# Quasiclassical Theory for Thermodynamic Properties of the Two-Band Superconductor under Magnetic Fields

###### Abstract

Consistent account is provided of strongly anisotropic properties in thermodynamics of under magnetic field using approximate analytic solution in the quasiclassical formalism. With the strength of the pairing interactions and the Fermi velocities of two bands as fitting parameters, it is possible to reproduce in wide regions of the phase diagram the - and temperature dependences of relevant quantities such as the zero-energy density of states (ZEDOS), two energy gaps, the specific heat and the anisotropy. It is demonstrated that the velocity ratio of two bands affects considerably the low-field behavior of the ZEDOS and energy gaps. The ratio also affects the high-temperature behavior of the anisotropy. The anisotropy of the coherence length in the band depends weakly on and , whereas the coherence length in the band remains isotropic even in the presence of the inter-band coupling.

## 1 Introduction

Superconductivity in [1] has attracted much interest not only for its high transition temperature K but for the unique character coming from distinct Fermi surfaces in and bands. Owing to the presence of multiple bands, the superconducting state has two different energy gaps, although the pairing is triggered by the electron-phonon coupling in the band. The two-band nature of superconductivity has been supported by various experiments such as point contact spectroscopy[2, 3], specific heat[4, 5, 6], and the angle-resolved photoemission spectroscopy (ARPES)[7]. In addition to providing two different gaps, the and bands have different dimensionality, i.e., the former is quasi two-dimensional and the latter is rather three-dimensional. The interplay of the two gaps with different size and anisotropy should lead to characterisitic behaviors, particularly in thermodynamic quantities in a mixed state. Indeed, the strong temperature dependence of the anisotropy of the upper critical field, , has been observed[8, 9, 10] and has been discussed theoretically as a consequence of such interplay[12, 11, 13, 14, 15].

There have been a number of theoretical investigations on various thermodynamic properties in the mixed state, such as the zero-temperature specific-heat coefficient [16, 17] and the field dependence of the energy gaps[18, 19, 20]. However, validity of each theoretical tool used in these studies is limited to either or . Hence consistent account of the whole - phase diagram within the single framework is highly desired.

Recently, one of the present authors has developed a theoretical tool to investigate thermodynamic properties[22] using the approximate analytic solution in the quasiclassical formalism[24]. The approximate solution was originally proposed by Brandt, Pesch and Tewordt (BPT) in the Gor’kov formalism[23], and is most suitable to strongly type-II superconductors near . However, a comparison with reliable numerical calculations indicates that the present approximation is applicable semi-quantitatively to the wide region of the - phase diagram[22]. The purpose of this paper is to give a consistent account of thermodynamic properties based on the approximate analytic solution, especially of anisotropic behaviors in the mixed phase.

The paper is organized as follows. In the next section, we briefly introduce the method of approximate analytic solution. In 3, we first determine the plausible strength of pairing interactions at , and discuss the ratio of the Fermi velocities of the two bands focusing on the temperature dependence of the anisotropy of . Then, using the parameters obtained, we compare and dependence of the specific heat with experimental results. It is shown that the low- behavior of the zero-energy density of states (ZEDOS) and the gaps is sensitive to the ratio of the velocities of two bands. The and dependence of the anisotropy of the coherence length is also discussed. The last section summarizes the paper.

## 2 The BPT approximation

To discuss thermodynamics under magnetic fields we have to solve the Eilenberger equation, the Maxwell equation and the gap equation self-consistently. It is tedious to solve these equations as they stand. However, if we approximate the spatial dependence of the internal field and the diagonal part of the quasiclassical Green function by their spatial average, and , and assume the Abrikosov lattice for the spatial variation of the gap function, we obtain the analytic solution of the quasiclassical Green function. Using hereafter, the result is given by [22]

(1) |

where is the band index (either or ), is the fermionic Matsubara frequency, is the Faddeeva function, and is the spatial average of the energy gap in the -band. We have assumed the isotropic -wave gap for both bands. In eq.(1) we have introduced the quantity

(2) |

where is the normalized Fermi wave vector, and is related to the Fermi velocity perpendicular to the applied field . We shall give explicit expression of later.

The Maxwell equation and the gap equation follow from the stationary condition of the free energy with respect to and , respectively. In the BPT approximation the free energy measured from that of the normal state in the clean limit[22] is given by

(3) |

where

(4) |

and denotes the average over the Fermi surface. In eq.(3), is the DOS in the normal state, and is the Euler’s constant. The cut-off frequency is used in the present paper. We have assumed since is the strongly type-II superconductor with [25].

For simplicity, we use the cylindrical (spherical) Fermi surface for the two-dimensional (three-dimensional) () band. Then the Fermi velocities are given by

(5) | |||

(6) |

where () is the azimuthal (polar) angle of . Here the quasi two-dimensionality in band is taken into account simply by with . We introduce by

(7) |

for , and

(8) |

for . Here characterizes the anisotropy of the coherence length perpendicular to the field. Thus, for . For there is strong anisotropy in band due to the quasi two-dimensionality. In this case, we determine variationally by minimizing the free energy[11, 26]. In the case of the Ginzburg-Landau theory, .

Due to the coupling between and bands, it may be expected that differs from unity even if band is isotropic. It turns out, however, that the variational solution has no anisotropy in within the numerical accuracy, whereas has a strong anisotropy. This result is consistent with that obtained by Ichioka et al., in which the local DOS(LDOS) of the band is nearly isotropic while that of the band has strong anisotropy[20]. On the experimental side, almost isotropic vortex core in the band is observed by scanning tunneling spectroscopy(STS) at low fields, even in the case of [21]. On the other hand, the use of the Abrikosov lattice with different ’s in two bands results in unphysical situation since the structure of penetrating flux differs in two bands. However, after spatial average over a larger scale, this deficiency of the solution may not be serious as far as we are concerned with homogeneous properties. Since only spatially averaged quantities have a meaning in the BPT approximation, we use the variational solution, , throughout this paper.

## 3 Anisotropy of thermodynamic quantities

Let us first determine the strength of the pairing interaction in the matrix,

(9) |

by fitting the specific heat data at . Note that the present approximation recovers the ordinary two-band BCS theory[30, 31] at . We use and ( is the total DOS) from the specific heat measurement [6]. These values are in good agreement with the band structure calculation.[28] Figure 1 shows the best fitting of the dependence of the specific heat at . The coupling constants are determined to be , and . The solid circle is taken from the experimental data by Wang et al.[4] We obtain the gap magnitude at as meV and meV, which are in good agreement with those determined by tunneling spectroscopy [2, 3]. Our coupling constants are slightly different from those determined by the band structure calculations[32, 33] due to the simplification of the Fermi surfaces in the present study. In this work we try to reproduce of overall dependence of the gap magnitude, i.e., specific heat, rather than determining the coupling constants uniquely. In fact, different choice of coupling constants may give similar behaviors in [15, 29].

Next, we determine the Fermi velocities of two bands by fitting the anisotropy of the upper critical fields, . The solid line in Fig. 2 represents our result for and , while the solid circles are experimental data taken from Lyard et al.[10] The result for , is also shown in Fig. 2 (dashed line) for comparison. The slight modification of the velocity ratio gives a considerable change in especially at high temperatures. The present result for is consistent with previous theoretical works[11, 12, 13, 14, 15]. It is emphasized that the strong temperature dependence of originates from the interplay of the two gaps since the dependence becomes much weaker in the absence of the interband coupling, . Note that is independent of in the single-band anisotropic GL theory. The values of and are in good agreement with those obtained from the band structure calculation[27, 28].

With these preliminaries, we discuss the field dependence using the coupling constants and the Fermi velocities determined above. Figure 3 shows the in-plane field dependence of two gaps and the ZEDOS at . The solid circle denotes the experimental data taken from Bouquet et al.[6] Owing to the smallness of as compared with , the coherence length in band is much larger than . Indeed, we obtain and for using the BCS formula. On the other hand, the upper critical field is determined mostly by the pairing in the band. Thus, the rapid rise of the ZEDOS should result as shown in Fig. 3(a). The contribution of the band to the ZEDOS saturates already at . Note that remains nonzero up to because of the interband coupling as shown in Fig. 3(b).

The results in the case of are shown in Fig.3(c) and (d) for comparison. The change of the velocity ratio only affects the low-field behaviors of the ZEDOS and of the gaps. On the contrary, of the ratio is affected much in higher temperatures as shown in Fig. 2. Experimentally Gonneli et al.[34] report the rapid suppression of the gaps, while Bugolavsky et al.[35] report no considerable suppression at low fields. There seems sample dependence in presumably due to the impurities. There are a number of investigations on the field dependence of the ZEDOS and the gaps both theoretically[19, 18, 20] and experimentally[34, 35]. All of these show similar behaviors except in low-field region.

Once we determine the necessary parameters, we calculate various quantities under magnetic fields without any free parameters. The dependences of the specific heat under magnetic fields with are shown in Fig. 4 for (a) and (b) for . For both field directions, the overall behavior of is consistent with the experimental results indicated in the figures. In particular, the observed specific heat jump at is reproduced very well for both field directions.

Finally, we discuss the and dependence of for in-plane fields. The dependence of at is shown in Fig. 5(a), where . At low temperatures the vortex cores elongate along -axis slightly more as compared with the GL theory. In the limit of , the anisotropy recovers the GL value. The vortex cores also elongate monotonously along -axis with increase of the field as shown in Fig. 5(b). It should be noted that the and dependence of does not come from the effect of two-band superconductivity, but comes purely from the quasi two-dimensionality of the band. This conclusion follows since there is no essential difference even in the absence of the inter-band interaction, .

## 4 Summary

We have investigated the thermodynamic properties and its anisotropy of the two-band superconductivity in on the basis of the approximate analytic solution in the quasiclassical formalism. Using the strength of the pairing interactions and the Fermi velocities of two bands as fitting parameters, we have reproduced the - and - dependence of the ZEDOS, the two gaps, the specific heat, and the anisotropy of the upper critical field in wide regions of the phase diagram.

The change of the velocity ratio affects considerably the low-field behaviors of the ZEDOS and the energy gaps. At the same time the velocity ratio determines the high-temperature behavior of the anisotropy . Thus, our theory predicts a strong correlation between the low-field behaviors of the ZEDOS and the high-temperature behavior of . We have shown that the anisotropy of the coherence length in the band has and dependence different from that in the GL theory. The isotropic nature of the coherence length in the band remains even if the inter-band coupling is present.

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