Elastic theory for the vortex-lattice melting in iron-based high-Tc superconductors

The vortex-lattice melting transitions in two typical iron-based high-Tc superconductor $Ba(Fe_{1-x}Co_{x})_{2}As_{2}$ (122-type) and$Nd(O_{1-x}F_{x})FeAs$ (1111-type) for magnetic fields both parallel and perpendicular to the anisotropy axis are studied within the elastic theory. Using the parameters from experiments, the vortex-lattice melting lines in the H-T diagram are located systematically by various groups of Lindemann numbers. It is observed that the theoretical result for the vortex melting on $Ba(Fe_{1-x}Co_{x})_{2}As_{2}$ for parallel fields agrees well the recent experimental data. The future experimental results for the vortex melting can be compared with the present theoretical prediction by tuning reasonable Lindemann numbers.


Introduction
Recently, the newly discovered iron-based superconductors have attracted considerable scientific interest both experimentally [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] and theoretically [18,19,20,21,22,23]. First, the novel superconductivity in 1111 phase F eAs superconductors was reported experimentally, giving a new path to high temperature superconductivity. The LaOF eAs under doping with F − irons at the O 2− sites was found to exhibit superconductivity with T c = 26K [1], and the superconductivity was also observed with holes doping [2]. Then T c was surprisingly increased up to above 40K when La in LaO 1−x F x F eAs was substituted by other rare earth elements [3,4]. It was quickly observed later that T c is about 55K in SmO 1−x F x F eAs [5,6] and Gd 1−x T h x OF eAs [7]. It is a first non-copper-based superconductors in which the maximum critical temperature is much higher than the theoretical value predicted from BCS theory [24]. On the other hand, the 122 phase iron-based superconductor BaF e 2 As 2 was discovered more recently [8], and the superconducting critical temperature was found to be as high as 38K by holedoping. It is observed that the electron doping of which by Co [9] and Ni [10] also induce superconductivity. Although there are some theoretical studies on ironbased superconductors, the type-II superconductivity as well as the mechanism of supeconductivity are not well understood to date.
The vortex-lattice solid (glass with random pinning) state with zero linear resistivity is crucial for the application of high-T c superconductors, thus the melting of vortexlattice in bulk type-II superconductors is of great significance [25,26,27,28,29]. The main aspect of Lindemann criterion suggests that the lattice melts when the root mean square thermal displacements of the components of a lattice reach a certain fraction of the equilibrium lattice spacing, such a criterion was first adopted to study the vortexlattice melting transition in type-II superconductors with a magnetic field parallel to the anisotropic axis, then this approach was used to draw the melting lines in the case of magnetic field perpendicular to the anisotropic axis [30,31,32,33]. Since the upper critical field is also high [13,12] in iron-based high-Tc superconductors, the thermal fluctuation may drive the vortex-lattice to a vortex liquid in a field far below the upper critical field [30] through vortex melting. We will extend the elastic theory to study the vortex melting in these newly discovered superconductors.
In this paper, using the parameters measured in recent experiments, we study the vortex-lattice melting transitions for two typical iron-based layered superconductors Ba(F e 1−x Co x ) 2 As 2 (122-type) with low anisotropy [9,12] and Nd(O 1−x F x )F eAs (1111type) with high anisotropy [4,13], in the framework of the elastic theory. The melting lines for magnetic fields both parallel and perpendicular to the anisotropic axis are systematically located with different groups of Lindemann numbers. A comparison with current experimental findings is made. The present paper is organized as follows. In Section 2, we introduce the theoretical method used in this work, Section 3 presents the main results, finally, we give a short summary.

Elastic theory
We will consider the field B both parallel and perpendicular to the anisotropy axis (i.e. c-axis in this paper), the elastic theories in both cases are presented respectively.

Thermal fluctuations
Whether the field is parallel or perpendicular to c-axis, for ideal triangular vortex-line lattice, the free energy in elastic theory can be expressed in an unified way with quadratic terms of the deviation vector u = (u x , u y ) describing the fluctuations of vortices from their equilibrium positions [25,26,30,31,32,33] The matrix C for fields parallel to the c-axis is different from that for fields perpendicular to it. We denote C c and C ab to be the elastic matrix for the fields parallel and perpendicular to c-axis, respectively, which are given as follows and In matrix C c , k 2 ⊥ = k 2 x + k 2 y , c 66 , c L , and c 44 are the wave-vector-dependent shear, bulk, and tilt elastic moduli [26,34,35], respectively, which are determined as follows and Where we have defined m 2 M is a quasi-particle effective mass in xy-plane, M z describes along c-axis, b = B/B c2 with B c2 is the upper critical field, and κ = λ ⊥ /k ⊥ . The bulk modulus is c L = c 11 − c 66 with shear modulus The matrix C ab also contains some elastic moduli. Since the anisotropy exists here, tilt and shear modulus are not isotropic any more. For example, c h 44 (k) (c e 44 (k)) is tilt modulus along (perpendicular to) c-axis. Similarly, c h 66 (c e 66 ) represents shear modulus parallel to (perpendicular to) the anisotropy axis c. For the detailed expressions for elastic moduli, one may refer to Ref. [33] The thermal fluctuations of the vortices are given by inverting the kernel C(k) as follows The integrations in Eq. (7) are over 0 < k z < ∞ and within the first Brillouin zone (BZ) for k x and k y . To be specific, we consider a lattice structure as the low temperature equilibrium state as shown in Fig. 1 of Ref. [33], where the lattice spacings are a x = a and a y = a 3/2 for fields parallel to c-axis, a x = a/ √ γ with x c and a y = a √ 3γ/2 for fields perpendicular to c-axis with γ 2 = m c /m ab and a = 2Φ 0 / √ 3B. For parallel fields, we consider the mean-square displacement of a vortex lattice from the equilibrium d 2 (T ) = u 2 x + u 2 y which can be written as For convenience, we introduce the dimensionless wave vector q = (k x /Λ x , k y /Λ y , k z /Λ), where Λ x,y are the wave numbers at the edges of the first BZ. Explicitly, they are given as Λ x = 4π/3a and Λ y = 2π/a √ 3 for fields parallel to c-axis and Λ x = 4π √ γ/3a and Λ y = 2π/a √ 3γ for fields perpendicular to c-axis. The unit of wave number in k z direction is taken as Λ = 4πB/Φ 0 = 8π/ √ 3a 2 . The thermal fluctuations are then expressed as and

Lindemann criterion
The Lindemann criterion presumes that the lattice melts, when the root mean square thermal displacements of the components of a lattice reach some fraction of the equilibrium lattice spacing. For two cases, we write the usual isotropic Lindemann criterion for parallel fields, and the anisotropic one for perpendicular fields, with c x and c y are two Lindemann numbers for two transverse directions. Combining the Lindemann criterion and the elastic theory, we can get the melting equations for two cases, and with ε the Ginzburg parameter

Layer pinning effect
When fields perpendicular to c-axis, the effect of layer pinning reduces fluctuations in both directions and induces an additional momentum-independent term to the elastic matrix in Eq.

Vortex melting in
We study the vortex melting in Ba(F e 1−x Co x ) 2 As 2 (x=0.1) as a representative of 122type iron-based superconductors [11]. The parameter measured from a most recent experiment [12] gives T c = 22K, λ ⊥ = 160nm, ξ ⊥ = 2.44nm, H c c2 (T = 0) = 50T, H ab c2 (T = 0) = 70T, and s = 23A. The anisotropic parameter γ falls from 2.0 to 1.5 with the decrease of temperature. In our calculation, we observe that the melting lines determined in the framework of elastic theory only change slightly with the variation of γ (in the range of 1.5 ∼ 2.0). So we set γ = 2 in our calculations.
We first calculate the melting line for the parallel fields with various Lindemann numbers. It is interesting to note in Fig. 1 that the vortex melting line for Ba(F e 1−x Co x ) 2 As 2 obtained in this paper is well consistent with the irreversible line measured experimentally [12] with Lindemann number c = 0.2. The irreversibility line in superconductor is usually regarded as the melting line. [30] Then we study the vortex melting for perpendicular fields. Due to the laking of the experimental data until now, three groups of Lindemann numbers are selected to     Setting C x = C y , we obtain two curves as shown in Figs. 2(a-c), as in Ref. [31]. As pointed out in Ref. [32], to interpret these two curves as two " melting lines " and thus reach to a conclusion of an intermediate phase is unphysical, since the elastic theory can give at best one melting line. To impose the same Lindemann number along the two directions is of no physical basis. In order to achieve a single melting line, one can tune the Lindemann numbers in the two directions. A good collapse of the melting lines in two directions can be achieved if setting the ratio c x /c y ≈ 1.33. It is interesting to note that this ratio is very close to 1.37 observed in Ref. [33] using parameters of cuprate superconductors.
The vortex melting for fields perpendicular to the c-axis is also influenced by the layer pinning. In order to study this intrinsic layer pinning effect, the matrix (15) is used to calculate the thermal fluctuations along two transverse directions. The melting lines for two groups of Lindemann number (C x = 0.1, C y = 0.1, C y = 0.075; C x = 0.2, C y = 0.2, C y = 0.15) are collected in Fig. 3. A good collapse can also be obtained by setting the same ration c x /c y ≈ 1.33. The future experimental data can be also compared with these values.
Similarly, we first calculate the melting lines for the fields parallel to c-axis, as shown in Fig. 4. Since the melting line has not been explicitly reported, the data at which the resistance reaches the small percentage (0.5%) of the normal-state resistance in Ref. [15] is regarded the approximated melting one here, which is also collected in Fig. 4. Interestingly, the theoretical melting line with c = 0.18 is consistent with the experimental data in a considerable wide temperature range. The discrepancy between experiment and theory becomes larger at high magnetic fields and low temperatures. A trend of temperature independent melting data is shown at high magnetic fields in experiments, implying that the dimensional crossover occurs at this regime. If the vortex system is of two dimensional nature at high magnetic fields, the melting line is almost independent of the temperature [30], Next, We calculate the melting lines when the fields perpendicular to c-axis, which are shown in Fig. 5. For perpendicular fields, three different groups of Linderman We also investigate the intrinsic layer pinning effect on the vortex melting in Nd(O 1−x F x )F eAs sample. The results are exhibited in Fig. 6. Surprisingly, we observe an intersection between two melting lines with two transverse directions for typical Lindemann numbers, which is not shown in the above 122 type. Obviously, a collapse of these two curves could not be achieved by setting any ration c x /c y . It is implied that the intermediate smectic phase [31,32,33] may exist in Nd(O 1−x F x )F eAs. However, its identification is beyond the phenomenological Lindemann theory. It is interesting to note that, when the exponentially weak intrinsic pinning was taken into account for a weakly anisotropic Ba(F e 1−x Co x ) 2 As 2 , the vortex melting behavior is only changed slightly, but for the strongly anisotropic Nd(O 1−x F x )F eAs, the intrinsic pinning would play a much more significant role.

Summary
In terms of the elastic theory, we have studied the thermal fluctuations in two typical iron-based superconductors Ba(F e 1−x Co x ) 2 As 2 and Nd(O 1−x F x )F eAs for magnetic fields parallel and perpendicular to the anisotropy axis. Using the parameters of these superconductors from experiments, the melting lines with various Lindemann numbers are composed. Interestingly, we can derive the melting line observed in both Ba(F e 1−x Co x ) 2 As 2 [12] and Nd(O 1−x F x )F eAs [15] for magnetic field parallel to the c−axis. Neglecting the layer pinning effect, it is found that thermal fluctuations normalized by vortex separations in the two transverse directions for parallel field are proportional to each other, similar to those observed in cuprate superconductors [32,33]. Interestingly, the intrinsic layer pinning play much more important role on the vortex melting in Nd(O 1−x F x )F eAs than in Ba(F e 1−x Co x ) 2 As 2 . More experimental works are motivated to confirm our prediction by comparing the melting dada with the interpolation one presented here. The present prediction may in turn provide a guide or reference to locate the melting line in future experiments.