Research Article | Open Access

Tsadik Kidanemariam, Gebregziabher Kahsay, "Theoretical Study of Upper Critical Magnetic Field () in Multiband Iron Based Superconductors", *Advances in Condensed Matter Physics*, vol. 2016, Article ID 5470429, 10 pages, 2016. https://doi.org/10.1155/2016/5470429

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

**Academic Editor:**Prasenjit Guptasarma

#### Abstract

This research work focuses on the theoretical investigation of the upper critical magnetic field, ; Ginzburg-Landau coherence length, ; and Ginzburg-Landau penetration depth, , for the two-band iron based superconductors , , and LiFeAs. By employing the phenomenological Ginzburg-Landau (GL) equation for the two-band superconductors , , and LiFeAs, we obtained expressions for the upper critical magnetic field, ; GL coherence length, ; and GL penetration depth, , 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, , and perpendicular, , to the symmetry axis (-direction) versus temperature are plotted. We also plotted the phase diagrams of the upper critical magnetic field, versus the angle . Similarly, the phase diagrams of the GL coherence length, , and GL penetration depth, , parallel and perpendicular to the symmetry axis versus temperature are drawn for the superconductors mentioned above. Our findings are in agreement with experimental observations.

#### 1. Introduction

The discovery of superconductivity in iron based materials by Kamihara et al. [1] at K in the iron-based superconductor was a welcomed surprise and this brought to an end the monopoly of cuprates as the only high temperature superconductors. The 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 which is the highest for noncuprate superconductors [3]. Among these new iron based superconductors, the iron based F-doped layered quaternary compound is a good candidate for this study since it is the same family as and has the same crystal structure. Previous extensive transport and heat capacity studies on 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: K at at ambient pressure [4].

Superconductivity was also obtained in the 122-family iron based multiband superconductors and great efforts were made to raise to temperatures higher than 38 K as detected in [5] and for optimally doped at at [6]. The isovalent doping of phosphorus for arsenic in suppresses magnetic ordering without changing the number of Fe(3d) electrons and induces superconductivity with a maximum value of . The isovalently P-doped 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 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 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 field 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 , 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 and when the magnetic field is applied within the -plane and -axis , 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) and orbitals [17]. One of the main features of two-band superconductors is the presence of two energy gaps, and , which vanish at the same critical temperature, . According to the microscopic theory, the presence of the two gaps is explained by the fact that in each band “” an intraband coupling constant and the interband coupling constant exist which on the one hand enhances pairing of electrons and on the other hand leads to the single critical temperature, .

#### 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 parameters and can be written as [18]wherewhere and are the free energies for each band. is the interaction free energy term between the two bands. is the energy stored in the local magnetic fields. and are superconducting order parameters. The coefficients and 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. and denote the effective mass of carriers for each band. is the GL temperature-dependent parameter and is GL temperature-independent parameter. is the external magnetic field and is the vector potential.

Now, substituting (2) into (1), we getIn 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 , , such that Near the critical temperature, , we have Thus, using (3) in (4), the first GL equations for the two bands can be obtained asEquation (5) can be expressed in terms of matrix product form asWe can compare (6) to the free-particle time independent Schrödinger equation with particle of mass and charge of moving in a magnetic field which is the same as eigenvalues of the harmonic oscillator. For the field in the direction of the -axis, we can linearize the GL equations as follows:Hence, (6) becomesThe energy eigenvalues of the quantum harmonic oscillator are given by In order to determine , we must choose which gives the upper critical magnetic field. The energy of the two bands for the state becomes and where the frequency of the oscillator is given by For a displacement in one direction, the vector potential is given by .

Thus, (8) can be written asBy considering , the upper critical magnetic field is obtained by setting the determinant of the matrix in (9) to zero; that is,from which we getFinally, the solution of the quadratic equation becomesNow let , , and , where , , and are the first band, second band, and interband effective coherence lengths, respectively, , where is the gradients of interband mixing of two order parameters in energy unit and and is the quantum flux.

Hence, (12) becomesNow, let us consider cases as follows.

*Case 1. *ConsiderIn 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. *ConsiderUsing Taylor’s binomial series expansions, for the negative value of the term in the square root in (13) we getFor the positive value of the square root in (13) we getDue 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 . 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 haveThe effect of anisotropy mass tensor on the upper critical magnetic field is included by replacing in (18) with coherence length tensor, .

Finally we have

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, ; GL coherence length, ; GL penetration depth, ; and GL characteristic parameter, .

Thus, the GL effective coherence length, , can be expressed asTherefore, (19) becomesIf the direction of the applied magnetic field is parallel to the -axis, then the expression for the upper critical magnetic field, , as function of temperature in (21) is given bywhere is the zero temperature upper critical field parallel to the -axis.

If the direction of the applied magnetic field is perpendicular to the -axis, then the expression for the upper critical magnetic field, , in (21) is given bywhere is the zero temperature upper critical field perpendicular to the -axis.

Using the experimental values and [19] at K, the mathematical expressions of the temperature dependence of the upper critical magnetic field parallel to the symmetry axis (in the -direction) and perpendicular to the symmetry axis in becomeGinzburg-Landau (GL) theory [20] can be used to determine the effect of an anisotropic effective mass on the angular dependence asFinally, the angular dependence of the upper critical magnetic field for at an angle from the -axis can be expressed asIt has been shown that a single band anisotropic model can properly describe the angular dependence of in a multiband system near [21].

Similarly, the current model/theory can be applied to 1111-family of iron based two-band superconductor 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 -axis) in for and [7] at K becomeThe angular dependence of the upper critical magnetic field at an angle from the -axis can be expressed asFurthermore, 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 -axis) in LiFeAs for and [22] at K becomeThe angular dependence of the upper critical magnetic field at an angle from the -axis can be expressed asThe expression for the anisotropy parameter is given by . Thus, we calculate the values of to be 2.14, 4.34, and 1.6 for , , 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, , is defined asThis parameter specifies the spatial width of the transition layer of the order parameter, , 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, , can be written as a function of temperature in the vicinity of the critical temperature, , asHere, is the temperature-independent positive constant.

Hence, the expression for the temperature dependent GL coherence length, , becomeswhere is the zero temperature coherence length. Using the experimental values of the zero temperature coherence lengths nm and nm [19], the temperature dependent coherence length in the -plane, , and along the -axis, , for becomesSimilarly, using the experimental values of the zero temperature coherence lengths nm and nm [24], the temperature dependent coherence length in the -plane, , and along the -axis, , for becomesFurthermore, using the experimental values of the zero temperature coherence lengths nm and nm [22], the temperature dependent coherence length in the -plane, , and along the -axis, , for LiFeAs becomesAccording 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, , and can be expressed in terms of the superconducting electron density, , asFor , (total electron density), thus, we haveIn the two-fluid theory of Gorter and Casimir [25] we can haveNow, combining (41), (42), and (43) we haveUsing the experimental value nm [26] for , the expressions for the temperature dependent penetration depth in the -plane, , and along the -axis, , becomeSimilarly, using the experimental value nm [24], the expressions for the temperature dependent penetration depth in the -plane, , and along the -axis, , for becomeUsing the experimental values nm and nm [27], the expressions for the temperature dependent penetration depth in the -plane, , and along the -axis, , for LiFeAs becomeThe GL parameters in the -axis and -plane are given by and , respectively [28].

Using the experimental values at K for , becomesUsing the experimental values at for , becomesSimilarly, at K for LiFeAs, becomesAs is well known, Abrikosov showed that the breakdown point between type I and type II superconductors is [12]. If the value of , then the superconductor is said to be type I and is type II if . Therefore, from (48) to (50), we conclude that the iron based superconductors , , 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 of and parallel and perpendicular to the symmetry axis by using the GL approach in two-band iron based superconductors , , and LiFeAs.

Firstly, using (27), (29), and (31) the phase diagram of versus is plotted for iron based superconductors , , and LiFeAs, respectively, as shown in Figure 1. As can be seen from Figure 1, the upper critical magnetic field is maximum in the -plane and minimum in the -axis for , , and LiFeAs.

Secondly, using (24), (28), and (30) and by taking some available experimental data, the phase diagrams of and versus temperature are plotted for , , 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 , , and LiFeAs.

Thirdly, the expressions for the temperature dependent and for , , 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 and 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: , for , , and LiFeAs.

Finally, using (45), (46), and (47) and taking the experimental values, we plotted the phase diagrams of and 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, for , , 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 superconductors , , 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 , , and LiFeAs. This indicates that the upper critical magnetic field along -plane, , is quite different from the upper critical magnetic field along -axis, . 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 , , 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 , , and LiFeAs as shown in Figure 1. Finally, we conclude that the model we used works for (122), (1111), and LiFeAs (111) and the results we obtained are in agreement with experimental findings [4, 19, 22, 27, 29–32].

#### Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

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#### Copyright

Copyright © 2016 Tsadik Kidanemariam and Gebregziabher Kahsay. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.