Critical Magnetic Field Ratio of Anisotropic Magnetic Superconductors

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1 Critical Magnetic Field Ratio of Anisotropic Magnetic Superconductors A.Changjan,,3 and P.Udomsamuthirun, Prasarnmitr Physics Research Unit,Department of Physics, Faculty of Science,Srinakharinwirot Uniersity angkok,thailand. Thailand Center of Excellence in Physics, Si Ayutthaya Road,angkok 4,Thailand. 3 Department of Mathematics and asic ience, Faculty of Science and Technology,Phathumwan Institute of Technology, angkok 33,Thailand. PACS :74..Fg; 74.7.Ad Abstract The upper critical field, the lower critical field and the critical magnetic field ratio of anisotropic magnetic superconductors are calculated by Ginzburg-Landau theory analytically. The effect of the Ginzburg- Landau parameter( κ ), magnetic sueptibility( χ ) and magnetic-toanisotropic parameter ratio(θ ) on the critical field ratio are considered. We find that the alue of critical magnetic field ratio with χ <, higher κ,and higher θ case are shown the higher alue. And, the diamagnetic superconductors with highly anisotropic and the ferromagnetic superconductors are shown the highest and the lowest, respectiely. Keywords: Upper critical field, Lower critical field, Ginzburg-Landau Theory, anisotropy magnetic superconductors,critical Magnetic Field Ratio

2 .Introduction The Ginzburg-Landau(GL) theory is widely used in framework for deribing metallic and magnetic superconductors in magnetic field [,,3]. For type I superconductors, the magnetic response is diamagnetic. It shows total exclusion of flux in low magnetic field. For type II superconductors, there are quantized flux penetrates the superconductors in high magnetic field. The magnetic response of these material can be different from type I. The properties of type-ii superconductors at low and high applied magnetic field,close to lower and upper critical field was studied by many researchers. Hampshire[4,5] studied the magnetic superconductors by GL theory including the spatial ariation and nonlinear magnetic response of magnetic in-field. The analytic formula of upper critical field is shown. Askerzade[6] studied two-band GL theory and apply to determine the temperature dependence of lower,upper and thermodynamic critical field for non-magnetic superconductors. Askerzade[7], Udomsamuthirun et al.[8],min-xia and Zi-Zhao[9] studied the upper critical field of anisotropy two-band superconductors by GL theory. In this paper, we calculate analytically the upper critical field, the lower critical field and the critical magnetic field ratio of the anisotropic magnetic superconductor by Ginzburg-Landau(GL) theory that the Gibbs free energy of magnetic superconductors of Hampshire[4,5] are used..model and Calculation According to the Ginzburg and Landau[] theory, the Helmholtz free energy of non-magnetic superconductors is of the form F s h H sd () m 4 ( H, T ) FN + αψ + β ψ + ( i ea) ψ + Here, F N,and F s are the Helmholtz free energy in the normal state and superconducting state, is the net field in the superconductiity region, ψ is the order parameter and ( ψ ) is proportional to the density of carriers, m denotes the effectie mass of the carriers, the coefficient α depends linearly on the temperature, while coefficient β is independent of temperature. The fourth term accounts for the kinetic energy of the carriers and the lasted term accounts for the energy stored in the local magnetic fields. Hampshire[4,5] proposed the Gibbs s free energy of magnetic G, T F, T μ, superconductors by assuming that ( ) ( ) HM s N

3 3 H M M + M and M χh s. Therefore Gibbs free s M, μ energy of magnetic superconductors can be written as G N m 4 (, T ) F + αψ + β ψ + ( ih ea) ψ + ( μ M ) ( μ M )M s For small change in -field, a series in is introduced γ + γ + γ μ d μ d μ ( μm ) ( μm ) M ( μ M ) M Here γ,γ and γ are coefficient parameters. We can get the γ by differentiating aboe equation twice and taking M to be small that γ dm d d d μ μ μ [( M ) M ] Here γ for non-magnetic superconductors. The coefficient parameter γ depends on the gradient of field and γ for the uniform applied field. Finally, the Gibbs s free energy of magnetic superconductors can be written as G 4 (, T ) F + αψ + β ψ + ( i ea) s N ψ + γ + γ + m () h γ (3) μ When the magnetic superconductors place in magnetic field, there are the field produced by ( μ M ), the applied field ( μ H ) and the field produced by the carriers ( μ M ) that can produce relathip between field as So that M μ H + μm + μ M (4) χ H T + χ H M H (5) c ( ) ( ) c ( χ χ ) H + μ ( + )( H ) μ χ + c M (6) where the differential sueptibility ( χ ) is ( χ ) is M H T, H Hc. M H T, H Hc and sueptibility y comparing Ginzburg-Landau theory to CS theory, we can get ψ ~ Δ that Δ is gap function from CS theory and ψ is the order

4 4 parameter from GL theory. And, the anisotropic gap function[] can be written as where ( k) Δ ( kˆ, T ) ~ f ( kˆ ) Δ( T ) f ˆ is the anisotropic function. Similarity, we can write anisotropic order parameter [8] ψ kˆ, T f kˆ ψ T (7) ( ) ( ) ( ) The anisotropy gap function has been propose by Haas and Maki[] : + a cos δ f ( δ ) and by Posazhennikoa et al.[] : f ( δ ), + a + a cos δ here δ is the polar angle and a is anisotropy parameter. In case of the symmetry order parameter f ( k ˆ ). Minimising G s of Eq.(3) with respect to ψ and A,here we use anisotropic order parameter in Eq.(7), they lead to the st and nd Ginzburg-Landau equation, which now include the effect of anisotropic function and magnetic as αψ f () ˆ 4 () ˆ k + β ψ ψ f k + ( i ea) ψ f ( kˆ ) m h (8) γ ie 4e A γ h ψ ( ) M ( ) f () kˆ + + μ χ ψ ψ ψ ψ μ m m (9) Where, M J is the supercurrent density and <... > is aeraged oer Fermi surface. Eq.(8) and Eq.(9) can be reduced to the isotropic magnetic superconductors by setting... to and γ. For the uniform applied field, the upper critical field can be found from the linearised st Ginzburg-Landau equation,eq.(8).then, we get h ( T ) ξ () mα φ μh ( χ, ) c T μh c () πξ T + χ + χ ( )( ) ( ) Here ξ is the coherence length, μh c is the upper critical field that πh magnetic flux quantum ( φ ) is e of nonmagnetic superconductors. and H c ( χ, ) is upper critical field T

5 5 Eqs.(-) show that the coherence length is unchanged by the presence of the magnetic and anisotropic function. And, the presence of the reduces the upper critical field strength by a factor ( + χ ),but the effect of anisotropic function is not found. For lower critical field s calculation, we choose a flux line in cylindrical coordinate ( r)zˆ where ( r) has its maximum alue at core and tends to zero at large radial. A ector potential in cylindrical coordinate is chosen as A [ μ ( χ χ ) H r + μ ( + χ )( H + M ) r] ˆ θ A( )θ ˆ c r From the solution for the wae function ψ ( T ) proided by iφ ( ) ψ ( T ) e () ψ T (3) where φ is the phase of order parameter. y using the releant Maxwell s equation, j (for the μ D magnetostatic case, ) and nd Ginzburg-Landau equation in uniform t h field, the ector potential and magnetic flux quantum are A φ and e πh Φ. The equation for the ector potential is taken the form e μ γ ( + χ ) 4 e α 4e ( A) f ( kˆ μ α ) f mβ γ mβ ( + χ ) Let our sample has a surface perpendicular to the x-axis and the external magnetic field is zˆ, so the internal magnetic field should be the form b b( x)zˆ. Then, the London equation is of the form ( x) d b b( x). Here, λ is the London penetration depth of anisotropic dx λ magnetic superconductors γ λ μ ( + χ ) ( 4e ) f () kˆ mβ α The lower critical field can be obtained as () kˆ (4) h 4 λ lnκ (5) c e

6 6 λ Where κ is Ginzburg-Landau parameter ; κ. The lower critical field ξ Eq.(5) is the same form of Abrikoso[] but it has the difference in κ that depend on λ in Eq.(4). We introduce a dimensionless parameter, the critical magnetic field ratio as c c η (6) Substitution Eq.() and Eq.(5) in Eq.(6), we can get κ η ( + χ)lnκ 4θκ (7) ( + χ ) lnθκ In caseκ >>, the approximated critical magnetic field ratio is ( + χ )( θ κ ) θκ η (8) Here κ is the isotropic non-magnetic Ginzburg-Landau parameter; λ κ and κ θκ that λ and ξ are the penetration depth and coherence ξ γ ( ) length of Abrikoso[]. The + χ θ is the magnetic-to-anisotropic f () kˆ parameter ratio. For the pure superconductors that is the isotropic nonmagnetic superconductors, Eq.(7) can be reduced to η that κ lnκ agreed with the Ginzburg-Landau relation. 3.Results and Diuss The effect of κ, χ and θ on the critical magnetic field ratio(η )are shown in Figure. Here, we use the κ 57,and κ 8 that are of Y 3 [3] and Hg 3[4] superconductor, respectiely. We consider in 3 cases the non-magnetic superconductor case( χ ), the diamagnetic case ( χ < ) and ferromagnetic case( χ > ). The η must be the positie alues so the alue of γ is between to. The θ represents the ratio of magnetic parameter to anisotropic parameter. And θ, θ >, θ < are the isotropic non-magnetic case, the highly anisotropic case and highly magnetic case, respectiely. We find that the case χ <, the higher critical field ratio, the higher κ with higher θ is found. The diamagnetic superconductors( χ < ) with highly anisotropic case ( θ > ) shows the

7 7 highest alue and the ferromagnetic superconductor( χ > ) with highly magnetic case ( θ < ) shows the lowest alue. We also find that in the ferromagnetic superconductors, the difference between the upper critical field and the lower critical field is smaller than diamagnetic superconductors. Yada and Paulose[5] measured the lower and upper critical magnetic field of FeTe.6Se. 4 superconductor. They found the upper critical magnetic field about 65 T for mid-point cure, the lower critical 4 magnetic field about 8 Oe or.9x T,and the magnetic sueptibility χ.35x emu/gm. From Eq.(6), we can estimate the critical magnetic 5 field ratio of this material as η 5.4x. According to this η, and Eq.(7), the estimated Ginzburg-Landua parameter of this material should be between κ 337 to κ 9 for θ 5. ecause FeTe.6Se. 4 shows the anisotropic and magnetic property, we think that the κ of material should be the same order Y3 material. 4.Conclus The upper critical, the lower critical field and the critical magnetic field ratio of anisotropic magnetic superconductors are calculated by Ginzburg-Landau theory analytically. The effect of the Ginzburg-Landau parameter, magnetic sueptibility and magnetic to anisotropic parameter ratio on the critical field ratio are considered. We find that the critical field ratio of χ < case, with the higher κ,and higher θ,the higher alue of is the critical field ratio found. The diamagnetic superconductor with highly anisotropic case is shown the highest alue and the ferromagnetic superconductor with highly magnetic is shown the lowest alue of the critical magnetic field ratio. We can use the critical magnetic ratio to predicted the magnetic parameters of superconductors. Acknowledgements The authors would like to thank Professor Dr.Suthat Yoksan for the useful diuss. The financial support of the Srinakharinwirot Uniersity, Phathumwan Institute of Technology and ThEP center are acknowledged. References [] V.L.Ginzburg and L.D.Landau.Zh.Eksp.Teor.Fiz.,64(957). []A.A.Abrikosso.So.Phys-JETP5,74(957). [3] L.P.Gorko,So.Phys.-JETP 37,55(958). [4] D.P.Hampshire,Physica C 34,(998). [5] D.P.Hampshire,J.Phys.:Condens.Matter 3,695(). [6] I.N.Askerzade,Physica C 397,99(3). [7] I.N.Askerzade.JETP Letter 8,583(5).

8 8 [8] P.Udomsamuthirun,A.Changjan,C.Kumongsa,S.Yoksan.Physica C 433,6(6). [9] L.Min-Zia and G.ZiZhao,Chinese Physics 6,86(7). [] A.L.Fetter and J.D.Walecka,Quantum Theory of Many-Particle System.McGRAW-Hill 995. [] S.Haas,K.Maki,Phys.Re.65()5(R). [] E.Posazhennikoa,T.Dahm,K.Maki,Europhys.Lett. 6(3)577. [3] Z.Hao,J.R.Clem,M.W.McElfresh,L.Ciate,A.P.Malozemoff and F.Holtzberg, Phys.Re. 43(99)884. [4] Mun-Seog Kim,Myoung-Kwang ae,sung-ik Lee and W.C.Lee,Chinese Journal of Physics 33(995)69. [5] C.S.Yada and P.L.Paulose New J.Phys. (October 9) κ 8,θ.5 κ 8,θ. κ 8,θ.5 κ 57,θ.5 κ 57,θ. κ 57,θ.5 η χ Figure. The critical magnetic field ratio ersus the critical magnetic sueptibility of the anisotropic magnetic superconductors.

There are two main theories in superconductivity: Ginzburg-Landau Theory. Outline of the Lecture. Ginzburg-Landau theory

Ginzburg-Landau Theory There are two main theories in superconductivity: i Microscopic theory describes why materials are superconducting Prof. Damian Hampshire Durham University ii Ginzburg-Landau Theory