Superconductivity in doped Dirac semimetals

Superconductivity in doped Dirac semimetals

Tatsuki Hashimoto Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan    Shingo Kobayashi Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan Institute for Advanced Research, Nagoya University, Nagoya 464-8601, Japan    Yukio Tanaka Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan    Masatoshi Sato Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
July 7, 2019
Abstract

We theoretically study intrinsic superconductivity in doped Dirac semimetals. Dirac semimetals host bulk Dirac points, which are formed by doubly degenerate bands, so the Hamiltonian is described by a matrix and six types of -independent pair potentials are allowed by the Fermi-Dirac statistics. We show that the unique spin-orbit coupling leads to characteristic superconducting gap structures and vectors on the Fermi surface and the electron-electron interaction between intra and interorbitals gives a novel phase diagram of superconductivity. It is found that when the inter-orbital attraction is dominant, an unconventional superconducting state with point nodes appears. To verify the experimental signature of possible superconducting states, we calculate the temperature dependence of bulk physical properties such as electronic specific heat and spin susceptibility and surface state. In the unconventional superconducting phase, either dispersive or flat Andreev bound states appear between point nodes, which leads to double peaks or single peak in the surface density of states, respectively. As a result, possible superconducting states can be distinguished by combining bulk and surface measurements.

pacs:
pacs

I Introduction

Unconventional superconductivity is one of the main topics in condensed matter physics. In the last decade, it has been revealed that surface states, called surface Andreev bound states (SABSs), and nodal structures in unconventional superconductors can be characterized by topological numbers of a bulk wave function Sato (2009, 2010); Sato et al. (2011); Brydon et al. (2011); Schnyder and Ryu (2011); Schnyder and Brydon (2015); Kitaev (2009); Ryu et al. (2010); Schnyder et al. (2008); Read and Green (2000); Tanaka et al. (2012); Alicea (2012); Sato and Fujimoto (2016).

The concept of topology has also been expanding widely in the normal state since the discovery of topological insulators (TIs) having a surface Dirac cone protected by time-reversal symmetry Hasan and Kane (2010); Qi and Zhang (2011); Ando (2013). Beside TIs, topological crystalline insulators Fu (2011); Hsieh et al. (2012), whose surface Dirac cones are protected by point-group symmetry instead of time-reversal symmetry, and Weyl semimetals Murakami (2007); Wan et al. (2011); Xu et al. (2011); Burkov and Balents (2011), which have the bulk Weyl cones, have generated great interest owing to their outstanding electronic properties and potential applications in electronic devices Plekhanov et al. (2014); Liu et al. (2014a); Zeljkovic et al. (2015).

Recent theoretical studies have revealed that the doped topological materials can be promising candidates to realize unconventional superconductivity due to their unique spin-orbit interaction Fu and Berg (2010); Kobayashi and Sato (2015) and robustness against the nonmagnetic impurities Michaeli and Fu (2012); Nagai (2015); Nagai et al. (2014); Foster et al. (2014). In particular, the superconductivity in the TIs has been studied a lot since the observation of zero-bias conductance peak suggesting the existence of SABSs in CuBiSe Sasaki et al. (2011); Hor et al. (2010); Hashimoto et al. (2013); Bay et al. (2012); Zocher and Rosenow (2013); Yamakage et al. (2012); Hao and Lee (2011); Hsieh and Fu (2012); Sasaki and Mizushima (2015); Nagai et al. (2012); Yip (2013); Hashimoto et al. (2014); Takami et al. (2014); Nagai (2014, 2015); Nagai et al. (2014); Fu and Berg (2010); Kriener et al. (2011); Brydon et al. (2014); Sasaki et al. (2014); Mizushima et al. (2014); Levy et al. (2013); Nakayama et al. (2015); Matano et al. (2015). Superconductivity in topological crystalline insulators has also been observed Bushmarina et al. (1986), and it has been predicted that exotic SABSs appear if fully gapped odd-parity superconductivity is realized Sasaki et al. (2012); Hashimoto et al. (2015). Furthermore, the realization of exotic superconductivity has been anticipated in doped Weyl semimetals Lu et al. (2015); Bednik et al. (2015); Cho et al. (2012).

In this paper, we study superconductivity in doped rotation symmetric Dirac semimetals (DSs). DSs are materials that host bulk Dirac cones Murakami et al. (2007); Murakami (2007). Several materials have been predicted to be DSs Wan et al. (2011); Wang et al. (2012, 2013); Schoop et al. (2015); Young et al. (2012), and CdAs Neupane et al. (2014); Borisenko et al. (2014); Neupane et al. (2015); Yi et al. (2014); Liu et al. (2014b); Jeon et al. (2014); He et al. (2014); Liang et al. (2015) and NaBi Liu et al. (2014c); Xu et al. (2015); Kushwaha et al. (2015) have been confirmed experimentally. Recently, superconductivity has been observed in CdAs Aggarwal et al. (2015); Wang et al. (2015); He et al. (2015). Moreover, point contact experiments for CdAs have suggested the existence of SABS Aggarwal et al. (2015); Wang et al. (2015), gathering great attention as a candidate of unconventional superconductor. In addition, two of the authors have revealed that a unique orbital texture in DSs suggests unconventional pairings ( and in this paper) Kobayashi and Sato (2015).

However, the physical properties of the superconducting states in doped DSs have not been examined systematically, and thus it has been difficult to identify the pairing symmetry experimentally. One of our purposes of this paper is to clarify the physical property and topological structure of possible superconducting states in doped DSs. Due to the presence of time-reversal symmetry and inversion symmetry, the electronic states near the Dirac points are minimally described by a 4 4 Dirac Hamiltonian with spin and orbital degrees of freedom. For the superconducting state, doubly degenerate bands allow six types of -independent Cooper pairs and the unique orbital texture favors an equal-spin pairing, giving rise to point nodes on the Fermi surface. From this view point, the superconductivity in doped DSs can be unconventional and its physical implication deserves further exploration. Superconductivity in Dirac systems has also been studied for doped TIs, Weyl semimetals, and bilayer Rashba systems. These superconducting states also show unconventional superconductivity, but we emphasize that the crystal symmetry and spin-orbit coupling of DSs are different from them. Hence the Cooper pairs respect different irreducible representations, implying that they can show unique superconducting gap structures and vectors on the Fermi surface.

Figure 1: (a) The energy dispersion of the Dirac semimetal. The bulk Dirac cones appear on the axis. (b) The double Fermi surface of the doped Dirac semimetal. Symmetry axis and plane focused in this paper are also shown.

To clarify these points, we derive an analytical formula of possible pair potentials in the band basis and reveal the superconducting gap structure and vector on the Fermi surface. It is found that the superconducting gap structure can be classified into four types, i.e., isotropic full gap, point node at poles, horizontal line node, and vertical line node. We also show that these superconducting gap structure can be interpreted from the orbital structure of the DS and possible pair potentials. Moreover, for the odd-parity pairings, the direction of the vector is either parallel to - plane or parallel to -direction. These characteristics of the superconducting states are completely different from those of other topological materials. We also solve the linearized gap equation to make a superconducting phase diagram, in which an unconventional superconducting state ( or in this paper) is realized when the inter-orbital attraction is sufficiently stronger than the intra-orbital one. To examine whether we can distinguish the possible pair potentials experimentally, we calculate the temperature dependence of the specific heat that reflects the superconducting gap structure and spin susceptibility that reflects the Van-Vleck effect and the direction of the vector. Furthermore, we calculate the surface state by using the recursive Green’s function method. The unconventional superconducting states show either dispersive or flat Andreev bound states on the surface depending on the parity of mirror-reflection symmetry. Using mirror-reflection symmetry, we discuss topological numbers relevant to zero-energy states. As a result, these physical implications conclude that the possible superconducting states can be distinguished by combining bulk and surface measurements.

This paper is organized as follows. First, we introduce a model Hamiltonian for DSs and consider possible pair potentials in Sec. II. In Sec. III, by transforming the pair potentials from the orbital basis to the band basis, we obtain a single-band description of the pair potentials. We also show the superconducting gap structure and vector on the Fermi surface. In Sec. IV, we show that the superconducting gap structure can be interpreted from the orbital structure of DSs. In Sec. V, we obtain the phase diagram for the superconducting state. Numerical results for the bulk and surface states are shown in Secs. VI and VII, respectively. In Sec. VIII, we discuss the difference between superconductivity in DSs and TIs and briefly mention superconductivity in other classes of Dirac semimetals. Finally, we summarize our results in Sec. IX.

Ii Model

DSs have both time-reversal symmetry and inversion symmetry, which lead to doubly degenerate bands. Thus, to construct a model with four-fold degenerate Dirac points, it is necessary to take into account the orbital degrees of freedom in addition to the spin degrees of freedom. In the broad sense, there are two types of DS, i.e., accidental ones and symmetry protected ones. The former type appears just on the topological phase transition point between a TI and a normal insulator Murakami (2007); Murakami et al. (2007); Tanaka et al. (2013); Souma et al. (2012). In this case, the bulk Dirac cones are easily gapped out. On the other hand, the latter type of DSs host bulk Dirac points protected by rotational symmetry on the rotational axis and topological surface states Yang and Nagaosa (2014); Yang et al. (2015). A representative example of the latter type is CdAs, where the relevant orbitals are and with a total angular momentum , the and bulk Dirac points are protected by the four fold symmetry. We discuss in the following the CdAs class DSs. In the basis set of , , , and , the low energy effective Hamiltonian for the DSs near point is described by

(1)

where and are the Pauli matrices in the spin and orbital space, respectively Wang et al. (2013); Yang and Nagaosa (2014). As summarized in Ref. Yang and Nagaosa (2014), the basis functions for four fold symmetric DSs are given as

(2)
(3)
(4)
(5)
(6)

where , , , , and are material dependent parameters. The energy dispersion is shown in Fig. 1 (a). By tuning the chemical potential, double Fermi surfaces appear as shown in Fig. 1 (b).

The crystals of CdAs belong to the point group and thus the Hamiltonian satisfies the following symmetries: (i) time-reversal symmetry: ,

(7)

(ii) inversion symmetry: ,

(8)

(iii) four-fold rotational symmetry along axis: ,

(9)

(iv) - mirror-reflection symmetry: ,

(10)

(v) - mirror-reflection symmetry: ,

(11)

(vi) (110) mirror-reflection symmetry: ,

(12)

The corresponding symmetry axis and planes are shown in Fig. 1 (b).

Next, we consider the superconducting state. We assume the following pair interaction Fu and Berg (2010):

(13)

where and are intra- and interorbital interactions, respectively, and () is the density operator for orbital . Then we construct the Bogoliubov de Gennes (BdG) Hamiltonian in the mean-field regime:

(14)
(15)

where and are the Pauli matrices in the Nambu (particle-hole) space, and denote the chemical potential and pair potential, respectively. Here, the basis is taken as . Then, we discuss possible pair potentials. For the two orbital system, there are sixteen combinations of two Pauli matrices, and ), but six combinations out of them satisfy the Fermi-Dirac statistics, which are described by , , , , , and . These pair potentials can be classified into inter- or intraorbital in addition to the spin-singlet or triplet classes. and are spin-singlet intraoribtal pairings. , , and are spin-triplet interoribtal pairings. is a spin-singlet inter-oribtal pairing. Moreover, these pair potentials are classified into four irreducible representations of the point group : ( and ), (), () and ( and ), which are summarized in Table 1. Symmetry properties of the pair potentials under the inversion , four-fold rotation , and mirror-reflection symmetry , and are also summarized in Table 1. As long as we consider the -independent pair potentials in this orbital basis, the parity of the intra- (inter-) orbital pair potentials is even (odd) under the inversion operation. It is noted that the matrix forms of the possible pair potentials are common in two-orbital or layer systems such as TIs Fu and Berg (2010), Weyl semimetals Bednik et al. (2015), and bilayer Rashba systems Nakosai et al. (2012). However, the superconductivity in the DSs is completely different from that in other materials since the normal state is different, as we see below.

spin orb. Rep.
singlet intra
singlet intra
triplet inter
triplet inter
singlet inter
triplet inter
Table 1: Possible pair potentials for the Dirac semimetals. Spin state, orbital state, irreducible representation and symmetry properties of each pairings are shown.

Iii Single band description of pair potentials : superconducting gap and spin structure

In this section, to understand the superconducting gap and spin structure on the Fermi surface, we derive the pair potentials in the band basis, where is diagonalized Yip (2013). Then, we extract the conduction or valence band components of the pair potentials in order to obtain a single-band description. First, we diagonalize the spin part of . The Hamiltonian reduces to

(16)

where and . The corresponding eigenvectors are given by

(17)
(18)

where and . By using the eigenvectors, the following relations are obtained as

(19)
(20)
(21)

where , () are the Pauli matrices for the spin helicity basis. Next, we diagonalize the orbital part. The eigenvalues are given by

(22)

where . The corresponding eigenvectors are

(23)
(24)

where , , . Finally, the eigenvectors for the normal Hamiltonian are obtained as

(25)

By using the eigenvectors , we obtain the pair potentials in the band and spin helicity basis. Then, with Eqs. (19) - (21), we transform the pair potentials from the band and spin helicity basis to the band and real spin basis: where and are band indices. The obtained results are

(26)
(27)
(28)
(29)
(30)
(31)
Figure 2: (a) - (f) Superconducting gap structure on the Fermi surface and (g) - (l) the bulk density of states for the possible pair potentials. The color on the Fermi surface indicates the magnitude of the energy gap ().
Figure 3: vector on the Fermi surface for the spin-triplet pairings in the band space. In the case of and , vectors are almost parallel to the - plane. On the other hand, in the case of and , the vectors are almost parallel to the direction. Here, we show the vector on the Fermi surface at , however, the dependence of the direction of the vector is quite small since this is induced by terms.

where are the Pauli matrices in the band basis and . In the case of the DS, the Fermi surface consists of either an electron or a hole band. If the chemical potential is large enough compared with the magnitude of the pair potential, , we can consider that the superconductivity occurs in either conduction or valence band (quasi-classical approximation). Thus the inter-band and valence (or conduction) band component can be ignored. Namely, it is sufficient to consider only the (1,1) [or (2,2)] component of . The conduction band components of pair potentials are as follows:

(32)
(33)
(34)
(35)
(36)
(37)

It is found that, in the single-band description, and are regarded as spin-singlet even-parity pairings, and , , and are spin-triplet odd-parity pairings. In the single-band description, the spin-singlet or triplet completely correspond to even or odd under the inversion operation for all pairings. Although is a spin-singlet inter-orbital pairing in the orbital basis, it is considered as a spin-triplet pairing in the band basis. In other words, the spin-triplet component of is induced by the spin-orbit interaction. In addition, if the parameters related to the spin-orbit interaction are absent , the odd-parity pairings , which means that the spin-orbit interaction is essential to realize unconventional superconductivity. As shown below, these single-band representations are useful to capture bulk superconducting properties such as the heat capacity and the spin susceptibility.

iii.1 Superconducting gap structure

In Fig. 2, we show the magnitude of the superconducting gap plotted on the Fermi surface (a)-(f) and the bulk density of state (DOS) (g)-(h). In the case of , the gap structure has an isotropic full gap, where the DOS diverges at and there is no state in . In the case of , line nodes exist in the horizontal direction. Therefore, the DOS is proportional to and divergence at is suppressed. It should be noted that the line nodes are accidental nodes, and thus, by tuning some parameters, we can remove the nodes without any topological phase transition. For and , the superconducting gap has point nodes on -axis. The DOS near is proportional to . As is seen from Fig. 2, the superconducting gaps of and are quite similar. This is because and are different only by terms, i.e., and in Eq. 1. For (), there are point nodes on the axis. In the absence of the terms, the point nodes become the line node at () for (). Although the terms change the superconducting gap structure, the superconducting gap structure of can be effectively considered as a line node, as is obvious from Figs. 2 (e) and 2 (f), since the gap opening effect of the terms is quite small compared with the -linear terms around point. Then, the DOS is proportional to at very near but the line shape is almost linear in the wide region of . The results for the superconducting gap with and without the terms considered are summarized in Table 2.

It has been revealed that if the pair potential satisfies the four fold rotational symmetry with non zero , the superconducting state inherits the invariant Dirac points of the normal state Kobayashi and Sato (2015). In the case of , , and , is non zero. Therefore, we can say that the point nodes on axis in and originate from the normal state, which are topologically protected.

SC gap SC gap -vector -vector
without terms with terms without terms with terms
FG FG none none
horizontal LNs horizontal LNs none none
PNs at poles PNs at poles - plane - component is dominant
PNs at poles PNs at poles - plane - component is dominant
vertical LN PNs at poles -axis -component is dominant
vertical LN PNs at poles -axis -component is dominant
Table 2: Superconducting (SC) gap structure and vector for the possible pair potential. FG, PN and LN stand for full gap, point node and line node, respectively.

iii.2 vector

For spin-triplet superconductors, the pair potentials can be described with vectors, which behave like three-dimensional vectors in spin space Balian and Werthamer (1963). In our basis, the vector is defined as

(38)

where For the possible pair potentials, is easily obtained from Eqs. (32) - (37). The direction of the vector on the Fermi surface is important to interpret the magnetic response of the superconductivity, the details of which are mentioned in Sec. VI. In Fig. 3, we show the vector of the spin-triplet pair potentials in the band basis, i.e., , , and . Note that we show the vector for , but the dependence of the direction of the -vector is negligible since it originates from the terms. In the case of and , the direction of the -vector is almost parallel to the - plane. Although there is an component, it is much smaller than the other components since it is induced by the terms. On the other hand, in the case of and , the direction of the vector is almost parallel to the axis, and - plane component is negligible for the same reason in the case of and .

Iv Interpretation of superconducting gap structure with orbital texture

In this section, we interpret the superconducting gap structures from the orbital texture of DSs. In the previous letter, two of the present authors have argued how the orbital texture is consistent with and Kobayashi and Sato (2015). Here we generalize the argument and explain the nodal structures of all possible pairing symmetries in terms of the orbit-momentum locking. For simplicity, we ignore the terms in this section.

First, we discuss the orbit-momentum locking of DSs. Consider the Hamiltonian in Eq. (1). The spin is already diagonalized in Eq. (1), so we can divide the Hamiltonian into spin-up and spin-down sectors as

(39)
(40)

These Hamiltonians have two characteristic features. First, the first terms in Eqs. (39) and (40) dominate on the axis. Because of the uniaxial rotational symmetry around the -direction in DSs, the orbital mixing second terms are not allowed on the axis. Second, the orbital mixing second terms become dominant away from the axis at each Dirac point. At Dirac points, both the first and the second terms vanish, but since the second terms are linear in while the first ones are quadratic, the second terms are dominant except on the axis. It should be noted that these two features are required by the symmetry of DSs.

The above features give rise to a unique orbital texture on the Fermi surfaces surrounding the Dirac points. Near the poles of the Fermi surface, the first terms in Eqs. (39) and (40) are dominant, so we have the -directed parallel orbital configuration shown in Fig. 4 (a). On the other hand, near the equators of the Fermi surfaces, the second terms are dominant, so we have the orbit-momentum locking structure in Fig. 4 (b).

Now consider the pairing states in DSs, and compare them with the orbital textures. According to the BCS theory, Cooper pairs form between electrons with opposite momenta, and , and, for , , , , (, ), they form between electrons in different (same) spin sectors. First, consider a Cooper pair between electrons in the different spin sectors. As illustrated in (i) of Fig. 4 (a), near the poles of the Fermi surface, the Cooper pair has a parallel orbital configuration in the -direction. On the other hand, on the equator of the Fermi surface, the Cooper pair has a parallel orbital configuration in the direction [(ii) in Fig. 4 (b)], or an antiparallel-orbital configuration in the direction [(iii) in Fig. 4 (b)]. It is found that these orbital configurations are consistent with . Note that is diagonal on the quantization basis of or but it is off diagonal on the quantization basis of foo (). This means that , which is proportional to , has parallel orbital configurations in and directions but an anti-parallel one in the direction, which is exactly the same as the aforementioned orbital structure of Cooper pairs in DSs. On the other hand, the other gap functions, , , and are not fully consistent with the orbital texture. As summarized in Table 3, () indicates an antiparallel-orbital pair in the () direction and a parallel-orbital pair in other directions, and indicates an antiparallel orbital pair in any direction. Therefore, for instance, is inconsistent with the orbital texture on the equator of the Fermi surfaces, so it has horizontal line nodes. In a similar manner, one can see that vertical line nodes of and come from the inconsistency between the pairings and the orbital texture of DSs.

For a Cooper pair between electrons in the same spin sector, the orbital texture in Fig. 4 (a) gives a parallel orbital configuration in the direction near the poles of the Fermi surface [(iv)], however, that in Fig. 4 (b) provides an anti-parallel orbital configuration in the and directions on the equator of the Fermi surfaces [(v), (vi)]. Since represents an antiparallel orbital pair in any direction, these orbital configurations are consistent with and on the equator of the Fermi surface, but not consistent with them near the poles. Consequently, there arise point nodes at the poles for these gap functions.

P AP P AP
AP P P AP
P P AP AP
Table 3: Orbital configuration of the possible pair potentials for each direction. P (AP) stands for parallel (anti-parallel) orbital configuration.
Figure 4: Orbital texture of the Dirac semimetal for spin-up and spin-down space. Single-headed arrows on the Fermi surface indicate orbital . Orbit-momentum locking by the first term in Eq. (1): plotted on the - plane (a). Orbit-momentum locking by the second term in Eq. (1): plotted on the - plane at (b). For a Cooper pair between electrons in the different spin sectors, the orbit-momentum locking provides the parallel (anti-parallel) orbital configuration in the - and - (-) directions, which are indicated by double-headed arrows (i) and (ii) [(iii)]. For an equal-spin Cooper pair, the orbit-momentum locking provides the parallel (anti-parallel) orbital configuration in the - (- and -) direction, which is indicated by a double-headed arrow (iv) [(v) and (vi)].

V Phase diagram

In this section, we obtain the - phase diagram by solving the linearized gap equation in a manner similar to the superconducting TI Fu and Berg (2010) and the bilayer Rashba system Nakosai et al. (2012). For simplicity, we ignore the terms in this section. The linearized gap equations for possible pair potentials are given by

(41)
(42)

Here, is the irreducible susceptibility:

(43)
(44)

where is the number of the unit cell, is the Boltzman constant, is the single-particle Green’s function for the normal state, with the projection operator onto the conduction band , and is the form factor originating from the orbital degrees of freedom. for each pair potential is given by , , , , and .

The - phase diagram is shown in Fig. 5. As is obvious from the form factors, the irreducible representations satisfy the relation: . Therefore, the - phase diagram consists of , and , and () cannot appear in the phase diagram. The phase boundary is given by

(45)

If the interorbital attraction is sufficiently stronger than the intraorbital attraction , the unconventional superconducting phase ( or ) is realized. We can expect that the Coulomb repulsion leads to stronger as is discussed in the superconducting TI Brydon et al. (2014). We can also interpret this phase diagram with the orbital structure. Since all inter-orbital pairings in Table 1 are odd under parity, one can naturally obtain an odd-parity pairing state if dominates. Then, among the odd-parity pairing states in Table 1, only and are consistent with the orbital texture on the equator of the Fermi surface. Consequently, we obtain and in the phase diagram.

Figure 5: - phase diagram for the superconducting DS, where () is the inter- (intra-) orbital attraction. The blue (green) region indicates the region where the and () has the highest stability.

Vi Bulk physical property

In this section, we obtain bulk physical properties for the possible pair potentials.

vi.1 Specific heat

Here, we calculate the temperature dependence of the electronic specific heat for the superconducting states. The temperature dependence of the specific heat reflects the superconducting gap structure Sigrist and Ueda (1991), which has been considered as useful information to determine the symmetry of pair potential experimentally Hashimoto et al. (2013); Kriener et al. (2011); Loram et al. (1993); Nomura and Yamada (2002); NishiZaki et al. (2000). The specific heat in the superconducting state is given by

(46)

where and is the eigenvalue for the BdG Hamiltonian Eq. (15) given as

(47)
(48)
(49)
(50)

Here,

(51)

and for each pair potential is

(52)
(53)
(54)
(55)
(56)
(57)

We assume that the superconducting gap has the following phenomenological temperature dependence obtained from the BCS theory:

(58)

where is the superconducting critical temperature.

In Fig. 6, we show the temperature dependence of the specific heat as a function of the temperature . In the case of , has an exponential behavior near , and the magnitude of the specific heat jump at is the largest among all possible pair potentials. On the other hand, in the case of , the superconducting gap has line nodes, which leads to a linear behavior of around . To satisfy the entropy balance, , the magnitude of the specific heat jump becomes smaller compared with . Moreover, the line shape of for is convex upward. The superconducting gap of and are different only by the term. Therefore, as is seen from Fig. 6 (b), the temperature dependencies of for