ; the field inside the slab depends only on z .Consider the first term in Eq. (18). Since v and m takeonly z values and n = x , it is readily seen that this termreduces to − c π ρ yy ∂ H x ∂z . (22)The term on the right of Eq. (18) can be treated similarlyto obtain cρ yy λ − yy ∂ z A y . Hence − ∂ H x ∂z + λ − yy H x + 4 πc ρ yy ∂H x ∂t = 0 . (23)This equation is equivalent to the isotropic Eq. (3) withthe same solution (9) for the slab, but now k x = λ − yy − iδ yy , δ yy = c πσ yy ω . (24)As expected, the decaying behavior of H x is determinedby characteristics of persistent and normal currents inthe y direction.Thus, the isotropic result for the susceptibility is di-rectly translated to this situation. In particular, if λ yy (cid:28) δ yy one obtains for the component χ xx of thesusceptibility tensor: χ xx + 14 π = λ yy πd + i λ yy πdδ yy . (25)If the applied field is directed along y , the same argumentleads to: χ yy + 14 π = λ xx πd + i λ xx πdδ xx . (26)These formulas cannot be used too close to T c where theinequality λ (cid:28) δ is violated.It is worth noting that the anisotropy of the penetra-tion depth is related to the anisotropy of susceptibility: γ λ = λ xx λ yy ≈ Re χ yy + 1 / π Re χ xx + 1 / π . (27)Taking the ratio of imaginary parts we obtainIm χ yy Im χ xx ≈ δ yy δ xx λ xx λ yy = γ σ γ λ , (28)where γ σ = σ xx /σ yy .
1. Angular dependence of susceptibility
Let the applied field at the sample surface be at an an-gle ϕ with the a axis, H = H ( ˆ x cos ϕ + ˆ y sin ϕ ). Since London and Maxwell equations are linear, the solution isthe superposition of two solutions for applied fields ori-ented along the principle directions: H = H (cid:20) ˆ x cos ϕ cosh( k x z )cosh( k x d/
2) + ˆ y sin ϕ cosh( k y z )cosh( k y d/ (cid:21) (29)where the factor e − iωt is omitted for brevity. It is worthnoting that since the decay lengths for the magnetic fieldalong ˆ x (on the order of 1 /k x ) differs from 1 /k y , thefield rotates with increasing depth z . In this situation,the magnetic moment µ will have not only the compo-nent parallel to the applied field, µ (cid:107) , but a perpendicularcomponent as well.One has for the electric field: E = iωH c (cid:20) ˆ x sin ϕ sinh( k y z ) k y cosh( k y d/ − ˆ y cosϕ sinh( k x z ) k x cosh( k x d/ (cid:21) . (30)The commonly measured susceptibility is defined as χ (cid:107) = µ (cid:107) H = µ x cos ϕ + µ y sin ϕH = χ xx cos ϕ + χ yy sin ϕ. (31)where χ xx and χ yy are given in Eqs (25) and (26). Thisgives:Re χ (cid:107) = − π + 12 πd (cid:0) λ yy cos ϕ + λ xx sin ϕ (cid:1) , Im χ (cid:107) = 12 πd (cid:32) λ yy δ yy cos ϕ + λ xx δ xx sin ϕ (cid:33) . (32)
2. Dissipation and torque
Given the fields at the surface z = ± d/
2, one eval-uates the Pointing vector, i.e. the energy flux into thesample and the dissipation power [7]. One obtains afterstraightforward algebra: S z = − c π Re( E × H ∗ ) z = d/ = ωH π λ xx δ xx (cid:34) sin ϕ + (cid:18) λ yy λxx (cid:19) (cid:18) δ xx δyy (cid:19) cos ϕ (cid:35) . (33)Here, S z denotes the time average over the period 2 π/ω .If the parameter p = (cid:18) λ yy λxx (cid:19) (cid:18) δ xx δyy (cid:19) > , (34)cos ϕ dominates and the dissipation has minimum at ϕ = π/
2, i.e, for H directed along y . If p <
1, the dissi-pation is minimal for the field H directed along x . Sincethe system prefers the state with minimum dissipation,one expects a torque for 0 < ϕ < π/ τ averaged over the AC period: τ z = 12 Re( µ × H ∗ ) = 12 Re( µ x H ∗ y − µ y H ∗ x ) (35)where µ x = χ xx H cos ϕ and µ y = χ yy H sin ϕ . We ob-tain: τ z = H πd ( λ yy − λ xx ) sin 2 ϕ. (36) III. DISCUSSION
Anisotropic London equations taking into account nor-mal currents are derived and applied for evaluation of thesurface impedance and susceptibility χ for a simple geom-etry in which sample surfaces coincide with the ab planesof orthorhombic crystal. In principle, applying the acfield along a and b crystal axes one can extract both χ aa and χ bb of the susceptibility tensor.In usual situation of the penetration depth small rela-tive to the skin depth, the deviation of real part of suscep-tibility from Meissner’s − / π depends only on λ , so thatthe ratio of these deviations for two principle directionsgives the anisotropy parameter γ λ = λ aa /λ bb , Eq. (27).Hence, γ λ can, in principle, be extracted from microwavesusceptibility data. The behavior of γ λ with tempera-ture is of intense interests in studies of new materialsand it remains to be seen whether or not experimentalcomplications related to finite size of actual samples canbe overcome [8].Given the slab geometry we consider in this paper,thick anisotropic films seems the best to check our for-mulas. Strongly anisotropic properties of cuprates makesthem good candidates for such measurements. One canfind plenty of information for these possibilities in Ref. 9.While deep in the superconducting state the con-tribution of normal quasiparticles to susceptibility ismuch smaller than the Meissner contribution by a fac-tor ∼ λ /δ , see Eq. (14), only the latter is frequency- dependent via the skin depth δ ( ω ). Therefore, one canmeasure the response as a function of ω and extract the ω -dependent part. In fact, Eq. (14) can be written as χ + 1 / π = A + iBω with ω -independent A, B . Thereforethe derivative of the response with respect to frequencywill provide the imaginary part of χ .Another quantity which can be extracted from the sus-ceptibility data is the conductivity of normal excitations σ . It coincides with the normal state conductivity near T c (for gapless superconductors, for all temperatures). How-ever, experimentally, little is known about this conduc-tivity away of T c . Still, this quantity is of interest, in par-ticular, given recent theoretical work of Smith, Andreev,and Spivak stating that the conductivity can be stronglyenhanced due to inelastic scattering [10]. The anisotropyof σ can, in principle, be extracted from the ratio of imag-inary parts of susceptibility and the anisotropy of thepenetration depth, Eq. (28).It should be noted that we applied the general Eqs. (18)to an infinite slab. In experiments, one deals with finitesamples. In this case, magnetic susceptibility measuredin the applied field along, say b -axis in addition to λ xx will also depend on λ zz . What is worse, the sample shapewill give an extra angular modulation when the angle ϕ of the applied field direction is swept. These and otherdifficulties which may arise in measurements of the sus-ceptibility of anisotropic samples and possible ways toovercome them are discussed elsewhere [8]. IV. ACKNOWLEDGEMENT
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