# Theoretical Study of Upper Critical Magnetic Field (HC2) in Multiband Iron Based Superconductors

Authors: Tsadik Kidanemariam; Gebregziabher Kahsay
Journal: Advances in Condensed Matter Physics
 (2016)

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## Abstract

This research work focuses on the theoretical investigation of the upper critical magnetic field,HC2; Ginzburg-Landau coherence length, ξGL(T); and Ginzburg-Landau penetration depth, λGL(T), for the two-band iron based superconductors BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs. By employing the phenomenological Ginzburg-Landau (GL) equation for the two-band superconductors BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs, we obtained expressions for the upper critical magnetic field, HC2; GL coherence length, ξGL; and GL penetration depth, λGL, as a function of temperature and the angular dependency of upper critical magnetic field. By using the experimental values in the obtained expressions, phase diagrams of the upper critical magnetic field parallel, HC2∥c, and perpendicular, HC2⊥c, to the symmetry axis (c-direction) versus temperature are plotted. We also plotted the phase diagrams of the upper critical magnetic field, HC2 versus the angle θ. Similarly, the phase diagrams of the GL coherence length, ξGL, and GL penetration depth, λGL, parallel and perpendicular to the symmetry axis versus temperature are drawn for the superconductors mentioned above. Our findings are in agreement with experimental observations.


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## Body

## 1. Introduction

The discovery of superconductivity in iron based materials by Kamihara et al. [1] at TC=26 K in the iron-based superconductor LaFeAsO1-xFx was a welcomed surprise and this brought to an end the monopoly of cuprates as the only high temperature superconductors. The TC of F-doped LaOFeAs was reported to increase with pressure and reach a maximum of about 43 K at 4 GPa, obtained from electrical resistance measurements under high pressure [2]. Up to now, the critical temperature for iron based superconductors has been raised to 57.3 K for Sm0.95La0.05O0.85F0.15FeAs which is the highest for noncuprate superconductors [3]. Among these new iron based superconductors, the iron based F-doped layered quaternary compound NdO1-xFxFeAs is a good candidate for this study since it is the same family as LaFeAsO1-xFx and has the same crystal structure. Previous extensive transport and heat capacity studies on NdO1-xFxFeAs single crystal revealed that at zero external magnetic fields the compound was found to undergo a transition from the normal state to the superconducting state at superconducting critical temperature: TC=51 K at x=0.18 at ambient pressure [4].Superconductivity was also obtained in the 122-family iron based multiband superconductors and great efforts were made to raiseTC to temperatures higher than 38 K as detected in (Ba0.6K0.4)Fe2As2 [5] and for optimally doped BaFe2(As1-xPx)2 at TC=31K at x=0.32 [6]. The isovalent doping of phosphorus for arsenic in BaFe2(As1-xPx)2 suppresses magnetic ordering without changing the number of Fe(3d) electrons and induces superconductivity with a maximum value of TC=31K. The isovalently P-doped BaFe2(As1-xPx)2 system is particularly suitable to study the detailed evolution of the electronic properties. The isovalent doping does not add any charge carriers to the system and the dopant changes the electronic structure mainly because of differences in ion size.The discovery of superconductivity in the 111-family of iron based superconductor, LiFeAs, is reported by Wang et al. [7]. The compound crystallizes into a structure containing [FeAs] conducting layer that is interlaced with Li charge reservoir. Superconductivity was observed with TC up to 18 K [7] in this compound. The two-band nature of the order parameter in LiFeAs is confirmed by heat capacity measurements [8] and by ab initio calculations [9]. Several methods have been used to determine the temperature dependence and anisotropy of HC2 for LiFeAs and the first results were obtained from resistivity measurements [10]. Among the iron based superconductors, LiFeAs shows a relatively small upper critical magnetic field.Soon, after the discovery of superconductivity in iron based superconductors, it was found that the iron based superconductors possess a large superconducting critical magnetic fieldHC2 and low anisotropy [11] when compared to the cuprates. The theory of upper critical magnetic field in superconductors was essentially developed right after Abrikosov proposed the idea of type II superconductors [12]. Determining the upper critical magnetic field HC2, where superconductivity ceases in a type II superconductor, is one of the most important steps for gathering and understanding of unconventional superconductivity including the pairing mechanism, the pairing strength, and the coherence length. In an anisotropic type II superconductors, the magnetic field destroys superconductivity at the upper critical magnetic fields HC2ab and HC2c when the magnetic field is applied within the ab-plane H∥ab and c-axis H∥c, respectively. It is well known that the Ginzburg-Landau theory remains a powerful instrument for the study of magnetic phase diagram of superconductors [13]. The free energy functional of two-band superconductors can be expressed by the power series of order parameters in the vicinity of the critical temperature and minimization of the free energy function gives the GL equations that can describe the field distribution in superconductors.In conventional superconductors, the electron pairing is mediated by an electron-phonon interaction and can be well understood within the microscopic-model developed by Bardeen, Cooper, and Schrieffer (BCS) in the theory of superconductivity [14]. The electron-phonon coupling mechanism is not sufficient to explain the superconductivity in iron based superconductors and their role is rather speculated by spin-fluctuation coupling mechanism.The main specialty of iron based superconductors is their multiband nature. In the multiband model of superconductivity, several intraband and interband interaction terms are present. The importance of interband pairing in multiband models has been emphasized by de Menezes [15]. The Fermi surface, which has contributions from all five possible Fe(3d) orbitals [16], is very similar to the different iron based compounds and all the five Fe(3d) orbitals pass through the Fermi surface. The first-principle calculations have shown that the energy bands near the Fermi surface are mainly contributed by the Fe(3d) dxz and dyz orbitals [17]. One of the main features of two-band superconductors is the presence of two energy gaps, Δ1 and Δ2, which vanish at the same critical temperature, TC. According to the microscopic theory, the presence of the two gaps is explained by the fact that in each band “i” an intraband coupling constant Vii and the interband coupling constant Vij exist which on the one hand enhances pairing of electrons and on the other hand leads to the single critical temperature, TC.


## 2. Theoretical Formulations

### 2.1. Calculation of Upper Critical Magnetic Field

In the presence of two order parameters in a bulk isotropic s-wave superconductor, the phenomenological GL free energy density functional for two-band superconducting order parametersΨ1 and Ψ2 can be written as [18](1)FSC=F1+F2+F12+H28π,where(2)F1=-ℏ22m1∗∇-2ie∗AℏcΨ12-α1Ψ12+β12Ψ14,F2=-ℏ22m2∗∇-2ie∗AℏcΨ22-α2Ψ22+β22Ψ24,F12=ϵΨ1Ψ2∗+Ψ2Ψ1∗+ϵ1∇+2ie∗Aℏc∗Ψ1∗∇-2ie∗AℏcΨ2+∇+2ie∗Aℏc∗Ψ2∗∇-2ie∗AℏcΨ1,where F1 and F2 are the free energies for each band. F12 is the interaction free energy term between the two bands. H2/8π is the energy stored in the local magnetic fields. Ψ1 and Ψ2 are superconducting order parameters. The coefficients ϵ and ϵ1 describe the interband interaction of the two order parameters (proximity effect) and the interband mixing of gradients of two order parameters (drag effect) [18], respectively. m1∗ and m2∗ denote the effective mass of carriers for each band. α is the GL temperature-dependent parameter and βi is GL temperature-independent parameter. H=∇×A is the external magnetic field and A is the vector potential.Now, substituting (2) into (1), we get(3)FSC=-ℏ22m1∗∇-2ie∗AℏcΨ12-α1Ψ12+β12Ψ14-ℏ22m2∗∇-2ie∗AℏcΨ22-α2Ψ22+β22Ψ24+ϵΨ1Ψ2∗+Ψ2Ψ1∗+ϵ1∇+2ie∗Aℏc∗Ψ1∗∇-2ie∗AℏcΨ2+∇+2ie∗Aℏc∗Ψ2∗∇-2ie∗AℏcΨ1+H28π.In order to derive the GL equations for the two-gap superconductor, we minimize the free energy function given in (3) with respect to variations in the complex conjugate of the order parameters Ψ1∗, Ψ2∗, such that (4)∂FSC∂Ψ1∗=0,∂FSC∂Ψ2∗=0.Near the critical temperature, TC, we have Ψ1,23→0. Thus, using (3) in (4), the first GL equations for the two bands can be obtained as(5)-ℏ22m1∗∇-2ie∗Aℏc2Ψ1-α1Ψ1+ϵΨ2+ϵ1∇-2ie∗Aℏc2Ψ2=0,-ℏ22m2∗∇-2ie∗Aℏc2Ψ2-α2Ψ2+ϵΨ1+ϵ1∇-2ie∗Aℏc2Ψ1=0.Equation (5) can be expressed in terms of matrix product form as(6)H11H12H21H22Ψ1Ψ2=-ℏ22m1∗∇-2ie∗Aℏc2-α1ϵ+ϵ1∇-2ie∗Aℏc2ϵ+ϵ1∇-2ie∗Aℏc2-ℏ22m2∗∇-2ie∗Aℏc2-α2Ψ1Ψ2=0.We can compare (6) to the free-particle time independent Schrödinger equation with particle of mass mi and charge of e∗ moving in a magnetic field which is the same as eigenvalues of the harmonic oscillator. For the field in the direction of the c-axis, we can linearize the GL equations as follows:(7)H11Ψ1=-ℏ22m1∗∇-2ie∗Aℏc2Ψ1=E11Ψ1,H22Ψ2=-ℏ22m2∗∇-2ie∗Aℏc2Ψ2=E22Ψ2.Hence, (6) becomes(8)E11-α1ϵ+ϵ1∇-2ie∗Aℏc2ϵ+ϵ1∇-2ie∗Aℏc2E22-α2Ψ1Ψ2=0.The energy eigenvalues of the quantum harmonic oscillator are given by En=n+1/2ℏωi. In order to determine HC2, we must choose n=0 which gives the upper critical magnetic field. The energy of the two bands for the n=0 state becomes E11=(1/2)ℏω1 and E22=(1/2)ℏω2 where the frequency of the oscillator is given by ωi=2e∗HC2/mi∗c. For a displacement x in one direction, the vector potential is given by A=HC2x.Thus, (8) can be written as(9)ℏe∗HC2m1∗c-α1ϵ-ϵ12e∗HC2ℏcϵ-ϵ12e∗HC2ℏcℏe∗HC2m2∗c-α2Ψ1Ψ2=0.By considering m1∗=m2∗=m∗, the upper critical magnetic field HC2 is obtained by setting the determinant of the matrix in (9) to zero; that is,(10)ℏe∗HC2m∗c-α1ϵ-ϵ12e∗HC2ℏcϵ-ϵ12e∗HC2ℏcℏe∗HC2m∗c-α2=0,from which we get(11)e∗ℏm∗c2-4e∗2ϵ12ℏ2c2HC22-e∗ℏm∗cα1+α2-4e∗ϵϵ1ℏcHC2+α1α2-ϵ2=0.Finally, the solution of the quadratic equation becomes(12)HC2=e∗ℏ/m∗cα1+α2-4e∗ϵϵ1/ℏc±e∗2ℏ2/m∗2c2α12+α22-2e∗2ℏ2/m∗2c2α1α2-8e∗2ϵϵ1/m∗c2α1+α2+16e∗2ϵ12/ℏ2c2α1α2+4e∗2ℏ2ϵ2/m∗2c22e∗ℏ/m∗c2-4e∗2ϵ12/ℏ2c2.Now let α1=ℏ2/2m∗ξGL12, α2=ℏ2/2m∗ξGL22, and ϵ=ℏ2/2m∗ξGL122, where ξGL1, ξGL2, and ξGL12 are the first band, second band, and interband effective coherence lengths, respectively, ϵ1=υℏ2/2m∗, where υ is the gradients of interband mixing of two order parameters in energy unit and Φ0=2πℏ/e∗ and is the quantum flux.Hence, (12) becomes(13)HC2=Φ02π1-υ212ξGL12+12ξGL22-υξGL122±Φ02π1-υ212ξGL12+12ξGL22-υξGL1222-1-υ21ξGL12ξGL22-1ξGL1241/2.Now, let us consider cases as follows.Case 1.
Consider(14)12ξGL12+12ξGL22-υξGL1222≪1-υ21ξGL12ξGL22-1ξGL124,In this case the term in the square root in (13) becomes complex. Therefore, we can conclude that the first case is not valid since magnetic field is a real quantity.Case 2.
Consider(15)12ξGL12+12ξGL22-υξGL1222≫1-υ21ξGL12ξGL22-1ξGL124.Using Taylor’s binomial series expansions, for the negative value of the term in the square root in (13) we get(16)HC2-=Φ02π121/ξGL12ξGL22-1/ξGL1241/2ξGL12+1/2ξGL22-υ/ξGL122-18Φ02π1-υ21/ξGL12ξGL22-1/ξGL12421/2ξGL12+1/2ξGL22-υ/ξGL1223.For the positive value of the square root in (13) we get(17)HC2+=Φ02π1-υ212ξGL12+12ξGL22-υξGL122-Φ02π121/ξGL12ξGL22-1/ξGL1241/2ξGL12+1/2ξGL22-υ/ξGL122+18Φ02π1-υ21/ξGL12ξGL22-1/ξGL12421/2ξGL12+1/2ξGL22-υ/ξGL1223.Due to the drag effect, the GL theory for a two-band superconductor can be reduced to the GL theory for an effective single-band superconductor. Two-band model can be reduced to one band model if we set α2=ϵ=ϵ1=0. We observe that (16) cannot be reduced to one band model. So it does not have a physical meaning. But (17) can be reduced to one band model and is the analytical equation of upper critical magnetic field and thus we have(18)HC2=Φ02πξGL122ξGL122ξGL12+ξGL122ξGL22-2υ11-υ2-ξGL122/ξGL12ξGL22-1ξGL122/ξGL12+ξGL122/ξGL22-2υ2+1-υ2ξGL124/ξGL12ξGL22-12ξGL122/ξGL12+ξGL122/ξGL22-2υ4.The effect of anisotropy mass tensor on the upper critical magnetic field is included by replacing ξ in (18) with coherence length tensor, ξ.
Finally we have(19)HC2=Φ02πξGL122sin2⁡θ+γ2cos2⁡θξGL122ξGL12+ξGL122ξGL22-2υ11-υ2-ξGL122/ξGL12ξGL22-1ξGL122/ξGL12+ξGL122/ξGL22-2υ2+1-υ2ξGL124/ξGL12ξGL22-12ξGL122/ξGL12+ξGL122/ξGL22-2υ4.The interband mixing of gradients of two order parameters in an isotropic bulk two-band superconductor plays important role. The two-band superconductor is characterized with a single upper critical magnetic field,HC2; GL coherence length, ξGL; GL penetration depth, λGL; and GL characteristic parameter, κ.Thus, the GL effective coherence length,ξGLeff, can be expressed as(20)1ξGLeff2=1ξGL122ξGL122ξGL12+ξGL122ξGL22-2υ11-υ2-ξGL122/ξGL12ξGL22-1ξGL122/ξGL12+ξGL122/ξGL22-2υ2+1-υ2ξGL124/ξGL12ξGL22-12ξGL122/ξGL12+ξGL122/ξGL22-2υ4.Therefore, (19) becomes(21)HC2T=Φ02πξGLeff2Tsin2⁡θ+γ2cos2⁡θ.If the direction of the applied magnetic field is parallel to the c-axis, then the expression for the upper critical magnetic field, HC2∥c, as function of temperature in (21) is given by(22)HC2∥cT=Φ02πξGLab201-TTC2,where HC2∥c(0)=Φ0/2πξGLab20 is the zero temperature upper critical field parallel to the c-axis.If the direction of the applied magnetic field is perpendicular to thec-axis, then the expression for the upper critical magnetic field, Hc2⊥c(T), in (21) is given by(23)HC2⊥cT=Φ02πξGLab0ξGLc01-TTC2,where HC2⊥c(0)=Φ0/2πξGLab0ξGLc0 is the zero temperature upper critical field perpendicular to the c-axis.Using the experimental valuesHC2∥c(0)=36T and HC2⊥c(0)=77T [19] at TC=31 K, the mathematical expressions of the temperature dependence of the upper critical magnetic field parallel to the symmetry axis (in the c-direction) HC2∥c(T) and perpendicular to the symmetry axis HC2⊥c(T) in BaFe2(As1-xPx)2 become(24)HC2∥cT=36Tesla1-TTC2,HC2⊥cT=77Tesla1-TTC2.Ginzburg-Landau (GL) theory [20] can be used to determine the effect of an anisotropic effective mass m∗ on the angular dependence as(25)HC2GLθsin⁡θHC2⊥c2+HC2GLθcos⁡θHC2∥c2=1.Finally, the angular dependence of the upper critical magnetic field for BaFe2(As1-xPx)2 at an angle θ from the c-axis can be expressed as(26)HC2GLθ=HC2∥ccos2⁡θ+γ-2sin2⁡θ,(27)HC2GLθ=36cos2⁡θ+0.2186sin2⁡θ.It has been shown that a single band anisotropic model can properly describe the angular dependence of HC2(θ) in a multiband system near TC [21].Similarly, the current model/theory can be applied to 1111-family of iron based two-band superconductorNdO1-xFxFeAs by considering its experimental values. The expressions of the temperature dependence of the upper critical magnetic field parallel and perpendicular to the symmetry axis (in the c-axis) in NdO1-xFxFeAs for HC2∥c(0)=70T and HC2⊥c(0)=304T [7] at TC=51 K become(28)HC2∥cT=70Tesla1-TTC2,HC2⊥cT=304Tesla1-TTC2.The angular dependence of the upper critical magnetic field at an angle θ from the c-axis can be expressed as(29)HC2GLθ=70cos2⁡θ+0.053sin2⁡θ.Furthermore, the model/theory can be applied to the 111-family of iron based superconductor LiFeAs by considering its experimental values. The expressions of the temperature dependence of the upper critical magnetic field parallel and perpendicular to the symmetry axis (in the c-axis) in LiFeAs for HC2∥c(0)=15T and HC2⊥c(0)=24.2T [22] at TC=18 K become(30)HC2∥cT=15Tesla1-TTC2,HC2⊥cT=24.2Tesla1-TTC2.The angular dependence of the upper critical magnetic field at an angle θ from the c-axis can be expressed as(31)HC2GLθ=15cos2⁡θ+0.38sin2⁡θ.The expression for the anisotropy parameter (γ) is given by γ=HC2⊥c/HC2∥c. Thus, we calculate the values of γ to be 2.14, 4.34, and 1.6 for BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs, respectively.


### 2.2. Calculation of Ginzburg-Landau Coherence Length and Penetration Depth

There are two important temperature dependent material parameters which arise from the GL model that characterize the phenomenological properties of a superconductor. The coherence length,ξGL, is defined as(32)ξGLT=ℏ22m∗αT1/2.This parameter specifies the spatial width of the transition layer of the order parameter, Ψ, in the neighborhood of the boundary between a normal region and a superconducting region.Using the assumption of Changjan and Udomsamuthirun [23], the GL temperature-dependent parameter, α, can be written as a function of temperature in the vicinity of the critical temperature, TC, as(33)αiT≈α01-TTC2.Here, α0 is the temperature-independent positive constant.Hence, the expression for the temperature dependent GL coherence length,ξGL(T), becomes(34)ξGLT=ξGL01-TTC2-1/2,where ξGL(0) is the zero temperature coherence length. Using the experimental values of the zero temperature coherence lengths ξGLab0=3.2 nm and ξGLc0=1.3 nm [19], the temperature dependent coherence length in the ab-plane, ξGLab, and along the c-axis, ξGLc, for BaFe2(As1-xPx)2 becomes(35)ξGLabT=3.2nm1-TTC2-1/2,(36)ξGLcT=1.3nm1-TTC2-1/2.Similarly, using the experimental values of the zero temperature coherence lengths ξGLab0=3.7 nm and ξGLc0=0.9 nm [24], the temperature dependent coherence length in the ab-plane, ξGLab, and along the c-axis, ξGLc, for NdO1-xFxFeAs becomes(37)ξGLabT=3.7nm1-TTC2-1/2,(38)ξGLcT=0.9nm1-TTC2-1/2.Furthermore, using the experimental values of the zero temperature coherence lengths ξGLab0=4.8 nm and ξGLc0=1.7 nm [22], the temperature dependent coherence length in the ab-plane, ξGLab, and along the c-axis, ξGLc, for LiFeAs becomes(39)ξGLabT=4.8nm1-TTC2-1/2,(40)ξGLcT=1.7nm1-TTC2-1/2.According to the Meissner effect, an external magnetic field is expelled completely from the interior of a superconductor. But there was experimental evidence that magnetic fields penetrate a superconductor and a surface current flows in a very thin layer of thickness which is called the GL penetration depth, λGL, and can be expressed in terms of the superconducting electron density, ns, as(41)λGLT=m∗c216πe∗2ns1/2.For T→0, ns→n (total electron density), thus, we have(42)λGL0=m∗c216πe∗2n1/2.In the two-fluid theory of Gorter and Casimir [25] we can have(43)nsn=1-TTC4.Now, combining (41), (42), and (43) we have(44)λGLT=λGL01-TTC4-1/2.Using the experimental value λGLab0=101 nm [26] for BaFe2(As1-xPx)2, the expressions for the temperature dependent penetration depth in the ab-plane, λGLab, and along the c-axis, λGLc, become(45)λGLabT=101nm1-TTC4-1/2,λGLcT=216nm1-TTC4-1/2.Similarly, using the experimental value λGLab0=200 nm [24], the expressions for the temperature dependent penetration depth in the ab-plane, λGLab, and along the c-axis, λGLc, for NdO1-xFxFeAs become(46)λGLabT=200nm1-TTC4-1/2,λGLcT=868.6nm1-TTC4-1/2.Using the experimental values λGLab0=198.4 nm and λGLc0=250 nm [27], the expressions for the temperature dependent penetration depth in the ab-plane, λGLab, and along the c-axis, λGLc, for LiFeAs become(47)λGLabT=198.4nm1-TTC4-1/2,λGLcT=250nm1-TTC4-1/2.The GL parameters in the c-axis and ab-plane are given by κGLc=λGLab/ξGLab and κGLab=λGLabλGLc/ξGLabξGLc, respectively [28].Using the experimental values atT=0 K for BaFe2(As1-xPx)2, κGL becomes(48)κGLc=31.6,κGLab=72.4.Using the experimental values at T=0K for NdO1-xFxFeAs, κGL becomes(49)κGLc=54,κGLab=228.4.Similarly, at T=0 K for LiFeAs, κGL becomes(50)κGLc=41.3,κGLab=78.As is well known, Abrikosov showed that the breakdown point between type I and type II superconductors is κ=1/2 [12]. If the value of κ<1/2, then the superconductor is said to be type I and is type II if κ>1/2. Therefore, from (48) to (50), we conclude that the iron based superconductors BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs are type II superconductors.


## 2.1. Calculation of Upper Critical Magnetic Field

In the presence of two order parameters in a bulk isotropic s-wave superconductor, the phenomenological GL free energy density functional for two-band superconducting order parametersΨ1 and Ψ2 can be written as [18](1)FSC=F1+F2+F12+H28π,where(2)F1=-ℏ22m1∗∇-2ie∗AℏcΨ12-α1Ψ12+β12Ψ14,F2=-ℏ22m2∗∇-2ie∗AℏcΨ22-α2Ψ22+β22Ψ24,F12=ϵΨ1Ψ2∗+Ψ2Ψ1∗+ϵ1∇+2ie∗Aℏc∗Ψ1∗∇-2ie∗AℏcΨ2+∇+2ie∗Aℏc∗Ψ2∗∇-2ie∗AℏcΨ1,where F1 and F2 are the free energies for each band. F12 is the interaction free energy term between the two bands. H2/8π is the energy stored in the local magnetic fields. Ψ1 and Ψ2 are superconducting order parameters. The coefficients ϵ and ϵ1 describe the interband interaction of the two order parameters (proximity effect) and the interband mixing of gradients of two order parameters (drag effect) [18], respectively. m1∗ and m2∗ denote the effective mass of carriers for each band. α is the GL temperature-dependent parameter and βi is GL temperature-independent parameter. H=∇×A is the external magnetic field and A is the vector potential.Now, substituting (2) into (1), we get(3)FSC=-ℏ22m1∗∇-2ie∗AℏcΨ12-α1Ψ12+β12Ψ14-ℏ22m2∗∇-2ie∗AℏcΨ22-α2Ψ22+β22Ψ24+ϵΨ1Ψ2∗+Ψ2Ψ1∗+ϵ1∇+2ie∗Aℏc∗Ψ1∗∇-2ie∗AℏcΨ2+∇+2ie∗Aℏc∗Ψ2∗∇-2ie∗AℏcΨ1+H28π.In order to derive the GL equations for the two-gap superconductor, we minimize the free energy function given in (3) with respect to variations in the complex conjugate of the order parameters Ψ1∗, Ψ2∗, such that (4)∂FSC∂Ψ1∗=0,∂FSC∂Ψ2∗=0.Near the critical temperature, TC, we have Ψ1,23→0. Thus, using (3) in (4), the first GL equations for the two bands can be obtained as(5)-ℏ22m1∗∇-2ie∗Aℏc2Ψ1-α1Ψ1+ϵΨ2+ϵ1∇-2ie∗Aℏc2Ψ2=0,-ℏ22m2∗∇-2ie∗Aℏc2Ψ2-α2Ψ2+ϵΨ1+ϵ1∇-2ie∗Aℏc2Ψ1=0.Equation (5) can be expressed in terms of matrix product form as(6)H11H12H21H22Ψ1Ψ2=-ℏ22m1∗∇-2ie∗Aℏc2-α1ϵ+ϵ1∇-2ie∗Aℏc2ϵ+ϵ1∇-2ie∗Aℏc2-ℏ22m2∗∇-2ie∗Aℏc2-α2Ψ1Ψ2=0.We can compare (6) to the free-particle time independent Schrödinger equation with particle of mass mi and charge of e∗ moving in a magnetic field which is the same as eigenvalues of the harmonic oscillator. For the field in the direction of the c-axis, we can linearize the GL equations as follows:(7)H11Ψ1=-ℏ22m1∗∇-2ie∗Aℏc2Ψ1=E11Ψ1,H22Ψ2=-ℏ22m2∗∇-2ie∗Aℏc2Ψ2=E22Ψ2.Hence, (6) becomes(8)E11-α1ϵ+ϵ1∇-2ie∗Aℏc2ϵ+ϵ1∇-2ie∗Aℏc2E22-α2Ψ1Ψ2=0.The energy eigenvalues of the quantum harmonic oscillator are given by En=n+1/2ℏωi. In order to determine HC2, we must choose n=0 which gives the upper critical magnetic field. The energy of the two bands for the n=0 state becomes E11=(1/2)ℏω1 and E22=(1/2)ℏω2 where the frequency of the oscillator is given by ωi=2e∗HC2/mi∗c. For a displacement x in one direction, the vector potential is given by A=HC2x.Thus, (8) can be written as(9)ℏe∗HC2m1∗c-α1ϵ-ϵ12e∗HC2ℏcϵ-ϵ12e∗HC2ℏcℏe∗HC2m2∗c-α2Ψ1Ψ2=0.By considering m1∗=m2∗=m∗, the upper critical magnetic field HC2 is obtained by setting the determinant of the matrix in (9) to zero; that is,(10)ℏe∗HC2m∗c-α1ϵ-ϵ12e∗HC2ℏcϵ-ϵ12e∗HC2ℏcℏe∗HC2m∗c-α2=0,from which we get(11)e∗ℏm∗c2-4e∗2ϵ12ℏ2c2HC22-e∗ℏm∗cα1+α2-4e∗ϵϵ1ℏcHC2+α1α2-ϵ2=0.Finally, the solution of the quadratic equation becomes(12)HC2=e∗ℏ/m∗cα1+α2-4e∗ϵϵ1/ℏc±e∗2ℏ2/m∗2c2α12+α22-2e∗2ℏ2/m∗2c2α1α2-8e∗2ϵϵ1/m∗c2α1+α2+16e∗2ϵ12/ℏ2c2α1α2+4e∗2ℏ2ϵ2/m∗2c22e∗ℏ/m∗c2-4e∗2ϵ12/ℏ2c2.Now let α1=ℏ2/2m∗ξGL12, α2=ℏ2/2m∗ξGL22, and ϵ=ℏ2/2m∗ξGL122, where ξGL1, ξGL2, and ξGL12 are the first band, second band, and interband effective coherence lengths, respectively, ϵ1=υℏ2/2m∗, where υ is the gradients of interband mixing of two order parameters in energy unit and Φ0=2πℏ/e∗ and is the quantum flux.Hence, (12) becomes(13)HC2=Φ02π1-υ212ξGL12+12ξGL22-υξGL122±Φ02π1-υ212ξGL12+12ξGL22-υξGL1222-1-υ21ξGL12ξGL22-1ξGL1241/2.Now, let us consider cases as follows.Case 1.
Consider(14)12ξGL12+12ξGL22-υξGL1222≪1-υ21ξGL12ξGL22-1ξGL124,In this case the term in the square root in (13) becomes complex. Therefore, we can conclude that the first case is not valid since magnetic field is a real quantity.Case 2.
Consider(15)12ξGL12+12ξGL22-υξGL1222≫1-υ21ξGL12ξGL22-1ξGL124.Using Taylor’s binomial series expansions, for the negative value of the term in the square root in (13) we get(16)HC2-=Φ02π121/ξGL12ξGL22-1/ξGL1241/2ξGL12+1/2ξGL22-υ/ξGL122-18Φ02π1-υ21/ξGL12ξGL22-1/ξGL12421/2ξGL12+1/2ξGL22-υ/ξGL1223.For the positive value of the square root in (13) we get(17)HC2+=Φ02π1-υ212ξGL12+12ξGL22-υξGL122-Φ02π121/ξGL12ξGL22-1/ξGL1241/2ξGL12+1/2ξGL22-υ/ξGL122+18Φ02π1-υ21/ξGL12ξGL22-1/ξGL12421/2ξGL12+1/2ξGL22-υ/ξGL1223.Due to the drag effect, the GL theory for a two-band superconductor can be reduced to the GL theory for an effective single-band superconductor. Two-band model can be reduced to one band model if we set α2=ϵ=ϵ1=0. We observe that (16) cannot be reduced to one band model. So it does not have a physical meaning. But (17) can be reduced to one band model and is the analytical equation of upper critical magnetic field and thus we have(18)HC2=Φ02πξGL122ξGL122ξGL12+ξGL122ξGL22-2υ11-υ2-ξGL122/ξGL12ξGL22-1ξGL122/ξGL12+ξGL122/ξGL22-2υ2+1-υ2ξGL124/ξGL12ξGL22-12ξGL122/ξGL12+ξGL122/ξGL22-2υ4.The effect of anisotropy mass tensor on the upper critical magnetic field is included by replacing ξ in (18) with coherence length tensor, ξ.
Finally we have(19)HC2=Φ02πξGL122sin2⁡θ+γ2cos2⁡θξGL122ξGL12+ξGL122ξGL22-2υ11-υ2-ξGL122/ξGL12ξGL22-1ξGL122/ξGL12+ξGL122/ξGL22-2υ2+1-υ2ξGL124/ξGL12ξGL22-12ξGL122/ξGL12+ξGL122/ξGL22-2υ4.The interband mixing of gradients of two order parameters in an isotropic bulk two-band superconductor plays important role. The two-band superconductor is characterized with a single upper critical magnetic field,HC2; GL coherence length, ξGL; GL penetration depth, λGL; and GL characteristic parameter, κ.Thus, the GL effective coherence length,ξGLeff, can be expressed as(20)1ξGLeff2=1ξGL122ξGL122ξGL12+ξGL122ξGL22-2υ11-υ2-ξGL122/ξGL12ξGL22-1ξGL122/ξGL12+ξGL122/ξGL22-2υ2+1-υ2ξGL124/ξGL12ξGL22-12ξGL122/ξGL12+ξGL122/ξGL22-2υ4.Therefore, (19) becomes(21)HC2T=Φ02πξGLeff2Tsin2⁡θ+γ2cos2⁡θ.If the direction of the applied magnetic field is parallel to the c-axis, then the expression for the upper critical magnetic field, HC2∥c, as function of temperature in (21) is given by(22)HC2∥cT=Φ02πξGLab201-TTC2,where HC2∥c(0)=Φ0/2πξGLab20 is the zero temperature upper critical field parallel to the c-axis.If the direction of the applied magnetic field is perpendicular to thec-axis, then the expression for the upper critical magnetic field, Hc2⊥c(T), in (21) is given by(23)HC2⊥cT=Φ02πξGLab0ξGLc01-TTC2,where HC2⊥c(0)=Φ0/2πξGLab0ξGLc0 is the zero temperature upper critical field perpendicular to the c-axis.Using the experimental valuesHC2∥c(0)=36T and HC2⊥c(0)=77T [19] at TC=31 K, the mathematical expressions of the temperature dependence of the upper critical magnetic field parallel to the symmetry axis (in the c-direction) HC2∥c(T) and perpendicular to the symmetry axis HC2⊥c(T) in BaFe2(As1-xPx)2 become(24)HC2∥cT=36Tesla1-TTC2,HC2⊥cT=77Tesla1-TTC2.Ginzburg-Landau (GL) theory [20] can be used to determine the effect of an anisotropic effective mass m∗ on the angular dependence as(25)HC2GLθsin⁡θHC2⊥c2+HC2GLθcos⁡θHC2∥c2=1.Finally, the angular dependence of the upper critical magnetic field for BaFe2(As1-xPx)2 at an angle θ from the c-axis can be expressed as(26)HC2GLθ=HC2∥ccos2⁡θ+γ-2sin2⁡θ,(27)HC2GLθ=36cos2⁡θ+0.2186sin2⁡θ.It has been shown that a single band anisotropic model can properly describe the angular dependence of HC2(θ) in a multiband system near TC [21].Similarly, the current model/theory can be applied to 1111-family of iron based two-band superconductorNdO1-xFxFeAs by considering its experimental values. The expressions of the temperature dependence of the upper critical magnetic field parallel and perpendicular to the symmetry axis (in the c-axis) in NdO1-xFxFeAs for HC2∥c(0)=70T and HC2⊥c(0)=304T [7] at TC=51 K become(28)HC2∥cT=70Tesla1-TTC2,HC2⊥cT=304Tesla1-TTC2.The angular dependence of the upper critical magnetic field at an angle θ from the c-axis can be expressed as(29)HC2GLθ=70cos2⁡θ+0.053sin2⁡θ.Furthermore, the model/theory can be applied to the 111-family of iron based superconductor LiFeAs by considering its experimental values. The expressions of the temperature dependence of the upper critical magnetic field parallel and perpendicular to the symmetry axis (in the c-axis) in LiFeAs for HC2∥c(0)=15T and HC2⊥c(0)=24.2T [22] at TC=18 K become(30)HC2∥cT=15Tesla1-TTC2,HC2⊥cT=24.2Tesla1-TTC2.The angular dependence of the upper critical magnetic field at an angle θ from the c-axis can be expressed as(31)HC2GLθ=15cos2⁡θ+0.38sin2⁡θ.The expression for the anisotropy parameter (γ) is given by γ=HC2⊥c/HC2∥c. Thus, we calculate the values of γ to be 2.14, 4.34, and 1.6 for BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs, respectively.


## 2.2. Calculation of Ginzburg-Landau Coherence Length and Penetration Depth

There are two important temperature dependent material parameters which arise from the GL model that characterize the phenomenological properties of a superconductor. The coherence length,ξGL, is defined as(32)ξGLT=ℏ22m∗αT1/2.This parameter specifies the spatial width of the transition layer of the order parameter, Ψ, in the neighborhood of the boundary between a normal region and a superconducting region.Using the assumption of Changjan and Udomsamuthirun [23], the GL temperature-dependent parameter, α, can be written as a function of temperature in the vicinity of the critical temperature, TC, as(33)αiT≈α01-TTC2.Here, α0 is the temperature-independent positive constant.Hence, the expression for the temperature dependent GL coherence length,ξGL(T), becomes(34)ξGLT=ξGL01-TTC2-1/2,where ξGL(0) is the zero temperature coherence length. Using the experimental values of the zero temperature coherence lengths ξGLab0=3.2 nm and ξGLc0=1.3 nm [19], the temperature dependent coherence length in the ab-plane, ξGLab, and along the c-axis, ξGLc, for BaFe2(As1-xPx)2 becomes(35)ξGLabT=3.2nm1-TTC2-1/2,(36)ξGLcT=1.3nm1-TTC2-1/2.Similarly, using the experimental values of the zero temperature coherence lengths ξGLab0=3.7 nm and ξGLc0=0.9 nm [24], the temperature dependent coherence length in the ab-plane, ξGLab, and along the c-axis, ξGLc, for NdO1-xFxFeAs becomes(37)ξGLabT=3.7nm1-TTC2-1/2,(38)ξGLcT=0.9nm1-TTC2-1/2.Furthermore, using the experimental values of the zero temperature coherence lengths ξGLab0=4.8 nm and ξGLc0=1.7 nm [22], the temperature dependent coherence length in the ab-plane, ξGLab, and along the c-axis, ξGLc, for LiFeAs becomes(39)ξGLabT=4.8nm1-TTC2-1/2,(40)ξGLcT=1.7nm1-TTC2-1/2.According to the Meissner effect, an external magnetic field is expelled completely from the interior of a superconductor. But there was experimental evidence that magnetic fields penetrate a superconductor and a surface current flows in a very thin layer of thickness which is called the GL penetration depth, λGL, and can be expressed in terms of the superconducting electron density, ns, as(41)λGLT=m∗c216πe∗2ns1/2.For T→0, ns→n (total electron density), thus, we have(42)λGL0=m∗c216πe∗2n1/2.In the two-fluid theory of Gorter and Casimir [25] we can have(43)nsn=1-TTC4.Now, combining (41), (42), and (43) we have(44)λGLT=λGL01-TTC4-1/2.Using the experimental value λGLab0=101 nm [26] for BaFe2(As1-xPx)2, the expressions for the temperature dependent penetration depth in the ab-plane, λGLab, and along the c-axis, λGLc, become(45)λGLabT=101nm1-TTC4-1/2,λGLcT=216nm1-TTC4-1/2.Similarly, using the experimental value λGLab0=200 nm [24], the expressions for the temperature dependent penetration depth in the ab-plane, λGLab, and along the c-axis, λGLc, for NdO1-xFxFeAs become(46)λGLabT=200nm1-TTC4-1/2,λGLcT=868.6nm1-TTC4-1/2.Using the experimental values λGLab0=198.4 nm and λGLc0=250 nm [27], the expressions for the temperature dependent penetration depth in the ab-plane, λGLab, and along the c-axis, λGLc, for LiFeAs become(47)λGLabT=198.4nm1-TTC4-1/2,λGLcT=250nm1-TTC4-1/2.The GL parameters in the c-axis and ab-plane are given by κGLc=λGLab/ξGLab and κGLab=λGLabλGLc/ξGLabξGLc, respectively [28].Using the experimental values atT=0 K for BaFe2(As1-xPx)2, κGL becomes(48)κGLc=31.6,κGLab=72.4.Using the experimental values at T=0K for NdO1-xFxFeAs, κGL becomes(49)κGLc=54,κGLab=228.4.Similarly, at T=0 K for LiFeAs, κGL becomes(50)κGLc=41.3,κGLab=78.As is well known, Abrikosov showed that the breakdown point between type I and type II superconductors is κ=1/2 [12]. If the value of κ<1/2, then the superconductor is said to be type I and is type II if κ>1/2. Therefore, from (48) to (50), we conclude that the iron based superconductors BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs are type II superconductors.


## 3. Results and Discussion

In this study we obtained expressions for the angular dependence of upper critical magnetic field, temperature dependence of the upper critical magnetic field parallel and perpendicular to the symmetry axis, the temperature dependence ofξGL and λGL parallel and perpendicular to the symmetry axis by using the GL approach in two-band iron based superconductors BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs.Firstly, using (27), (29), and (31) the phase diagram of HC2 versus θ is plotted for iron based superconductors BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs, respectively, as shown in Figure 1. As can be seen from Figure 1, the upper critical magnetic field is maximum in the ab-plane ±90° and minimum in the c-axis 0° for BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs.Figure 1
Variation of upper critical magnetic fieldHC2 with the angle θ for BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs.Secondly, using (24), (28), and (30) and by taking some available experimental data, the phase diagrams of HC2∥c and HC2⊥c versus temperature are plotted for BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs, respectively, as shown in Figure 2. As can be seen from Figure 2, the upper critical magnetic field decreases with increasing temperature and has nonlinear dependence for BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs.Figure 2
The temperature dependence of the upper critical magnetic fieldsHC2⊥c (solid line) and HC2∥c (dashed line) for BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs.Thirdly, the expressions for the temperature dependentξGLab and ξGLc for BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs are given in (35) and (36); (37) and (38); (39) and (40), respectively. Using those equations and by considering some available experimental data, the phase diagrams of ξGLab and ξGLc versus temperature are plotted as shown in Figure 3. As can be seen in Figure 3, the GL coherence length increases with temperature which subsequently diverges at the critical temperature: ξGL(T→TC)→∞, for BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs.Figure 3
The temperature dependence of the GL coherence lengthsξGLab (solid line) and ξGLc (dashed line) for BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs.Finally, using (45), (46), and (47) and taking the experimental values, we plotted the phase diagrams of λGLab and λGLc versus temperature as shown in Figure 4. As can be seen in Figure 4, the GL penetration depth increases with temperature and diverges at the critical temperature, λGL(T→TC)→∞ for BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs.Figure 4
The temperature dependence of the GL penetration depthsλGLc (solid line) and λGLab (dashed line) for BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs.


## 4. Conclusion

In this research work, we used two-band GL approach in order to determine the upper critical magnetic field, GL coherence length, and GL penetration depth for the two-band iron based superconductorsBaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs. According to the mathematical computations, the coherence length is strongly temperature dependent, so that it can cause temperature dependency and anisotropy nature on the upper critical magnetic field of BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs. This indicates that the upper critical magnetic field along ab-plane, HC2⊥c, is quite different from the upper critical magnetic field along c-axis, HC2∥c. We plotted upper critical magnetic field versus temperature phase diagrams and the critical magnetic field decays with increasing temperature as shown in Figure 2. We also plotted GL coherence length and GL penetration depth versus temperature for BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs and we obtained that the characteristic length increases with temperature and each diverges to infinite at its critical temperature as shown in Figures 3 and 4. Furthermore, we plotted the angular dependence of the upper critical magnetic field for BaFe2(As1-xPx)2, NdO1-xFxFeAs, and LiFeAs as shown in Figure 1. Finally, we conclude that the model we used works for BaFe2(As1-xPx)2 (122), NdO1-xFxFeAs (1111), and LiFeAs (111) and the results we obtained are in agreement with experimental findings [4, 19, 22, 27, 29–32].



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