Two-step flux penetration in layered antiferromagnetic superconductor
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l Two-step flux penetration in layeredantiferromagnetic superconductor
T. Krzyszto´n
Institute of Low Temperature and Structure Research, Polish Academy of Sciences,50-950 Wroc law, P.O.Box 1410, Poland
Abstract
A layered antiferromagnetic superconductor in the mixed state may posses magneticdomains created along the Josephson vortices. This may happen when an externalmagnetic field is strong enough to flip over magnetic moments, lying in the phasecore of the Josephson vortex, from their ground state configuration. The formationof the domain structure of the vortices modifies the surface energy barrier of thesuperconductor. During this process the entrance of the flux is stopped and a newlycreated state exhibits perfect shielding. Such behavior should be visible as a plateauon the dependence of flux density as a function of the external magnetic field. Theend of the plateau determines the critical field, which we call the second criticalfield for flux penetration.
Key words:
High- T C superconductivity, mixed state, magnetic superconductors PACS:
INTRODUCTION
Among classical magnetic superconductors there are three groups of clus-ter compounds, REMo S , REMo Se , and RERh B ( RE=rare-earth) whichhave been the primary systems for study of the interplay between supercon-ductivity and long-range magnetic order [1,2]. Although good quality singlecrystals of these materials have been available and measured for a long timea very interesting phenomenon was recently discovered in DyMo S due tovery carefully conducted experiment [3]. This phenomenon predicted in [4]and named two-step flux penetration was previously observed solely on ( bct )ErRh B [5]. The present work is inspired by this discovery and the hope thatthe same behavior could possibly be observed in some of the layered supercon-ducting structures. The specific feature caused by the long antiferromagneticorder in the mixed state of a superconductor is the possibility of creation of Preprint submitted to Elsevier 6 November 2018 he spin-flop (SF)(or metamagnetic) domains along the vortices [6]. This iseasy to understand by taking two sublattices antiferromagnet as an example.An infinitesimal magnetic field applied perpendicular to the easy axis makesthe ground antiferromagnetic (AF) state unstable against the phase transfor-mation to the canted phase (SF). On the contrary, if the magnetic field isapplied parallel to the easy axis the antiferromagnetic configuration is stableup to the thermodynamic critical field H T . When the field is further increaseda canted phase develops in the system. Let us assume that in an antiferro-magnetic superconductor the lower critical field fulfils the relation H c < H T and that the external field, H c < H < H T , is applied parallel to the easyaxis. Then the superconducting vortices appear in the ground antiferromag-netic state. When the field is increased above H pl (see Fig.3) approximatelyequal to H T the phase transition to the canted phase originates in the vortexcore. The spatial distribution of the field around the vortex is a decreasingfunction of the distance from its center. Hence the magnetic field intensity inthe neighborhood of the core is less then H T . Therefore, the rest of the vortexremains in the antiferromagnetic configuration. The radius of the SF domaingrows as the field is increased. The above considerations apply to the classicalsuperconducting Chevrel phases as well as to the high T c superconductors,where antiferromagnetic order is produced by the regular lattice of RE ionsoccupying isolating layers. In this paper we consider the structure shown on n+1n C z x y Fig. 1. Schematic drawing of a piece of the layered superconductor. The shadedareas (n,n+1) represent superconducting layers. The bold arrows represent magneticmoments of RE ions lying in the isolating layers. The axes of the reference frameare shown.
Fig. 1 that we believe simulates a real structure of many antiferromagnetic lay-ered superconductors. A good candidate to show the above behavior shouldpossess the isolating layers with the magnetic moments of RE ions runningparallel and antiparallel to the direction (easy axis) lying in the ab plane. Atypical example of such system is ErBa Cu O . This compound has tetragonalunit cell with small orthorombic distortion in the ab plane. The Er ions formtwo sublattices antiferromagnetic structure of magnetic moments laying par-allel and antiparallel to the b direction [7]. Another example may be Er nickelboride-carbides [8,9,10]. The layered structure of RE nickel boride-carbidesis reminiscent of that of the high- T c oxide superconductors and consists of2E-carbon layers separated by Ni B sheets. BASIC EQUATIONS
We start description of our problem in terms of the Lawrence-Doniach en-ergy functional. In this approach a layered superconductor is described by thesuperconducting planes with the interlayer distance d, as shown on Fig. 1.The antiferromagnetic subsystem consisting of RE ions is confined to the in-sulating layers. The magnetic moments are running parallel and antiparallelto the x-axis (easy axis). The Lawrence-Doniach functional is obtained fromthe standard Ginzburg-Landau energy by discretization of the kinetic energyin the z-direction. F S = Z d rd X n ¯ h m (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − i ∇ ( ) + 2 ie ¯ h A ( ) (cid:19) Ψ n (cid:12)(cid:12)(cid:12)(cid:12) + a | Ψ n | + 12 b | Ψ n | + ¯ h M d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ n +1 exp ei ¯ h Z ( n +1) dnd A z dz ! − Ψ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1)The quantity ¯ h, e, m, denote Planck constant, charge of the electron and massof the current carrier in the ab plane, whereas M denotes mass of the currentcarrier in the z-direction. The antiferromagnetic two sublattices subsystemwith single ion anisotropy is described with the following energy density func-tional f M = X n (cid:26) J M n · M n + K X i =1 ( M xin ) − | γ | X i =1 X j = x,y,z ( ∇ M jin ) (cid:27) . (2)where M n = M n + M n is the sum of the magnetization vectors of the sublat-tices in the n-th insulating layer, M xin is the component of the magnetizationsublattice vector along the anisotropy axis in the n-th layer, J denotes theexchange constant between two sublattices, K is the single ion anisotropyconstant, q | γ | is the magnetic stiffness length, and M = | M n | = | M n | .Since in the following we analyze the phenomena with characteristic length-scales much larger then the interatomic distance it is justified to omit thegradient term in f M . The components of the total magnetization vector M have the following form in both sublattices: M iy = M sin θ i , M ix = M cos θ i ,where θ i (canted spin angle) is the angle between the magnetization in thesublattice and the external magnetic field directed along the x -axis. The AF( θ = 0 , θ = π ) and SF phases ( θ = − θ = θ ) are in thermodynamic equilib-rium in an applied field equal to the thermodynamic critical field [11] H T = M [ K ( J − K )] / . (3)3he canted spin angle of the SF phase is then expressed ascos θ = KM H T . (4)Finally we add the magnetic field energy to obtain the free energy of the entiresystem F = F s + Z (cid:26) f M + µ H (cid:27) dV . (5)According to experiments the antiferromagnetic order is very weak affectedby the presence of superconductivity, then it is reasonable to neglect the effectof superconductivity on the exchange interaction in F . Instead we introduceelectromagnetic coupling between the magnetic and superconducting subsys-tem. This means that both order parameters Ψ n and M are coupled throughthe vector potential A B = rot A = µ H + M , (6) j s = rot H , (7)where B is the vector of the magnetic flux density (magnetic induction) and H is the vector of the thermodynamic magnetic field intensity. The functional (5)can be treated in the London approximation by assuming a constant modulusΨ n within the planes and allowing only for phase ( ϕ n ) degree of freedom. Theequilibrium conditions of the whole system can be obtained via minimizationthe Gibbs free energy functional G = F − Z ( BH ) dV . Performing this taskwith respect to vector potential A and ϕ n provides us with the fundamentalequations for currents and phases. X n dλ (cid:18) φ π ∇ ( ) ϕ n − A ( ) (cid:19) δ ( z − nd ) = µ j ( ) = rot (2) ( B − M ) (8) X n (cid:18) φ π λ j d sin χ n +1 ,n (cid:19) Θ( z − dn )Θ[ d ( n + 1) − z ] = µ j z = rot z ( B − M ) (9) ∇ ( ) (cid:18) ∇ ( ) ϕ n − πφ A ( ) (cid:19) = 1 r j (cid:18) sin χ n +1 ,n − sin χ n,n − (cid:19) (10)where δ ( z − nd ) is the Dirac delta function, Θ( z − dn ) Heaviside step func-tion, λ denotes London penetration depth in the superconducting plane, λ j = λ q M /m , r j = d q M /m and χ n +1 ,n = ϕ n +1 − ϕ n + 2 ei ¯ h Z ( n +1) dnd A z dz is thegauge invariant phase difference. In the following we shall investigate the prob-lem of a single vortex line lying parallel to the Josephson coupled supercon-ducting layers, separated by the insulating antiferromagnetic layers.4 F B(y) yz x
Fig. 2. Single Josephson vortex lying in the ab plane along the ˆ x -axis. The shadedarea shows induced SF domain along the phase core. SINGLE JOSEPHSON VORTEX
We assume that the vortex center, located in the central n = 0 layer, is parallelto the x -axis. The relation between the magnetic field of the vortex and thegauge-invariant phase difference χ n +1 ,n ( y ) of the superconducting wave func-tion across layers n and n + 1 can be found by integrating the vector potentialgiven by equations (8) and (9) around a rectangular, semi-infinite contour C,as shown in Fig. 1. This contour, parallel to the yz plane, is located apartfrom the central junction n = 0, where nonlinearities of the phase differencemust be taken into account. The magnetic flux within this contour in givenas. Φ( y ) = d Z ∞ y dy ′ B ( y ′ , z ) = I C A d l Differentiating the result with respect to y one obtains B = λ ∂ (cid:16) B − M (cid:17) ∂z + λ j ∂ (cid:16) B − M (cid:17) ∂y (11)The above equation shows that in the Josephson vortex the screening currentsvanish on a length scale λ j along y axis, and a scale λ along z direction. Onthese scales the Josephson and Abrikosov vortices in an anisotropic supercon-ductor are roughly equivalent apart from small corrections in the current flow,Fig. 2, and the magnetic field pattern due to the layered structure. But in con-trast to the Abrikosov vortex, where the large current flow near the core leadsto complete supression of the order parameter, the supression of the orderparameter in the superconducting layers is only weak in the so called phasecore of the Josephson vortex. Within the distances r j along y , and d along z we have to take into account the nonlinearity and discretness of Eq. (10).On these length scales the phase is changing rapidly and the current density j z reaches its maximum value j c = φ πµ λ j d . In the region of the phase coreLondon model fails.To make the problem simpler we assume that the magnetization in the isolated5ortex is constant across the SF phase domain. Thus, we can write | M | = M if ρ ≤ ρ m ρ > ρ m , (12)where ρ m is dimensionless radius of the SF domain in the cylindrical referenceframe x = x ; y = λ j ρ sin θ ; z = λρ sin θ . Then the solution of Eq. (11) fora single Josephson vortex is given by the modified Bessel functions K and I b SF = C K ( ρ ) + C I ( ρ ) , for ρ j < ρ ≤ ρ m b AF = C K ( ρ ) , for ρ > ρ m , (13)( ρ j denotes the dimensionless phase coherence length ) with the followingboundary conditions: b SF ( ρ m ) = µ H T + M = B T b AF ( ρ m ) = µ H T . (14)These conditions, together with the flux quantization condition, are used tocalculate the arbitrary constants in Eq. (13). C = B T ρ m I ( ρ m ) − " µ H T ρ m K ( ρ m ) K ( ρ m ) − ϕ πλλ j I ( ρ m ) ρ m K ( ρ m ) I ( ρ m ) − I ( ρ m ) + ρ m K ( ρ m ) I ( ρ m ) C = B T [ ρ m K ( ρ m ) −
1] + " µ H T ρ m K ( ρ m ) K ( ρ m ) − ϕ πλλ j K ( ρ m ) ρ m K ( ρ m ) I ( ρ m ) − I ( ρ m ) + ρ m K ( ρ m ) I ( ρ m ) C = µ H T K ( ρ m ) . (15)Finally we write free energy of the isolated vortex ε = λ j λ µ I σ d σ { [ b SF ( r ) − M ] × rot b SF ( r ) } + λ j λ µ I σ d σ { b AF ( r ) × rot b AF ( r ) } , (16)where r = ( yλ j , zλ ) is the position of the vortex line, σ denotes the surfaceof the phase core, and σ the surface of the SF domain respectively. Theintegrals in Eq. (16) performed as line integrals along the contours of the cross6ections of the appropriate surfaces give ε - the line tension of the vortex.The minimum of ε with respect to ρ m determines ρ m = 5 φ πλλ j B T . (17) FREE ENERGY OF THE LATTICE
Equation (11), in the new coordinates, can be rewritten for the lattice ofvortices in the following way: B + rotrot B = φ λλ j X m δ ( r − r m ), (18)where r m specify the positions of the phase cores of the vortices. The solutionof Eq. (18) is then a superposition B ( r ) = X m B m ( r − r m )of the solutions B m ( r − r m ) of isolated vortices at points r m . The free energyof the system can thus be written as F = λλ j µ I σ d σ ( B × rot B ) (19)The above symbolic surface integral is taken over the surfaces of the phasecores and surfaces of the SF domains. The energy of the Meissner state ischosen as zero of the energy scale. Again, when the surface integrals are re-placed by contour ones over appropriate cross sections we get line energy ofthe system. This, in turn, multiplied by vortex density n gives f -free energydensity of the system. After some transformations one can derive the followingformula f = nε + nφ H T (ln β ) − X m K ( r m ), (20) β = s πλλ j B T φ , here the sum is over all vortices excluding the one in the origin, and r m denotesthe distance of a vortex from the origin. The lattice sum may now be replacedby integral in the yz -plane over a smoothed vortex density, excluding the area n − associated with the single flux line in the origin. The free energy densitythen reduces to f = nε + B (cid:18) H T B T (cid:19)(cid:18) β ln β (cid:19) + B H T β s λ j λ ln (cid:18) a q λλ j (cid:19) (21)7 a q λλ j (cid:19) = 1 β (cid:18) B T B (cid:19)s λ j λ , here a = | a | denotes the length of the basal vector of the nonequilateraltriangular unit cell, and 2 | a | = a √ α ( α is the angle between bothvectors), tan α = q λλ j [12]. To determine the equilibrium state it is necessaryto minimize the Gibbs free energy density with respect to magnetic induction.The result yields an implicit equation for the constitutive relation B = B ( H ) H − ε φ = B (cid:18) H T B T (cid:19)(cid:18) β ln β (cid:19) + H T β s λ j λ ln (cid:18) a q λλ j (cid:19) (22) FLUX PENETRATION
Consider semi-infinite specimen in the half space y ≥
0, the vortex and theexternal magnetic field running parallel to the surface in the x direction. Thepresence of a surface of the superconductor leads to a distortion of the fieldand current of any vortex located within a distance of the order of penetrationdepth from the surface. To fulfill the requirement that the currents cannot flowacross the surface of the superconductor we need to introduce an image vortex,with vorticity opposite to the real one. Both vortices, direct and image, interactas real ones except that the interaction is attractive. In the low flux densityregime, Clem [13] has shown that there exist two regions: a vortex-free region ofthe width y ff near the surface of the sample, and a constant flux density regionfor y > y ff . Within the vortex-free area one can introduce the locally averagedmagnetic field B M which is a linear superposition of the Meissner screeningfield, the averaged direct vortices flux density exponentially decreasing towardsthe surface from its interior value B at y = y ff , and averaged image vorticesflux density. In our problem the x component of this superposition can beapproximated by B M = B cosh y ff − yλ j ! . (23)The boundary condition B M (0) = µ H determines the thickness of thevortex-free region y ff = λ j cosh − (cid:18) µ H B (cid:19) . (24)We assume that the test vortex line is lying within vortex free region at apoint r = ( yλ j , r = ( − yλ j ,
0) outside the superconductor.Now the local field of the test vortex can be understood as a superposition ofthe following fields 8 SF = b SF ( r ) − b AF (2 r ) + ˆ xB M ( r ff − r ) B AF = b AF ( r ) − b AF (2 r ) + ˆ xB M ( r ff − r ) , (25)where r ff = ( y ff λ j , x denotes the unit vector in the x direction. Havingdetermined the local magnetic field we can write the Gibbs free energy of thetest vortex line as G = λλ j µ I σ d σ { [ B SF ( r ) − µ H − M ] × rot B SF ( r ) } + λλ j µ I σ d σ { [ B AF ( r ) − µ H ] × rot B AF ( r ) } + λλ j µ I σ d σ { ˆ xB M ( r ff − r ) × rot B AF ( r ) } . (26)After some transformations [13,14] one can obtain the Gibbs free energy perunit length G G = G + G ′ + G M , (27)where G = ε − λλ j π µ D b AF (2 r ) G ′ = − λλ j π µ D [ b AF ( r ff ) − b AF ( r ff + r )] G M = − λλ j π µ [ D µ H − D B M ( r ff − r )] , (28)and D = − ρ j db SF ( ρ ) dρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ j − ρ m db SF ( ρ ) dρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ m − ρ m db AF ( ρ ) dρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ m D = − ρ j db SF ( ρ ) dρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ j − ρ m db SF ( ρ ) dρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ m − ρ m db AF ( ρ ) dρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ m (29) G describes the interaction of the test vortex with its image, G ′ is a correctionterm introduced by Clem [13], and G M describes the interaction energy of thetest vortex with the modified Meissner field. To find the conditions of thevortex entrance and exit, one has to solve a force balance equation for the testvortex, at the surface of the sample, and at the edge of the flux-filled area,respectively. A calculation using G and G M alone gives non vanishing force onthe test vortex at r = r ff . However, the force should be zero there, because G M is supposed to account for all the image vortices. To avoid double countingthe image vortex one can subtract from the self-energy a contribution of the9xcess image fixed at r = − r ff . One can easily check that G ′ is negligible atthe surface of the sample and has no influence on the conditions of the fluxentrance. When the flux starts to enter the sample, H = H en ( B ), y ff = y en = λ j cosh − (cid:18) µ H en ( B ) B (cid:19) , (30)and the energy barrier is moved toward the surface within ρ m . Thus, one canderive from the force balance equation − D D db AF ( ρ ) dρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ m = B sinh (cid:18) y en λ j (cid:19) . (31)The left hand side of the above equation gives H en (0) = H T β (2 ln β ) − . Thisfield may be thought as the second critical field for flux penetration calculatedin the single vortex approximation [14]. Combining Eqs. (30) and (31) wefinally obtain H en ( B ) = s B + (cid:18) µ H T β β (cid:19) . (32)In the opposite case, when the flux exits the sample, the surface energy barriertends to the edge of the flux-filled zone. Similar considerations as the aboveshow that µ H ex ( B ) ≃ B . (33)The measure of the height of the energy barrier against flux entrance is∆ H en ( B ) = | H en ( B ) − H eq ( B ) | , and against flux exit ∆ H ex ( B ) = | H eq ( B ) − H ex ( B ) | , where H eq is given by Eq. (22). DISCUSSION OF THE RESULTS
Let us make a short summary of the calculations and visualize the results onschematic magnetization curve shown in the Fig. 3. When the external fieldis not strong enough to create the SF domains inside vortices, than the mag-netization process of the sample being entirely in the AF phase is as follows.The vortices without magnetic structure start to enter the specimen at H en .When the field is increased up to the value H pl , which is of the order of H T ,the SF domains are created. Now, the screening current must redistribute itsflow in order to keep constant the flux carried by the vortex. This featureis easily seen from Eqs.(13 - 15). The redistribution of the screening current10 n1 en2 H plpl B H H HB
Fig. 3. Schematic drawing of the magnetization process. H en denotes the firstpenetration field for vortices without magnetic structure, H pl is the applied fieldwhich originates SF transitions inside vortices, and B pl is the corresponding fluxdensity. H en is the entrance field for the vortices possessing magnetic structure. changes the surface energy barrier preventing vortices from entering the sam-ple as expressed in Eq. (28). It means that the density of vortices n is keptconstant. Consequently the averaged flux density in the sample B = nϕ re-mains constant when the external field is increased. In Fig. 3 this feature isvisible as a plateau on the B ( H ) curve, or alternatively as a second negativeslope on the M ( H ) curve. The vortices start to penetrate the sample when theexternal field reaches the right edge of the plateau. We call this value, givenby Eq. (32), second critical field for flux penetration H en .To find the thermodynamic critical field H T , and then to calculate H en ( B ) thefollowing argumentation is proposed. At low fields, in the vicinity of the lowercritical field H c , the intensity of the field in the vortex core is 2 H c [15]. Whenthe external field is increased the field intensity in the vortex core increasesbecause of the superposition of the fields of the surrounding vortices. The fieldintensity in the core must reach H T in order to originate a transition to theSF phase. Thus, taking into account only the nearest neighbors we can writefor the nonunilateral triangular lattice H T = 2 H c + z ϕ πλλ j µ K aλ j ! + 2 K a λ j vuut λλ j , (34)here a corresponds to the value B pl of the flux density for which the penetrationprocess stops, see Fig.( 3). From the relation B pl = 2 ϕ o q λ j / ( a √ λ ) one cancompute a , which in turn may be inserted back into Eq. (34). It is easy toestimate the saturation magnetization M taking into account the volume ofthe elementary cell. Then, Eqs. (3) and (4) can be used to calculate M in the11F-phase domain M = 2 M cos θ = 2 KM H T . (35) CONCLUSION
The layered antiferromagnetic superconductor may reveal below T N a veryinteresting behavior in the magnetic field applied parallel to the supercon-ducting planes. When the sample is in the virgin state, initially it magnetizeslike ordinary type II superconductor. Upon the applied magnetic field of in-tensity equal to the critical field for flux penetration the sample undergoesa transformation from the Meissner to the mixed state. Then, the magneti-zation may proceed in an unusual way. When the field is further increased anew state may appear in which vortices possesses the spin-flop phases createdaround the cores. We have assumed that in this new state vortices undergometamorphosis to the shape shown in Fig. 2. This state is characterized bythe plateau on the magnetization curve, shown in the Fig. 3. It means thatthe magnetic flux density inside the sample is unaffected by an increased ex-ternal field. This perfect shielding should occur until the applied field reachescertain value of intensity, we call it second critical field for flux penetration.Then the vortices possessing magnetic structure enter into the sample. Thisphenomenon we named two-step flux penetration. ACKNOWLEDGEMENTS
The author would like to thank P. Tekiel and K. Rogacki for helpful discus-sions. This work was supported by the State Committee for Scientific Research(KBN) within the Project No. 2 P03B 125 19.
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