Identification of pairing mechanism leading to ferromagnetic superconductivity is one of the most challenging issues in condensed matter physics. Although different models have been proposed to explain this phenomenon, a quantitative understanding about this pairing is yet to be achieved. Using the localized-itinerant model, we find that in ferromagnetic superconducting materials both triplet pairing and singlet pairing of electrons are possible through magnon exchange depending upon whether the Debye cut off frequency of magnons is greater or lesser than the Hund's coupling (*J*) multiplied by average spin (*S*) per site. Taking into account the repulsive interaction due to the existence of paramagnons, we also find an expression for effective interaction potential between a pair of electrons with opposite spins. We apply the developed formalism in case of UGe_{2} and URhGe. The condition of singlet pairing is found to be fulfilled in these cases, as was previously envisaged by Suhl [Suhl, Phys. Rev. Lett. 87, 167007 (2001)]. We compute the critical temperatures of URhGe at ambient pressure and of UGe_{2} under different pressures for the first time through BCS equation. Thus, this work outlines a very simple way to evaluate critical temperature in case of a superconducting system. A close match with the available experimental results strongly supports our theoretical treatment.

## I. INTRODUCTION

The discovery of superconductivity in ferromagnetic UGe_{2} under high pressure is of special significance in the study of condensed matter,^{1,2} since ferromagnetism and superconductivity have been considered to be mutually exclusive phenomena. Subsequently, a number of materials like ZrZn_{2,} URhGe are found to exhibit ferromagnetism as well as superconductivity.^{3,4} This phenomenon is not explicable from the viewpoint of standard BCS theory. Thus it sets a foundation for the investigation of the underlying theory of superconductivity in presence of spin fluctuations. Different theoretical models have been proposed to explain this fact but the mechanism of Cooper pairing remains elusive till now. Initially, it was supposed that the same electrons are responsible for ferromagnetism as well as superconductivity. In this direction, Fay and Appel tried to describe the phenomenon as coupling due to the longitudinal spin fluctuation or exchange of paramagnons.^{5} Nevertheless, this theory meets some difficulties and fails to explain the absence of superconductivity in the paramagnetic phase.^{6} To account for the properties of ferromagnetic superconductors, Karchev puts forward a theory of magnon exchange superconductivity on the basis of itinerant model of electrons.^{7} The order parameter in this case is a spin antiparallel component of a spin-1 triplet with zero spin projection. According to Karchev, attractive interaction between electrons occurs due to magnon exchange, whereas repulsive interaction originates from the exchange of paramagnons. The competition between these two mechanisms leads to the existence of two successive phases, namely ferromagnetism and paramagnetism.

UGe_{2} is considered as the archetypal material where we see a coexistence of ferromagnetism and superconductivity at elevated pressure. It has been argued that the 5*f* electrons in Uranium compounds are of both localized and itinerant nature, which is also known as dualism of 5*f* electrons. In the case of UGe_{2}, this fact has been established by muon-spin-relaxation studies (*μ*SR) performed by different groups.^{8,9} Moreover, this dualism is confirmed by magnetic susceptibility, magnetization, electrical resistivity, magnetoresistivity, and specific-heat measurements in single crystal of UGe_{2}, carried out in a wide range of temperature and magnetic field.^{10} In this scenario, Karchev's proposal of considering electrons in UGe_{2} are of completely itinerant nature is quite ambiguous. Suhl and Abrikosov suggested that s-wave superconductivity may result from the electron interaction mediated by ferromagnetically aligned localized spins.^{11,12} This is analogous to the magnetic ordering arising from the coupling of two localized spins via one conduction electron. Unlike UGe_{2}, another ferromagnetic material URhGe shows superconductivity at ambient pressure. Despite of significant difference in space group and detailed structure, there are a number of similarities in UGe_{2} and URhGe. Both the materials can be viewed as arising out of zigzag arrangements of uranium atoms.^{13} Neutron scattering study with URhGe reveals that the magnetization is almost entirely due to uranium 5*f* electrons.^{13}

In this work, we adopted the localized-itinerant model and considered the interaction of itinerant electrons through the localized electrons having spins aligned in the ferromagnetic ordering. We find that this interaction potential may give rise to unconventional superconductivity with triplet or singlet pairing of electrons depending upon the Debye cut off frequency of magnons. Since the electrons near the Fermi surface mostly take part in Cooper pair formation, we approximate the interaction potential to a constant value in a very short range of electronic energy near the Fermi surface and find an expression for the critical temperature of superconductivity using the BCS equation. Taking UGe_{2} as a representative of ferromagnetic superconductors, we compute the Hund's coupling and the nearest neighbor exchange coupling and density of state (DOS) at the Fermi surface under normal pressure and also in higher pressures by varying the atomic positions according to the diffraction pattern.

This paper is organized as follows. First we find the interaction potential between electrons via magnon using localized-itinerant model in section II. The analytical expression for the critical temperature of superconductivity is also derived in a more general way. In section III we compute the Hund's coupling and the nearest neighbor exchange coupling and density of state (DOS) at Fermi surfaces of URhGe and UGe_{2} under normal pressure and also in higher pressures by varying the atomic positions. Finally, the computation of critical temperatures of the ferromagnetic superconductor UGe_{2} and URhGe are performed using the developed formalism within density functional theory (DFT) framework. The superconducting critical temperature of URhGe is in very well agreement with experimental result. The maximum critical temperature of superconductivity of UGe_{2} and its dependence on pressure is also found to be in good agreement with experimentally determined values.

## II. MAGNON MEDIATED COOPER PAIR FORMATION

It has already been mentioned that the 5*f* electrons in UGe_{2} are known to have both itinerant and localized nature. Hence, UGe_{2} can be viewed as a two-subset electronic system. The present work considers the interaction of itinerant electrons to form the Cooper pair, in the background of ferromagnetically ordered localized electrons. Here, the localized 5*f* electrons are responsible for the ferromagnetic moment. The transverse fluctuations in this array of localized spin-moments generate magnons. These magnons are considered to mediate the coupling of the itinerant electrons, leading to an unconventional superconducting behavior.^{10} Taking the electron-magnon interaction into account one can write the Hamiltonian of the electron-magnon system in terms of electron and magnon creation and annihilation operators as^{14}

where *H*_{0} is the unperturbed Hamiltonian given by the following expression

The interaction Hamiltonian, *H*_{em} is given by

with

In the above, |$\varepsilon _{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over k} \sigma }$|$\u025bk\u21c0\sigma $ is the energy of electrons with spin (σ) measured from the Fermi level, *E*_{F} is energy of the electrons at the Fermi level, |$\omega _{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over q} }$|$\omega q\u21c0$ is the magnon energy with wave vector |$\bar q$|$q\xaf$, *J* is the Hund's coupling constant at a spin site, *N* is the number of sites and *m** is the effective mass of the electron. In ferromagnetic substances the magnon frequency can be written in the following way

Here |$\gamma _{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over q} } = z^{ - 1} \sum\limits_{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over a} } {e^{i\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over q}.\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over a} } }$|$\gamma q\u21c0=z\u22121\u2211a\u21c0eiq\u21c0.a\u21c0$, *J*_{nn} is the exchange coupling constant between nearest neighbor spin sites at a distance *a* and *z* is the number of nearest neighbors. In case of straight or zigzag chain systems, from the expression of magnon frequency given by Eq. (6), one can infer that Debye cut off frequency of magnons, i.e., the maximum frequency of magnons can be written as

The Hamiltonian is conveniently diagonalized through a canonical transformation *e*^{−A}*He*^{A}, with |$\langle {n| A |m} \rangle = \frac{{\langle {n| {H_{em} } |m} \rangle }}{{E_m - E_n }}$|$\u27e8n|A|m\u27e9=\u27e8n|Hem|m\u27e9Em\u2212En$, to obtain the effective electron-electron interaction via magnon.^{15} For a ferromagnetic system at absolute zero temperature we take either |*n*⟩ or |*m*⟩ as vacuum state (|0⟩). Using the relations |$\varepsilon _{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over k} + \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over q} \downarrow } \approx \varepsilon _{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over k} \downarrow } + \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over v} _F \bullet \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over q}$|$\u025bk\u21c0+q\u21c0\u2193\u2248\u025bk\u21c0\u2193+v\u21c0F\u2022q\u21c0$ and |$\varepsilon _{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over k} \downarrow } - \varepsilon _{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over k} \uparrow } = JS$|$\u025bk\u21c0\u2193\u2212\u025bk\u21c0\u2191=JS$, for coupling of electrons with opposite wave vectors, the diagonalized Hamiltonian of the system can be written as

with |$\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over q} = \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over k} {}^\prime - \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over k}$|$q\u21c0=k\u21c0\u2032\u2212k\u21c0$. The wave function |$b_{k,\sigma \sigma ^\prime } = \langle {c_{ - \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over k} \sigma } c_{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over k} \sigma ^\prime } } \rangle$|$bk,\sigma \sigma \u2032=\u27e8c\u2212k\u21c0\sigma ck\u21c0\sigma \u2032\u27e9$ of a Cooper pair with spins *σ* and *σ′* respectively can be separated into an orbital part and a spin part as |$b_{k,\sigma \sigma ^\prime } = \phi (\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over k}, - \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over k})\chi _{\sigma \sigma ^\prime }$|$bk,\sigma \sigma \u2032=\varphi (k\u21c0,\u2212k\u21c0)\chi \sigma \sigma \u2032$. For singlet pairing with antiparallel spins, the spin part |$\chi _{ \uparrow \downarrow } = ( { \uparrow \downarrow - \downarrow \uparrow } )/\sqrt 2$|$\chi \u2191\u2193=(\u2191\u2193\u2212\u2193\u2191)/2$ is antisymmetric with respect to the interchange of spins, while the orbital part |$\phi (\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over k}, - \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over k})$|$\varphi (k\u21c0,\u2212k\u21c0)$ is symmetric with respect to the interchange of wave vectors. On the other hand, for triplet pairing with antiparallel spins the symmetric spin part is |$\chi _{ \uparrow \downarrow } = ( { \uparrow \downarrow + \downarrow \uparrow } )/\sqrt 2$|$\chi \u2191\u2193=(\u2191\u2193+\u2193\u2191)/2$ and the orbital part is antisymmetric. Using these symmetries one can write the interaction potential for singlet and triplet pairing of electrons with antiparallel spins as

This magnon mediated electron-electron interaction can be represented through Figure 1.

In itinerant model of electrons, the existence a repulsive interaction between electrons due to longitudinal spin fluctuations (i.e., paramagnons) has already been pointed out.^{7} In our formalism under the framework of localized-itinerant model, we are interested in evaluation of the interaction between itinerant electrons. Hence, we consider the effect of the repulsive interaction. The repulsive interaction between electrons with opposite spins can be expressed in terms of the interaction potential as

with |$\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over q} = \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over k} {}^\prime - \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over k}$|$q\u21c0=k\u21c0\u2032\u2212k\u21c0$, *δ* is a constant and the parameter ‘*r’* is the inverse of the static susceptibility (χ). Since, we are interested in the systems near the critical point of transition between paramagnetic and ferromagnetic phases, the static susceptibility is taken as the Pauli paramagnetic susceptibility, which is given by the relation χ = {ρ_{↑}(*E*_{F}) + ρ_{↑}(*E*_{F})}/2, with ρ_{↑}(*E*_{F}) and ρ_{↓}(*E*_{F}) are the DOS at the Fermi level for electrons with spins up and down respectively.^{16,17} Cooper pairing with anti parallel spins is possible if the magnon contribution to the interaction potential is negative and is greater in magnitude than the potential due to paramagnons. It is noteworthy from the above discussions that the interaction potentials given in Eqs (9) and (10) lead to electron-electron attraction under two different conditions. These cases are discussed in the following two subsections.

### A. Case-1: ω_{D} > *JS*

In the first case, the interaction potential due to triplet pairing may be attractive for |$\omega _{\vec q} > JS$|$\omega q\u20d7>JS$, with electron energy lying in the range |$( {\omega _{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over q} } - JS} ) > \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over v} _F \bullet \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over q} > - ( {\omega _{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over q} } - JS} )$|$(\omega q\u21c0\u2212JS)>v\u21c0F\u2022q\u21c0>\u2212(\omega q\u21c0\u2212JS)$. Now, the electrons taking part in Cooper pairing are mostly near the Fermi surface and hence the electronic energy |$\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over v} _F\cdot \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over q}$|$v\u21c0F\xb7q\u21c0$ in the expression of interaction potential can be neglected. Taking average over the magnon frequencies, the interaction potential can be written approximately as

where |$\langle {\frac{1}{{\omega _{\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over q} } - JS}}} \rangle$|$\u27e81\omega q\u21c0\u2212JS\u27e9$ means the average value over the magnon frequencies. In the present case the role played by the magnons is the same as that of the phonons in case of conventional superconductors. Following the same procedure as in the case of BCS theory and taking we can write the expression for the critical temperature of the ferromagnetic system as follows

where λ is the coupling constant which is given by the expression |$\lambda = \rho ( {E_F } )\,| {V_{eff}^T } |$|$\lambda =\rho (EF)|VeffT|$, with |$\rho ( {E_F } )\break = \sqrt {\rho _ \uparrow ( {E_F } )\rho _ \downarrow ( {E_F } )}$|$\rho (EF)=\rho \u2191(EF)\rho \u2193(EF)$ and the parameter *μ* is introduced to take into account the effect of the coulomb repulsive interaction between electrons. Here, ρ_{↑}(*E*_{F}) is the density of states of spin up electrons, while ρ_{↓}(*E*_{F}) signify the density of states of spin down electrons. Apart from the customary role that DOS performs in the BCS theory, it also tries to enhance the repulsive interaction and thus lessens the critical temperature. From Eq. (12) it is clear that large Hund's coupling constant makes effective interaction potential high. Nevertheless, from Eq. (13) it appears that high value of *J* is not useful for enhancing the critical temperature of superconductivity as it simultaneously makes the effective range of electronic energy to be narrow. On the other hand, the role of nearest neighbor exchange coupling is opposite to that of Hund's coupling in affecting the interaction potential and the effective range of electronic energy.

### B. Case-2: ω_{D} < *JS*

In the second case where the Debye cut off frequency is found to be less than *JS* (ω_{D} < *JS*), the interaction potential for singlet pairing is negative for the electronic energy in the range |$( {JS - \omega _q } )\break > \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over v} _F \bullet \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over q} > - ( {JS - \omega _q } )$|$(JS\u2212\omega q)>v\u21c0F\u2022q\u21c0>\u2212(JS\u2212\omega q)$. The electrons involved in pairing are mainly near the Fermi surface and hence the value of |$\mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over v} _F \bullet \mathord{\buildrel{\lower3pt\hbox{\scriptscriptstyle\rightharpoonup}}\over q}$|$v\u21c0F\u2022q\u21c0$ is very small. The effective interaction potential in this case reduces to

The critical temperature for superconductivity can be written similar to the case of triplet pairing as

with the coupling constant |$\lambda = \rho ( {E_F } )\,| {V_{eff}^S } |$|$\lambda =\rho (EF)|VeffS|$. In this case DOS at Fermi surface plays the same role as in the case of triplet pairing. However, the roles of Hund's coupling and nearest neighbor exchange coupling are altered and the critical temperature of superconductivity is found to increase with an increase in the parameters. Consequently, in this case it is easier to raise the critical temperature of superconductivity compared to the previous case by increasing Hund's coupling and the nearest neighbor exchange coupling constants.

## III. APPLICATION IN UGe_{2}

### A. Computational Details

All the calculations are carried out in the framework of Density Functional Theory (DFT) as implemented within the OpenMX v.3.5 code.^{18,19} To avoid oversimplification of the treatment of electron correlation in *f* electron systems, a multiband Hubbard type augmentation of the energy correction has been prescribed.^{20} We have employed LSDA+U method which is reported to be successful in predicting the electronic structure of strongly correlated systems.^{21} In a number of previous studies, it has been shown that the LSDA+U approach reproduces the magnetism of such heavy fermion systems with improved accuracy.^{22} Moreover the partial localization of the 5*f* electrons in uranium atoms are taken into account within the theory with the help of the “*dual*” version of the occupation number matrix in the LSDA+U method as implemented in OpenMX code.^{23} This method assures the partial localization of the 5*f* electrons and partial delocalization through proper treatment of nearest neighbor overlap. Basis functions are used in the form of linear combination of localized pseudoatomic orbitals (LCPAO).^{24} Norm-conserving Troullier and Martine (TM) type pseudopotentials were used in the calculation. U-6.0-s2p1d2f1 and Ge-5.5-s1p1d1 were used as basis functions,^{25} where U and Ge designate the atom type, followed by the cutoff radius (in Bohr radius units) in the confinement scheme,^{26,27} and the set of symbols define primitive orbitals taken into account. Fully relativistic pseudopotentials are used in the computation with partial core correction.^{25,28} The energy convergence criterion (Self Consistent Field) was set to 10^{−6} Hartree. Numerical integrations use the energy cutoff of 150 Ry with the real space grid techniques of Junquera *et al.*^{29,30}

Structural studies on UGe_{2} depict an orthorhombic structure with full inversion symmetry. UGe_{2} crystallizes into *Cmmm* symmetry with crystal parameters *a* = 4.0089 Å, *b* = 15.0889 Å, *c* = 4.095 Å.^{31} The structure can be envisaged as arising out of the zigzag arrangements of U atoms along *a* direction in the *a-b* plane (Figure 2(a)). The atomic positions in the crystal under normal pressure are given in Table I. A unit cell contains four Uranium atoms numbered 1,5,6,7 and form zigzag lines with spins, while eight Germanium atoms remain spinless. Based on the single crystal magnetization measurements and neutron powder diffraction data, a collinear magnetic structure with ferromagnetic ordering is predicted.^{32,33} In case of UGe_{2}, superconductivity is reported to appear in the pressure range of 10-16 kbar with maximum critical temperature of superconductivity *T*_{C} = 0.8 K at pressure 12.5 kbar. The Curie temperature is *T*_{Curie} = 52 K at ambient pressure which subsequently decreases with applied pressure and is found to vanish at 16 kbar. Hence, the superconducting phase is enclosed within the ferromagnetic phase and disappears in paramagnetic state.

Atom . | x
. | y
. | z
. |
---|---|---|---|

U | 0 | 0.14092 | 0 |

Ge | 0 | 0.30762 | 0.5 |

Ge | 0 | 0 | 0.5 |

Ge | 0.5 | 0 | 0 |

Atom . | x
. | y
. | z
. |
---|---|---|---|

U | 0 | 0.14092 | 0 |

Ge | 0 | 0.30762 | 0.5 |

Ge | 0 | 0 | 0.5 |

Ge | 0.5 | 0 | 0 |

This invokes a systematic study of the effect of pressure on the magnetic and superconducting behaviors of UGe_{2}. The decrease in the intensity of the <312> and <310> peaks in diffraction pattern indicates displacement of atoms from the zero pressure position along *b*-axis resulting in a small increase in either or both of *y*_{U} and *y*_{Ge} of the order 10^{−3}/kbar.^{2} Thus a straightening of the buckled Uranium chains with applied pressure is realized. To examine the superconductivity under pressure we take *b* = 15.04 Å and the shift in position of Uranium and Germanium along *b*-axis as δ*y*_{U} = δ*y*_{Ge} = *P* × 10^{−3}, with pressure *P* kbar. The computed results along with the computational details are discussed in the following.

The presumption that superconductivity could also appear in ferromagnets was stimulate by the observation of superconductivity in ferromagnet URhGe at ambient pressure. The physical properties of URhGe at ambient pressure is highly in similarity with those of UGe2 at higher pressures (10-16 kbar) where superconductivity is realized.^{34} In similarity with UGe_{2}, URhGe also have orthorhombic crystallization and contain zigzag chains of spin carrying U atoms numbered as 1, 4, 5 and 6.^{35,36} Recent experiments on single crystals of URhGe reveal that URhGe has the orthorhombic *Pnma* crystal structure. The zigzag arrangement of the U atoms along the *a* axis is observable from Figure 2(b). Table II reports the atomic positions in the URhGe crystal.

### B. Computation of *T*_{C}

#### 1. In case of UGe_{2}

##### a. Under normal pressure.

Under normal pressure all the four Uranium atoms have spin 1.13 each. This spin is near to the experimentally observed value is 1.17.^{37} As is evident from Figure 2, the U atoms numbered 1&7 and 5&6 are nearest neighbors. Hund's coupling at four sites is same with value 2821.979 cm^{−1} and the nearest neighbor exchange coupling is 72.79 cm^{−1}. This gives ω_{D} < *JS* and hence the condition of singlet superconductivity is fulfilled. The density of state (DOS) plot at normal pressure is given in Figure 3(a). It is observed that the DOS at the Fermi level has a characteristic peak at the normal pressure. Hence a high DOS at the Fermi level will result in an increase in the repulsive potential, which can easily be interpreted from Eq. (11). The critical temperature of superconductivity is computed with the help of the method developed and discussed in the previous paragraphs. Setting the parameter μ = 0.07, *T*_{C} is found to be approximately 0 K, which indicates the fact that UGe_{2} does not exhibit superconductivity under normal pressure.

##### b. Under High Pressure.

The *T*_{C} values are computed considering the system experiencing external pressures of magnitude 12.5, 14kbar. Under high pressure the Hund's coupling and spins at different sites are changed. In both cases, the condition of singlet superconductivity is fulfilled. The change in the nearest neighbor exchange coupling constant is more prominent with respect to spin and Hund's coupling. The change in the Hund's coupling and nearest neighbor exchange coupling with pressure results in a concomitant increase in the attractive potential. On the other hand, it is also obvious from Figure 3 that with the increase in pressure the density of states at Fermi level decreases leading to the reduction in the repulsive potential.

The computational estimate of Hund's coupling (*J*) and nearest neighbor exchange coupling (*J*_{nn}) are given in Table III. Hund's coupling does not change much with pressure. On the other hand, a drastic change in *J*_{nn} values is observed as is previously envisaged by Karchev comparing results from experimental reports.^{38} With the increase in pressure a concomitant change in the DOS for spin up electrons is also noticed. This increase in the *J*_{nn} sets the route to an associated increase in the Debye cutoff frequency (ω_{D}) which can be found in Table III. An increased ω_{D} in its turn raises the critical temperature of superconductivity. The calculated critical temperatures of superconductivity of UGe_{2} under different pressure are shown in Table III. The critical temperature of superconductivity is found to be 0.87 K at pressure 14 kbar which is well in agreement with experimentally observed value 0.8 K at pressure 12.5 kbar. It is also evident that the variation in the *T*_{C} is an outcome of the alteration of the exchange coupling between the spin centers of the UGe_{2} lattice with enhanced external pressure. Therefore, the superconducting behavior of the substance UGe_{2} is readily comprehensible from the magnon exchange mechanism in the framework of the localized-itinerant model.

. | . | . | . | . | . | Interaction Potential . | . | . | . |
---|---|---|---|---|---|---|---|---|---|

Pressure . | Spin/U . | J in cm^{−1}
. | J_{nn} in cm^{−1}
. | ρ_{↑}/site
. | ρ_{↓}/site
. | in eV . | ω_{D} in K
. | λ . | T_{C} in K
. |

normal | 1.13 | 2821.98 | 72.79 | 4.57 | 0.34 | −0.135979 | 946.848 | 0.1695 | 0 |

12.5 kb | 1.16 | 2853.67 | 99.45 | 2.98 | 0.39 | −0.176319 | 1327.845 | 0.1901 | 0.17 |

14 kb | 1.17 | 2865.44 | 106.32 | 2.84 | 0.41 | −0.187841 | 1431.871 | 0.2027 | 0.87 |

. | . | . | . | . | . | Interaction Potential . | . | . | . |
---|---|---|---|---|---|---|---|---|---|

Pressure . | Spin/U . | J in cm^{−1}
. | J_{nn} in cm^{−1}
. | ρ_{↑}/site
. | ρ_{↓}/site
. | in eV . | ω_{D} in K
. | λ . | T_{C} in K
. |

normal | 1.13 | 2821.98 | 72.79 | 4.57 | 0.34 | −0.135979 | 946.848 | 0.1695 | 0 |

12.5 kb | 1.16 | 2853.67 | 99.45 | 2.98 | 0.39 | −0.176319 | 1327.845 | 0.1901 | 0.17 |

14 kb | 1.17 | 2865.44 | 106.32 | 2.84 | 0.41 | −0.187841 | 1431.871 | 0.2027 | 0.87 |

#### 2. In case of URhGe

We also compute the T_{C} value of URhGe at its geometry in the ambient pressure. All the four U atoms in URhGe have spin 1.3 each. As evident from Figure 2(b) U atoms numbered 1 and 5 are nearest neighbors. Hund's coupling (*J*) at four sites is same with value 3376.72 cm^{−1} and the nearest neighbor exchange coupling (*J*_{nn}) is 51.66 cm^{−1}. This gives ω_{D} < *JS* and hence the condition of singlet superconductivity is fulfilled in this case. The density of state (DOS) plot is given in Figure 4. From the reduction in DOS at Fermi level it is evident that the repulsive potential is diminished thus leading to revelation of superconductivity at ambient pressure. The critical temperature of superconductivity in case of URhGe is found to be 0.2 K as shown in Table IV. This estimation is in agreement with the experimental observed result of 0.2 K.^{39}

. | . | . | . | . | Interaction Potential . | . | . | . |
---|---|---|---|---|---|---|---|---|

Spin/U . | J in cm^{−1}
. | J_{nn} in cm^{−1}
. | ρ_{↑}/site
. | ρ_{↓}/site
. | in eV . | ω_{D} in K
. | λ . | T_{C} in K
. |

1.3 | 3376.72 | 51.66 | 4.03 | 0.63 | −0.124848 | 773.0417 | 0.198932 | 0.2 |

. | . | . | . | . | Interaction Potential . | . | . | . |
---|---|---|---|---|---|---|---|---|

Spin/U . | J in cm^{−1}
. | J_{nn} in cm^{−1}
. | ρ_{↑}/site
. | ρ_{↓}/site
. | in eV . | ω_{D} in K
. | λ . | T_{C} in K
. |

1.3 | 3376.72 | 51.66 | 4.03 | 0.63 | −0.124848 | 773.0417 | 0.198932 | 0.2 |

## IV. CONCLUSION

To surmise, the interaction of itinerant electrons with localized electrons aligned in ferromagnetic ordering is assessed through localized-itinerant approach. The formation of the singlet or triplet Cooper pairs is also explained on the basis of Debye cut off frequency. It is found that if ω_{D} > *JS*, it is triplet pairing and on the other hand ω_{D} < *JS* leads to a singlet pairing situation. In case of singlet pairing the critical temperature of superconductivity increases with the increase in Hund's coupling (*J*) and the nearest neighbor exchange coupling (*J*_{nn}). Whereas, in case of triplet pairing no such parameter is found which effectively modulates the critical temperature of superconductivity. Apart from these, the density of states at Fermi level is also found to play a dual role. In the first case, DOS affects the electron-magnon coupling constant (λ), as λ appears as a product of DOS and the interaction potential, while in the other way DOS also affects the repulsive interaction. The genesis of superconductivity in ferromagnetic is justified through the singlet pairing of electrons on the basis of the present treatment taking UGe_{2} and URhGe as examples. The results validate pervious anticipation about the superconducting behavior of UGe_{2} by Suhl.^{11} The DOS vs energy plot of ferromagnetic UGe_{2} (Figure 3) reveals that under normal pressure there is a peak at Fermi level which subsequently vanishes at an elevated pressure. The reduction in DOS at Fermi level diminishes the repulsive potential leading to elucidation of superconductivity at high pressure. It is also evident from the DFT computation that the Hund's coupling does not change radically with pressure but the nearest neighbor exchange coupling constant (*J*_{nn}) increases markedly with the increase in pressure. This enhancement in *J*_{nn}, leads the critical temperature to rise through an increment of the Debye cut off frequency. The DOS characteristic of URhGe at ambient pressure (Figure 4) is found resemble with the high pressure behavior of UGe_{2}. The drop in DOS at the Fermi level is in accordance with the singlet superconductivity and concomitant elucidation of superconductivity at normal pressure. The critical temperature of superconductivity computed in this work for UGe_{2} under different pressures, and URhGe at normal pressure closely resemble the experimentally observed results. This agreement provides a strong support for the formalism employed here and also advocates its general applicability in materials where ferromagnetism and superconductivity coexist.

## ACKNOWLEDGMENT

Authors are thankful to DST, India for financial support. TG thanks CSIR, India for a senior research fellowship.