**1. Introduction**

In the beginning of this century the unconventional superconductivity of spin-triplet type had been experimentally discovered in several itinerant ferromagnets. Since then much experimental and theoretical research on the properties of these systems has been accomplished. Here we review the phenomenological theory of ferromagnetic unconventional superconductors with spin-triplet Cooper pairing of electrons. Some theoretical aspects of the description of the phases and the phase transitions in these interesting systems, including the remarkable phenomenon of coexistence of superconductivity and ferromagnetism are discussed with an emphasis on the comparison of theoretical results with experimental data.

The spin-triplet or *p*-wave pairing allows parallel spin orientation of the fermion Cooper pairs in superfluid 3He and unconventional superconductors [1]. For this reason the resulting unconventional superconductivity is robust with respect to effects of external magnetic field and spontaneous ferromagnetic ordering, so it may coexist with the latter. This general argument implies that there could be metallic compounds and alloys, for which the coexistence of spin-triplet superconductivity and ferromagnetism may be observed.

Particularly, both superconductivity and itinerant ferromagnetic orders can be created by the same band electrons in the metal, which means that spin-1 electron Cooper pairs participate in the formation of the itinerant ferromagnetic order. Moreover, under certain conditions the superconductivity is enhanced rather than depressed by the uniform ferromagnetic order that can generate it, even in cases when the superconductivity does not appear in a pure form as a net result of indirect electron-electron coupling.

The coexistence of superconductivity and ferromagnetism as a result of collective behavior of *f*-band electrons has been found experimentally for some Uranium-based intermetallic compounds as, UGe2 [2–5], URhGe [6–8], UCoGe [9, 10], and UIr [11, 12]. At low temperature (*T* ∼ 1 K) all these compounds exhibit thermodynamically stable phase of coexistence of spin-triplet superconductivity and itinerant (*f*-band) electron ferromagnetism (in short, FS

©2012 Uzunov, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Uzunov, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

phase). In UGe2 and UIr the FS phase appears at high pressure (*P* ∼ 1 GPa) whereas in URhGe and UCoGe, the coexistence phase persists up to ambient pressure (105Pa <sup>≡</sup> 1bar).

Experiments, carried out in ZrZn2 [13], also indicated the appearance of FS phase at *T* < 1 K in a wide range of pressures (0 < *P* ∼ 21 kbar). In Zr-based compounds the ferromagnetism and the *p*-wave superconductivity occur as a result of the collective behavior of the *d*-band electrons. Later experimental results [14, 15] had imposed the conclusion that bulk superconductivity is lacking in ZrZn2, but the occurrence of a surface FS phase at surfaces with higher Zr content than that in ZrZn2 has been reliably demonstrated. Thus the problem for the coexistence of bulk superconductivity with ferromagnetism in ZrZn2 is still unresolved. This raises the question whether the FS phase in ZrZn2 should be studied by surface thermodynamics methods or should it be investigated by considering that bulk and surface thermodynamic phenomena can be treated on the same footing. Taking into account the mentioned experimental results for ZrZn2 and their interpretation by the experimentalists [13–15] we assume that the unified thermodynamic approach can be applied. As an argument supporting this point of view let us mention that the spin-triplet superconductivity occurs not only in bulk materials but also in quasi-two-dimensional (2D) systems – thin films and surfaces and quasi-1D wires (see, e.g., Refs. [16]). In ZrZn2 and UGe2 both ferromagnetic and superconducting orders vanish at the same critical pressure *Pc*, a fact implying that the respective order parameter fields strongly depend on each other and should be studied on the same thermodynamic basis [17].

P

C

P c

N

Theory of Ferromagnetic Unconventional Superconductors with Spin-Triplet Electron Pairing 417

1

2

T

FM

TF (P)

FS <sup>3</sup>

P1

ferromagnetic superconductors, as indicated in the figure).

quantum (zero temperature) phase transitions [18, 19].

an additional Heisenberg exchange term [22].

topic was comprehensively discussed in Ref. [25].

type I type II

**Figure 1.** An illustration of *T* − *P* phase diagram of *p*-wave ferromagnetic superconductors (details are omitted): N – normal phase, FM – ferromagnetic phase, FS – phase of coexistence of ferromagnetic order and superconductivity, *TF*(*P*) and *TFS*(*P*) are the respective phase transition lines: solid lines correspond to second order phase transitions, dashed lines stand for first order phase transition; 1 and 2 are tricritical points; *Pc* is the critical pressure, and the circle *C* surrounds a relatively small domain of high pressure and low temperature, where the phase diagram may have several forms depending on the particular substance. The line of the FM-FS phase transition may extend up to ambient pressure (type I ferromagnetic superconductors), or, may terminate at *T* = 0 at some high pressure *P* = *P*<sup>1</sup> (type II

(0, *P*0*c*); *P*0*<sup>c</sup> Pc*. In this case, the *p*-wave ferromagnetic superconductor has three points of

These and other possible shapes of *T* − *P* phase diagrams are described within the framework of the general theory of Ginzburg-Landau (GL) type [18–20] in a conformity with the experimental data; see also Ref. [21]. The same theory has been confirmed by a microscopic derivation based on a microscopic Hamiltonian including a spin-generalized BCS term and

For all compounds, cited above, the FS phase occurs only in the ferromagnetic phase domain of the *T* − *P* diagram. Particularly at equilibrium, and for given *P*, the temperature *TF*(*P*) of the normal-to-ferromagnetic phase (or N-FM) transition is never lower than the temperature *TFS*(*P*) of the ferromagnetic-to-FS phase transition (FM-FS transition). This confirms the point of view that the superconductivity in these compounds is triggered by the spontaneous magnetization *M*, in analogy with the well-known triggering of the superfluid phase A1 in 3He at mK temperatures by the external magnetic field *H*. Such "helium analogy" has been used in some theoretical studies (see, e.g., Ref. [23, 24]), where Ginzburg-Landau (GL) free energy terms, describing the FS phase were derived by symmetry group arguments. The non-unitary state, with a non-zero value of the Cooper pair magnetic moment, known from the theory of unconventional superconductors and superfluidity in 3He [1], has been suggested firstly in Ref. [23], and later confirmed in other studies [7, 24]; recently, the same

Fig. 1 illustrates the shape of the *T* − *P* phase diagrams of real intermetallic compounds. The phase transition from the normal (N) to the ferromagnetic phase (FM) (in short, N-FM transition) is shown by the line *TF*(*P*). The line *TFS*(*P*) of the phase transition from FM to FS (FM-FS transition) may have two or more distinct shapes. Beginning from the maximal (critical) pressure *Pc*, this line may extend, like in ZrZn2, to all pressures *P* < *Pc*, including the ambient pressure *Pa*; see the almost straight line containing the point 3 in Fig. 1. A second possible form of this line, as known, for example, from UGe2 experiments, is shown in Fig. 1 by the curve which begins at *P* ∼ *Pc*, passes through the point 2, and terminates at some pressure *P*<sup>1</sup> > *Pa*, where the superconductivity vanishes. These are two qualitatively different physical pictures: (a) when the superconductivity survives up to ambient pressure (type I), and (b) when the superconducting states are possible only at relatively high pressure (for UGe2, *P*<sup>1</sup> ∼ 1 GPa); type II. At the tricritical points 1, 2 and 3 the order of the phase transitions changes from second order (solid lines) to first order (dashed lines). It should be emphasized

that in all compounds, mentioned above, *TFS*(*P*) is much lower than *TF*(*P*) when the pressure *P* is considerably below the critical pressure *Pc* (for experimental data, see Sec. 8).

In Fig. 1, the circle *C* denotes a narrow domain around *Pc* at relatively low temperatures (*T* 300 mK), where the experimental data are quite few and the predictions about the shape of the phase transition are not reliable. It could be assumed, as in the most part of the experimental papers, that (*T* = 0, *P* = *Pc*) is the zero temperature point at which both lines *TF*(*P*) and *TFS*(*P*) terminate. A second possibility is that these lines may join in a single (N-FS) phase transition line at some point (*T* 0, *P*� *<sup>c</sup> Pc*) above the absolute zero. In this second variant, a direct N-FS phase transition occurs, although this option exists in a very small domain of temperature and pressure variations: from point (0, *Pc*) to point (*T* 0, *P*� *<sup>c</sup> Pc*). A third variant is related with the possible splitting of the point (0, *Pc*), so that the N-FM line terminates at (0, *Pc*), whereas the FM-FS line terminates at another zero temperature point

2 Will-be-set-by-IN-TECH

phase). In UGe2 and UIr the FS phase appears at high pressure (*P* ∼ 1 GPa) whereas in URhGe and UCoGe, the coexistence phase persists up to ambient pressure (105Pa <sup>≡</sup> 1bar). Experiments, carried out in ZrZn2 [13], also indicated the appearance of FS phase at *T* < 1 K in a wide range of pressures (0 < *P* ∼ 21 kbar). In Zr-based compounds the ferromagnetism and the *p*-wave superconductivity occur as a result of the collective behavior of the *d*-band electrons. Later experimental results [14, 15] had imposed the conclusion that bulk superconductivity is lacking in ZrZn2, but the occurrence of a surface FS phase at surfaces with higher Zr content than that in ZrZn2 has been reliably demonstrated. Thus the problem for the coexistence of bulk superconductivity with ferromagnetism in ZrZn2 is still unresolved. This raises the question whether the FS phase in ZrZn2 should be studied by surface thermodynamics methods or should it be investigated by considering that bulk and surface thermodynamic phenomena can be treated on the same footing. Taking into account the mentioned experimental results for ZrZn2 and their interpretation by the experimentalists [13–15] we assume that the unified thermodynamic approach can be applied. As an argument supporting this point of view let us mention that the spin-triplet superconductivity occurs not only in bulk materials but also in quasi-two-dimensional (2D) systems – thin films and surfaces and quasi-1D wires (see, e.g., Refs. [16]). In ZrZn2 and UGe2 both ferromagnetic and superconducting orders vanish at the same critical pressure *Pc*, a fact implying that the respective order parameter fields strongly depend on each other and should

Fig. 1 illustrates the shape of the *T* − *P* phase diagrams of real intermetallic compounds. The phase transition from the normal (N) to the ferromagnetic phase (FM) (in short, N-FM transition) is shown by the line *TF*(*P*). The line *TFS*(*P*) of the phase transition from FM to FS (FM-FS transition) may have two or more distinct shapes. Beginning from the maximal (critical) pressure *Pc*, this line may extend, like in ZrZn2, to all pressures *P* < *Pc*, including the ambient pressure *Pa*; see the almost straight line containing the point 3 in Fig. 1. A second possible form of this line, as known, for example, from UGe2 experiments, is shown in Fig. 1 by the curve which begins at *P* ∼ *Pc*, passes through the point 2, and terminates at some pressure *P*<sup>1</sup> > *Pa*, where the superconductivity vanishes. These are two qualitatively different physical pictures: (a) when the superconductivity survives up to ambient pressure (type I), and (b) when the superconducting states are possible only at relatively high pressure (for UGe2, *P*<sup>1</sup> ∼ 1 GPa); type II. At the tricritical points 1, 2 and 3 the order of the phase transitions changes from second order (solid lines) to first order (dashed lines). It should be emphasized that in all compounds, mentioned above, *TFS*(*P*) is much lower than *TF*(*P*) when the pressure

*P* is considerably below the critical pressure *Pc* (for experimental data, see Sec. 8).

In Fig. 1, the circle *C* denotes a narrow domain around *Pc* at relatively low temperatures (*T* 300 mK), where the experimental data are quite few and the predictions about the shape of the phase transition are not reliable. It could be assumed, as in the most part of the experimental papers, that (*T* = 0, *P* = *Pc*) is the zero temperature point at which both lines *TF*(*P*) and *TFS*(*P*) terminate. A second possibility is that these lines may join in a single (N-FS)

variant, a direct N-FS phase transition occurs, although this option exists in a very small domain of temperature and pressure variations: from point (0, *Pc*) to point (*T* 0, *P*�

A third variant is related with the possible splitting of the point (0, *Pc*), so that the N-FM line terminates at (0, *Pc*), whereas the FM-FS line terminates at another zero temperature point

*<sup>c</sup> Pc*) above the absolute zero. In this second

*<sup>c</sup> Pc*).

be studied on the same thermodynamic basis [17].

phase transition line at some point (*T* 0, *P*�

**Figure 1.** An illustration of *T* − *P* phase diagram of *p*-wave ferromagnetic superconductors (details are omitted): N – normal phase, FM – ferromagnetic phase, FS – phase of coexistence of ferromagnetic order and superconductivity, *TF*(*P*) and *TFS*(*P*) are the respective phase transition lines: solid lines correspond to second order phase transitions, dashed lines stand for first order phase transition; 1 and 2 are tricritical points; *Pc* is the critical pressure, and the circle *C* surrounds a relatively small domain of high pressure and low temperature, where the phase diagram may have several forms depending on the particular substance. The line of the FM-FS phase transition may extend up to ambient pressure (type I ferromagnetic superconductors), or, may terminate at *T* = 0 at some high pressure *P* = *P*<sup>1</sup> (type II ferromagnetic superconductors, as indicated in the figure).

(0, *P*0*c*); *P*0*<sup>c</sup> Pc*. In this case, the *p*-wave ferromagnetic superconductor has three points of quantum (zero temperature) phase transitions [18, 19].

These and other possible shapes of *T* − *P* phase diagrams are described within the framework of the general theory of Ginzburg-Landau (GL) type [18–20] in a conformity with the experimental data; see also Ref. [21]. The same theory has been confirmed by a microscopic derivation based on a microscopic Hamiltonian including a spin-generalized BCS term and an additional Heisenberg exchange term [22].

For all compounds, cited above, the FS phase occurs only in the ferromagnetic phase domain of the *T* − *P* diagram. Particularly at equilibrium, and for given *P*, the temperature *TF*(*P*) of the normal-to-ferromagnetic phase (or N-FM) transition is never lower than the temperature *TFS*(*P*) of the ferromagnetic-to-FS phase transition (FM-FS transition). This confirms the point of view that the superconductivity in these compounds is triggered by the spontaneous magnetization *M*, in analogy with the well-known triggering of the superfluid phase A1 in 3He at mK temperatures by the external magnetic field *H*. Such "helium analogy" has been used in some theoretical studies (see, e.g., Ref. [23, 24]), where Ginzburg-Landau (GL) free energy terms, describing the FS phase were derived by symmetry group arguments. The non-unitary state, with a non-zero value of the Cooper pair magnetic moment, known from the theory of unconventional superconductors and superfluidity in 3He [1], has been suggested firstly in Ref. [23], and later confirmed in other studies [7, 24]; recently, the same topic was comprehensively discussed in Ref. [25].

For the spin-triplet ferromagnetic superconductors the trigger mechanism was recently examined in detail [20, 21]. The system main properties are specified by terms in the GL expansion of form *Miψjψk*, which represent the interaction of the magnetization *M* = {*Mj*; *j* = 1, 2, 3} with the complex superconducting vector field *ψ* = {*ψj*; *j* = 1, 2, 3}. Particularly, these terms are responsible for the appearance of superconductivity (|*ψ*| > 0) for certain *T* and *P* values. A similar trigger mechanism is familiar in the context of improper ferroelectrics [26].

a simple, yet comprehensive, modeling of *P* dependence of the free energy parameters, resulting in a very good compliance of our theoretical predictions for the shape the *T* − *P* phase diagrams with the experimental data (for some preliminary results, see Ref. [18, 19]). The theoretical analysis is done by the standard methods of phase transition theory [31]. Treatment of fluctuation effects and quantum correlations [31, 32] is not included in this study. But the parameters of the generalized GL free energy may be considered either in mean-field approximation as here, or as phenomenologically renormalized parameters which are affected

Theory of Ferromagnetic Unconventional Superconductors with Spin-Triplet Electron Pairing 419

We demonstrate with the help of present theory that we can outline different possible topologies for the *T* − *P* phase diagram, depending on the values of Landau parameters, derived from the existing experimental data. We show that for spin-triplet ferromagnetic superconductors there exist two distinct types of behavior, which we denote as Zr-type (or, alternatively, type I) and U-type (or, type II); see Fig. 1. This classification of the FS, first mentioned in Ref. [18], is based on the reliable interrelationship between a quantitative criterion derived by us and the thermodynamic properties of the ferromagnetic spin-triplet superconductors. Our approach can be also applied to URhGe, UCoGe, and UIr. The results shed light on the problems connected with the order of the quantum phase transitions at ultra-low and zero temperatures. They also raise the question for further experimental investigations of the detailed structure of the phase diagrams in the high-*P*/low-*T* region.

*<sup>f</sup>*S(*ψ*) + *<sup>f</sup>*F(*M*) + *<sup>f</sup>*I(*ψ*, *<sup>M</sup>*) + *<sup>B</sup>*<sup>2</sup>

where the fields *ψ*, *M*, and *B* are supposed to depend on the spatial vector *x* ∈ *V* in the volume *V* of the superconductor. In Eq. (1), the free energy density generated by the generic

where a summation over the indices (*i*, *j*) is assumed, the symbol *Dj* = (*h*¯ *∂*/*i∂xj* + 2|*e*|*Aj*/*c*) of covariant differentiation is introduced, and *Kj* are material parameters [1]. The free energy


<sup>2</sup> + *<sup>a</sup> <sup>f</sup> <sup>M</sup>*<sup>2</sup> +

density *f*F(*M*) of a standard ferromagnetic phase transition of second order [31] is

3 ∑ *j*=1

<sup>2</sup> + *bs* <sup>2</sup> <sup>|</sup>*ψ*<sup>|</sup> <sup>4</sup> + *us* <sup>2</sup> <sup>|</sup>*ψ*2<sup>|</sup> <sup>8</sup>*<sup>π</sup>* <sup>−</sup> *<sup>B</sup>***.***<sup>M</sup>*

3 ∑ *j*=1 |*ψj*|

<sup>2</sup> + *vs* 2

(*Diψi*)∗(*Djψj*)+(*Diψj*)∗(*Djψi*)

*b f*

, (1)

<sup>4</sup> , (2)

(3)

<sup>2</sup> *<sup>M</sup>*4, (4)

by additional physical phenomena, as for example, spin fluctuations.

**2. Theoretical framework**

with

Consider the GL free energy functional of the form

*V dx* 

*f*S(*ψ*) = *fgrad*(*ψ*) + *as*|*ψ*|

*fgrad*(*ψ*) = *K*1(*Diψj*)∗(*DiDj*) + *K*<sup>2</sup>

+*K*3(*Diψi*)∗(*Diψi*),

*f*F(*M*) = *c <sup>f</sup>*

*<sup>F</sup>*(*ψ*, *<sup>B</sup>*) =

superconducting subsystem (*ψ*) is given by

A crucial feature of these systems is the nonzero magnetic moment of the spin-triplet Cooper pairs. As mentioned above, the microscopic theory of magnetism and superconductivity in non-Fermi liquids of strongly interacting heavy electrons (*f* and *d* band electrons) is either too complex or insufficiently developed to describe the complicated behavior in itinerant ferromagnetic compounds. Several authors (see [20, 21, 23–25]) have explored the phenomenological description by a self-consistent mean field theory, and here we will essentially use the thermodynamic results, in particular, results from the analysis in Refs. [20, 21]. Mean-field microscopic theory of spin-mediated pairing leading to the mentioned non-unitary superconductivity state has been developed in Ref. [17] that is in conformity with the phenomenological description that we have done.

The coexistence of *s*-wave (conventional) superconductivity and ferromagnetic order is a long-standing problem in condensed matter physics [27–29]. While the *s*-state Cooper pairs contain only opposite electron spins and can easily be destroyed by the spontaneous magnetic moment, the spin-triplet Cooper pairs possess quantum states with parallel orientation of the electron spins and therefore can survive in the presence of substantial magnetic moments. This is the basic difference in the magnetic behavior of conventional (*s*-state) and spin-triplet superconductivity phases. In contrast to other superconducting materials, for example, ternaty and Chevrel phase compounds, where the effect of magnetic order on *s*-wave superconductivity has been intensively studied in the seventies and eighties of last century (see, e.g., Refs. [27–29]), in these ferromagnetic compounds the phase transition temperature *TF* to the ferromagnetic state is much higher than the phase transition temperature *TFS* from ferromagnetic to a (mixed) state of coexistence of ferromagnetism and superconductivity. For example, in UGe2 we have *TFS* ∼ 0.8 K versus maximal *TF* = 52 K [2–5]. Another important difference between the ternary rare earth compounds and the intermetallic compounds (UGe2, UCoGe, etc.), which are of interest in this paper, is that the experiments with the latter do not give any evidence for the existence of a standard normal-to-superconducting phase transition in zero external magnetic field. This is an indication that the (generic) critical temperature *Ts* of the pure superconductivity state in these intermetallic compounds is very low (*Ts* � *TFS*), if not zero or even negative.

In the reminder of this paper, we present general thermodynamic treatment of systems with itinerant ferromagnetic order and superconductivity due to spin-triplet Cooper pairing of the same band electrons, which are responsible for the spontaneous magnetic moment. The usual Ginzburg-Landau (GL) theory of superconductors has been completed to include the complexity of the vector order parameter *ψ*, the magnetization *M* and new relevant energy terms [20, 21]. We outline the *T* − *P* phase diagrams of ferromagnetic spin-triplet superconductors and demonstrate that in these materials two contrasting types of thermodynamic behavior are possible. The present phenomenological approach includes both mean-field and spin-fluctuation theory (SFT), as the arguments in Ref. [30]. We propose a simple, yet comprehensive, modeling of *P* dependence of the free energy parameters, resulting in a very good compliance of our theoretical predictions for the shape the *T* − *P* phase diagrams with the experimental data (for some preliminary results, see Ref. [18, 19]).

The theoretical analysis is done by the standard methods of phase transition theory [31]. Treatment of fluctuation effects and quantum correlations [31, 32] is not included in this study. But the parameters of the generalized GL free energy may be considered either in mean-field approximation as here, or as phenomenologically renormalized parameters which are affected by additional physical phenomena, as for example, spin fluctuations.

We demonstrate with the help of present theory that we can outline different possible topologies for the *T* − *P* phase diagram, depending on the values of Landau parameters, derived from the existing experimental data. We show that for spin-triplet ferromagnetic superconductors there exist two distinct types of behavior, which we denote as Zr-type (or, alternatively, type I) and U-type (or, type II); see Fig. 1. This classification of the FS, first mentioned in Ref. [18], is based on the reliable interrelationship between a quantitative criterion derived by us and the thermodynamic properties of the ferromagnetic spin-triplet superconductors. Our approach can be also applied to URhGe, UCoGe, and UIr. The results shed light on the problems connected with the order of the quantum phase transitions at ultra-low and zero temperatures. They also raise the question for further experimental investigations of the detailed structure of the phase diagrams in the high-*P*/low-*T* region.

### **2. Theoretical framework**

Consider the GL free energy functional of the form

$$F(\boldsymbol{\upmu}, \mathbf{B}) = \int\_{V} d\mathbf{x} \left[ f\_{\mathbf{S}}(\boldsymbol{\upmu}) + f\_{\mathbf{F}}(\mathbf{M}) + f\_{\mathbf{I}}(\boldsymbol{\upmu}, \mathbf{M}) + \frac{\mathbf{B}^{2}}{8\pi} - \mathbf{B}.\mathbf{M} \right],\tag{1}$$

where the fields *ψ*, *M*, and *B* are supposed to depend on the spatial vector *x* ∈ *V* in the volume *V* of the superconductor. In Eq. (1), the free energy density generated by the generic superconducting subsystem (*ψ*) is given by

$$f\_{\mathbf{S}}(\boldsymbol{\Psi}) = f\_{\mathrm{grad}}(\boldsymbol{\Psi}) + a\_{\mathrm{s}}|\boldsymbol{\Psi}|^{2} + \frac{b\_{\mathrm{s}}}{2}|\boldsymbol{\Psi}|^{4} + \frac{u\_{\mathrm{s}}}{2}|\boldsymbol{\Psi}^{2}|^{2} + \frac{v\_{\mathrm{s}}}{2} \sum\_{j=1}^{3} |\psi\_{j}|^{4} \tag{2}$$

with

4 Will-be-set-by-IN-TECH

For the spin-triplet ferromagnetic superconductors the trigger mechanism was recently examined in detail [20, 21]. The system main properties are specified by terms in the GL expansion of form *Miψjψk*, which represent the interaction of the magnetization *M* = {*Mj*; *j* = 1, 2, 3} with the complex superconducting vector field *ψ* = {*ψj*; *j* = 1, 2, 3}. Particularly, these terms are responsible for the appearance of superconductivity (|*ψ*| > 0) for certain *T* and *P* values. A similar trigger mechanism is familiar in the context of improper ferroelectrics [26]. A crucial feature of these systems is the nonzero magnetic moment of the spin-triplet Cooper pairs. As mentioned above, the microscopic theory of magnetism and superconductivity in non-Fermi liquids of strongly interacting heavy electrons (*f* and *d* band electrons) is either too complex or insufficiently developed to describe the complicated behavior in itinerant ferromagnetic compounds. Several authors (see [20, 21, 23–25]) have explored the phenomenological description by a self-consistent mean field theory, and here we will essentially use the thermodynamic results, in particular, results from the analysis in Refs. [20, 21]. Mean-field microscopic theory of spin-mediated pairing leading to the mentioned non-unitary superconductivity state has been developed in Ref. [17] that is in conformity with

The coexistence of *s*-wave (conventional) superconductivity and ferromagnetic order is a long-standing problem in condensed matter physics [27–29]. While the *s*-state Cooper pairs contain only opposite electron spins and can easily be destroyed by the spontaneous magnetic moment, the spin-triplet Cooper pairs possess quantum states with parallel orientation of the electron spins and therefore can survive in the presence of substantial magnetic moments. This is the basic difference in the magnetic behavior of conventional (*s*-state) and spin-triplet superconductivity phases. In contrast to other superconducting materials, for example, ternaty and Chevrel phase compounds, where the effect of magnetic order on *s*-wave superconductivity has been intensively studied in the seventies and eighties of last century (see, e.g., Refs. [27–29]), in these ferromagnetic compounds the phase transition temperature *TF* to the ferromagnetic state is much higher than the phase transition temperature *TFS* from ferromagnetic to a (mixed) state of coexistence of ferromagnetism and superconductivity. For example, in UGe2 we have *TFS* ∼ 0.8 K versus maximal *TF* = 52 K [2–5]. Another important difference between the ternary rare earth compounds and the intermetallic compounds (UGe2, UCoGe, etc.), which are of interest in this paper, is that the experiments with the latter do not give any evidence for the existence of a standard normal-to-superconducting phase transition in zero external magnetic field. This is an indication that the (generic) critical temperature *Ts* of the pure superconductivity state in these intermetallic compounds is very low (*Ts* � *TFS*),

In the reminder of this paper, we present general thermodynamic treatment of systems with itinerant ferromagnetic order and superconductivity due to spin-triplet Cooper pairing of the same band electrons, which are responsible for the spontaneous magnetic moment. The usual Ginzburg-Landau (GL) theory of superconductors has been completed to include the complexity of the vector order parameter *ψ*, the magnetization *M* and new relevant energy terms [20, 21]. We outline the *T* − *P* phase diagrams of ferromagnetic spin-triplet superconductors and demonstrate that in these materials two contrasting types of thermodynamic behavior are possible. The present phenomenological approach includes both mean-field and spin-fluctuation theory (SFT), as the arguments in Ref. [30]. We propose

the phenomenological description that we have done.

if not zero or even negative.

$$\begin{aligned} f\_{grad}(\boldsymbol{\upmu}) &= \boldsymbol{K}\_1 (\boldsymbol{D}\_i \boldsymbol{\upmu}\_j)^\* (\boldsymbol{D}\_i \boldsymbol{D}\_j) + \boldsymbol{K}\_2 \left[ (\boldsymbol{D}\_i \boldsymbol{\upmu}\_i)^\* (\boldsymbol{D}\_j \boldsymbol{\upmu}\_j) + (\boldsymbol{D}\_i \boldsymbol{\upmu}\_j)^\* (\boldsymbol{D}\_j \boldsymbol{\upmu}\_i) \right] \\ &+ \boldsymbol{K}\_3 (\boldsymbol{D}\_i \boldsymbol{\upmu}\_i)^\* (\boldsymbol{D}\_i \boldsymbol{\upmu}\_i) \end{aligned} \tag{3}$$

where a summation over the indices (*i*, *j*) is assumed, the symbol *Dj* = (*h*¯ *∂*/*i∂xj* + 2|*e*|*Aj*/*c*) of covariant differentiation is introduced, and *Kj* are material parameters [1]. The free energy density *f*F(*M*) of a standard ferromagnetic phase transition of second order [31] is

$$f\_{\mathbf{F}}(\mathbf{M}) = c\_f \sum\_{j=1}^{3} |\nabla \mathbf{M}\_j|^2 + a\_f \mathbf{M}^2 + \frac{b\_f}{2} \mathbf{M}^4,\tag{4}$$

#### 6 Will-be-set-by-IN-TECH 420 Superconductors – Materials, Properties and Applications Theory of Ferromagnetic Unconventional Superconductors with Spin-Triplet Electron Pairing <sup>7</sup>

with *c <sup>f</sup>* , *b <sup>f</sup>* > 0, and *a <sup>f</sup>* = *α*(*T* − *Tf*), where *α<sup>f</sup>* > 0 and *Tf* is the critical temperature, corresponding of the generic ferromagnetic phase transition. Finally, the energy *f*I(*ψ*, *M*) produced by the possible couplings of *ψ* and *M* is given by

$$f\_{\mathbf{I}}(\boldsymbol{\Psi}, \mathbf{M}) = i\gamma\_0 \mathbf{M}.(\boldsymbol{\Psi} \times \boldsymbol{\Psi}^\*) + \delta\_0 \mathbf{M}^2 |\boldsymbol{\Psi}|^2,\tag{5}$$

*M* = |*M*|. For this reason we should use the diagonal quadratic form [35] corresponding to the entire *ψ*2-part of the total free energy functional (1). The lowest energy term in this diagonal quadratic part contains a coefficient *<sup>a</sup>* of the form *<sup>a</sup>* = (*as* <sup>−</sup> *<sup>γ</sup>*0*<sup>M</sup>* <sup>−</sup> *<sup>δ</sup>M*2) [35]. Now the equation *a*(*T*) = 0 defines the critical temperature of the Meissner phase and the equation |*as*| = *μBM* stands for *Tc*2(*M*). It is readily seen that these two equations can be written

Theory of Ferromagnetic Unconventional Superconductors with Spin-Triplet Electron Pairing 421

the phase transition line corresponding to the vortex phase, described by the model (1) at zero external magnetic field and generated by the magnetization *M*, can be obtained from the phase transition line corresponding to the uniform superconducting phase by an effective change of the value of the parameter *γ*0. Both lines have the same shape and this is a particular property of the present model. The variation of the parameter *γ*<sup>0</sup> generates a family of lines. Now we propose a possible way of theoretical treatment of the *TFS*(*P*) line of the FM-FS phase transition, shown in Fig. (1). This is a crucial point in our theory. The phase transition line of the uniform superconducting phase can be calculated within the thermodynamic analysis of the uniform phases, described by the free energy (1). This analysis is done in a simple variant of the free energy (1) in which the fields *ψ* and *M* do not depend on the spatial vector *x*. The accomplishment of such analysis will give a formula for the phase transition line *TFS*(*P*) which corresponds a Meissner phase coexisting with the ferromagnetic order. The theoretical result for *TFS*(*P*) will contain a unspecified parameter *γ*0. If the theoretical line *TFS*(*P*) is fitted to the experimental data for the FM-FS transition line corresponding to a particular compound, the two curves will coincide for some value of *γ*0, irrespectively on the structure of the FS phase. If the FS phase contains a vortex superconductivity the fitting parameter

*γ*0(*eff*) should be identified as *γ*0. These arguments justify our approach to the investigation of the experimental data for the phase diagrams of intermetallic compounds with FM and FS

In the previous section we have justified a thermodynamic analysis of the free energy (1) in terms of uniform order parameters. Neglecting the *x*-dependence of *ψ* and *M*, the free energy per unit volume, *F*/*V* = *f*(*ψ*, *M*) in zero external magnetic field (*H* = 0), can be written in

1 gives the standard form of *a <sup>f</sup>* , and *n* = 2 applies for SFT [30] and the Stoner-Wohlfarth model [36]. Previous studies [20] have shown that the anisotropy represented by the *us* and *vs* terms in Eq. (6) slightly perturbs the size and shape of the stability domains of the phases,

reasons, in the present analysis we ignore the anisotropy terms, setting *us* = *vs* = 0, and consider *bs* ≡ *b* > 0 as an effective parameter. Then, without loss of generality, we are free to

<sup>2</sup> + *vs* 2

3 ∑ *j*=1 |*ψj*|

2.

<sup>4</sup> + *<sup>a</sup> <sup>f</sup> <sup>M</sup>*<sup>2</sup> +

*b f*

<sup>2</sup> *<sup>M</sup>*<sup>4</sup> (6)

*<sup>f</sup>* (*P*)], where *n* =

<sup>4</sup> term. For these

phases. In the remainder of this paper, we shall investigate uniform phases.

<sup>+</sup> *<sup>i</sup>γ*0*<sup>M</sup>* · (*<sup>ψ</sup>* <sup>×</sup> *<sup>ψ</sup>*∗) + *<sup>δ</sup>*0*M*2|*ψ*<sup>|</sup>

Here we slightly modify the parameter *<sup>a</sup> <sup>f</sup>* by choosing *<sup>a</sup> <sup>f</sup>* <sup>=</sup> *<sup>α</sup><sup>f</sup>* [*T<sup>n</sup>* <sup>−</sup> *<sup>T</sup><sup>n</sup>*

while similar effects can be achieved by varying the *bs* factor in the *bs*|*ψ*|

choose the magnetization vector to have the form *M* = (0, 0, *M*).

<sup>0</sup> but if the FS phase contains Meissner superconductivity,

<sup>0</sup> = (*γ*<sup>0</sup> − *μB*). Thus

in the same form, provided the parameter *γ*<sup>0</sup> in *a* is substituted by *γ*�

*γ*0(*eff*) should be interpreted as *γ*�

**3. Model considerations**

*f*(*ψ*, *M*) = *as*|*ψ*|

<sup>2</sup> + *bs* <sup>2</sup> <sup>|</sup>*ψ*<sup>|</sup> <sup>4</sup> + *us* <sup>2</sup> <sup>|</sup>*ψ*2<sup>|</sup>

the form

where the coupling parameter *γ*<sup>0</sup> ∼ *J* depends on the ferromagnetic exchange parameter *J* > 0, [23, 24] and *δ*<sup>0</sup> is the standard *M* − *ψ* coupling parameter, known from the theory of multicritical phenomena [31] and from studies of coexistence of ferromagnetism and superconductivity in ternary compounds [27, 28].

As usual, in Eq. (2), *as* = (*T* − *Ts*), where *Ts* is the critical temperature of the generic superconducting transition, *bs* > 0. The parameters *us* and *vs* and *δ*<sup>0</sup> may take some negative values, provided the overall stability of the system is preserved. The values of the material parameters *μ* = (*Ts*, *Tf* , *αs*, *α<sup>f</sup>* , *bs*, *us*, *vs*, *b <sup>f</sup>* , *Kj*, *γ*<sup>0</sup> and *δ*0) depend on the choice of the substance and on intensive thermodynamic parameters, such as the temperature *T* and the pressure *P*. From a microscopic point of view, the parameters *μ* depend on the density of states *UF*(*kF*) on the Fermi surface. On the other hand *UF* varies with *T* and *P*. Thus the relationships (*T*, *P*) *UF μ*, i.e., the functional relations *μ*[*UF*(*T*, *P*)], are of essential interest. While these relations are unknown, one may suppose some direct dependence *μ*(*T*, *P*). The latter should correspond to the experimental data.

The free energy (1) is quite general. It has been deduced by several reliable arguments. In order to construct Eq. (1)–(5) we have used the standard GL theory of superconductors and the phase transition theory with an account of the relevant anisotropy of the *p*-wave Cooper pairs and the crystal anisotropy, described by the *us*- and *vs*-terms in Eq. (2), respectively. Besides, we have used the general case of cubic anisotropy, when all three components *ψ<sup>j</sup>* of *ψ* are relevant. Note, that in certain real cases, for example, in UGe2, the crystal symmetry is tetragonal, *ψ* effectively behaves as a two-component vector and this leads to a considerable simplification of the theory. As shown in Ref. [20], the mentioned anisotropy terms are not essential in the description of the main thermodynamic properties, including the shape of the *T* − *P* phase diagram. For this reason we shall often ignore the respective terms in Eq. (2). The *<sup>γ</sup>*0-term triggers the superconductivity (*M*-triger effect [20, 21]) while the *<sup>δ</sup>*0*M*2|*ψ*<sup>|</sup> 2–term makes the model more realistic for large values of *M*. This allows for an extension of the domain of the stable ferromagnetic order up to zero temperatures for a wide range of values of the material parameters and the pressure *P*. Such a picture corresponds to the real situation in ferromagnetic compounds [20].

The total free energy (1) is difficult for a theoretical investigation. The various vortex and uniform phases described by this complex model cannot be investigated within a single calculation but rather one should focus on particular problems. In Ref. [24] the vortex phase was discussed with the help of the criterion [33] for a stability of this state near the phase transition line *Tc*2(*B*), ; see also, Ref. [34]. The phase transition line *Tc*2(*H*) of a usual superconductor in external magnetic field *H* = |*H*| is located above the phase transition line *Ts* of the uniform (Meissner) phase. The reason is that *Ts* is defined by the equation *as*(*T*) = 0, whereas *Tc*2(*H*) is a solution of the equation |*as*| = *μBH*, where *μ<sup>B</sup>* = |*e*|*h*¯ /2*mc* is the Bohr magneton [34]. For ferromagnetic superconductors, where *M* > 0, one should use the magnetic induction *B* rather than *H*. In case of *H* = 0 one should apply the same criterion with respect to the magnetization *M* for small values of |*ψ*| near the phase transition line *Tc*2(*M*);

*M* = |*M*|. For this reason we should use the diagonal quadratic form [35] corresponding to the entire *ψ*2-part of the total free energy functional (1). The lowest energy term in this diagonal quadratic part contains a coefficient *<sup>a</sup>* of the form *<sup>a</sup>* = (*as* <sup>−</sup> *<sup>γ</sup>*0*<sup>M</sup>* <sup>−</sup> *<sup>δ</sup>M*2) [35]. Now the equation *a*(*T*) = 0 defines the critical temperature of the Meissner phase and the equation |*as*| = *μBM* stands for *Tc*2(*M*). It is readily seen that these two equations can be written in the same form, provided the parameter *γ*<sup>0</sup> in *a* is substituted by *γ*� <sup>0</sup> = (*γ*<sup>0</sup> − *μB*). Thus the phase transition line corresponding to the vortex phase, described by the model (1) at zero external magnetic field and generated by the magnetization *M*, can be obtained from the phase transition line corresponding to the uniform superconducting phase by an effective change of the value of the parameter *γ*0. Both lines have the same shape and this is a particular property of the present model. The variation of the parameter *γ*<sup>0</sup> generates a family of lines.

Now we propose a possible way of theoretical treatment of the *TFS*(*P*) line of the FM-FS phase transition, shown in Fig. (1). This is a crucial point in our theory. The phase transition line of the uniform superconducting phase can be calculated within the thermodynamic analysis of the uniform phases, described by the free energy (1). This analysis is done in a simple variant of the free energy (1) in which the fields *ψ* and *M* do not depend on the spatial vector *x*. The accomplishment of such analysis will give a formula for the phase transition line *TFS*(*P*) which corresponds a Meissner phase coexisting with the ferromagnetic order. The theoretical result for *TFS*(*P*) will contain a unspecified parameter *γ*0. If the theoretical line *TFS*(*P*) is fitted to the experimental data for the FM-FS transition line corresponding to a particular compound, the two curves will coincide for some value of *γ*0, irrespectively on the structure of the FS phase. If the FS phase contains a vortex superconductivity the fitting parameter *γ*0(*eff*) should be interpreted as *γ*� <sup>0</sup> but if the FS phase contains Meissner superconductivity, *γ*0(*eff*) should be identified as *γ*0. These arguments justify our approach to the investigation of the experimental data for the phase diagrams of intermetallic compounds with FM and FS phases. In the remainder of this paper, we shall investigate uniform phases.

### **3. Model considerations**

6 Will-be-set-by-IN-TECH

with *c <sup>f</sup>* , *b <sup>f</sup>* > 0, and *a <sup>f</sup>* = *α*(*T* − *Tf*), where *α<sup>f</sup>* > 0 and *Tf* is the critical temperature, corresponding of the generic ferromagnetic phase transition. Finally, the energy *f*I(*ψ*, *M*)

*<sup>f</sup>*I(*ψ*, *<sup>M</sup>*) = *<sup>i</sup>γ*0*M*.(*<sup>ψ</sup>* <sup>×</sup> *<sup>ψ</sup>*∗) + *<sup>δ</sup>*0*M*2|*ψ*<sup>|</sup>

where the coupling parameter *γ*<sup>0</sup> ∼ *J* depends on the ferromagnetic exchange parameter *J* > 0, [23, 24] and *δ*<sup>0</sup> is the standard *M* − *ψ* coupling parameter, known from the theory of multicritical phenomena [31] and from studies of coexistence of ferromagnetism and

As usual, in Eq. (2), *as* = (*T* − *Ts*), where *Ts* is the critical temperature of the generic superconducting transition, *bs* > 0. The parameters *us* and *vs* and *δ*<sup>0</sup> may take some negative values, provided the overall stability of the system is preserved. The values of the material parameters *μ* = (*Ts*, *Tf* , *αs*, *α<sup>f</sup>* , *bs*, *us*, *vs*, *b <sup>f</sup>* , *Kj*, *γ*<sup>0</sup> and *δ*0) depend on the choice of the substance and on intensive thermodynamic parameters, such as the temperature *T* and the pressure *P*. From a microscopic point of view, the parameters *μ* depend on the density of states *UF*(*kF*) on the Fermi surface. On the other hand *UF* varies with *T* and *P*. Thus the relationships (*T*, *P*) *UF μ*, i.e., the functional relations *μ*[*UF*(*T*, *P*)], are of essential interest. While these relations are unknown, one may suppose some direct dependence *μ*(*T*, *P*). The latter

The free energy (1) is quite general. It has been deduced by several reliable arguments. In order to construct Eq. (1)–(5) we have used the standard GL theory of superconductors and the phase transition theory with an account of the relevant anisotropy of the *p*-wave Cooper pairs and the crystal anisotropy, described by the *us*- and *vs*-terms in Eq. (2), respectively. Besides, we have used the general case of cubic anisotropy, when all three components *ψ<sup>j</sup>* of *ψ* are relevant. Note, that in certain real cases, for example, in UGe2, the crystal symmetry is tetragonal, *ψ* effectively behaves as a two-component vector and this leads to a considerable simplification of the theory. As shown in Ref. [20], the mentioned anisotropy terms are not essential in the description of the main thermodynamic properties, including the shape of the *T* − *P* phase diagram. For this reason we shall often ignore the respective terms in Eq. (2). The *<sup>γ</sup>*0-term triggers the superconductivity (*M*-triger effect [20, 21]) while the *<sup>δ</sup>*0*M*2|*ψ*<sup>|</sup>

makes the model more realistic for large values of *M*. This allows for an extension of the domain of the stable ferromagnetic order up to zero temperatures for a wide range of values of the material parameters and the pressure *P*. Such a picture corresponds to the real situation

The total free energy (1) is difficult for a theoretical investigation. The various vortex and uniform phases described by this complex model cannot be investigated within a single calculation but rather one should focus on particular problems. In Ref. [24] the vortex phase was discussed with the help of the criterion [33] for a stability of this state near the phase transition line *Tc*2(*B*), ; see also, Ref. [34]. The phase transition line *Tc*2(*H*) of a usual superconductor in external magnetic field *H* = |*H*| is located above the phase transition line *Ts* of the uniform (Meissner) phase. The reason is that *Ts* is defined by the equation *as*(*T*) = 0, whereas *Tc*2(*H*) is a solution of the equation |*as*| = *μBH*, where *μ<sup>B</sup>* = |*e*|*h*¯ /2*mc* is the Bohr magneton [34]. For ferromagnetic superconductors, where *M* > 0, one should use the magnetic induction *B* rather than *H*. In case of *H* = 0 one should apply the same criterion with respect to the magnetization *M* for small values of |*ψ*| near the phase transition line *Tc*2(*M*);

2, (5)

2–term

produced by the possible couplings of *ψ* and *M* is given by

superconductivity in ternary compounds [27, 28].

should correspond to the experimental data.

in ferromagnetic compounds [20].

In the previous section we have justified a thermodynamic analysis of the free energy (1) in terms of uniform order parameters. Neglecting the *x*-dependence of *ψ* and *M*, the free energy per unit volume, *F*/*V* = *f*(*ψ*, *M*) in zero external magnetic field (*H* = 0), can be written in the form

$$f(\boldsymbol{\upmu}, \boldsymbol{\upmu}) = a\_s |\boldsymbol{\upmu}|^2 + \frac{b\_s}{2} |\boldsymbol{\upmu}|^4 + \frac{u\_s}{2} |\boldsymbol{\upmu}^2|^2 + \frac{v\_s}{2} \sum\_{j=1}^3 |\boldsymbol{\upmu}\_j|^4 + a\_f \boldsymbol{\upmu}^2 + \frac{b\_f}{2} \boldsymbol{\upmu}^4 \tag{6}$$

$$+ i \boldsymbol{\upgamma}\_0 \mathbf{M} \cdot (\boldsymbol{\upupmu} \times \boldsymbol{\upmu}^\*) + \delta\_0 \mathbf{M}^2 |\boldsymbol{\upupmu}|^2.$$

Here we slightly modify the parameter *<sup>a</sup> <sup>f</sup>* by choosing *<sup>a</sup> <sup>f</sup>* <sup>=</sup> *<sup>α</sup><sup>f</sup>* [*T<sup>n</sup>* <sup>−</sup> *<sup>T</sup><sup>n</sup> <sup>f</sup>* (*P*)], where *n* = 1 gives the standard form of *a <sup>f</sup>* , and *n* = 2 applies for SFT [30] and the Stoner-Wohlfarth model [36]. Previous studies [20] have shown that the anisotropy represented by the *us* and *vs* terms in Eq. (6) slightly perturbs the size and shape of the stability domains of the phases, while similar effects can be achieved by varying the *bs* factor in the *bs*|*ψ*| <sup>4</sup> term. For these reasons, in the present analysis we ignore the anisotropy terms, setting *us* = *vs* = 0, and consider *bs* ≡ *b* > 0 as an effective parameter. Then, without loss of generality, we are free to choose the magnetization vector to have the form *M* = (0, 0, *M*).

#### 8 Will-be-set-by-IN-TECH 422 Superconductors – Materials, Properties and Applications Theory of Ferromagnetic Unconventional Superconductors with Spin-Triplet Electron Pairing <sup>9</sup>

According to the microscopic theory of band magnetism and superconductivity the macroscopic material parameters in Eq. (6) depend in a quite complex way on the density of states at the Fermi level and related microscopic quantities [37]. That is why we can hardly use the microscopic characteristics of these complex metallic compounds in order to elucidate their thermodynamic properties, in particular, in outlining their phase diagrams in some details. However, some microscopic simple microscopic models reveal useful results, for example, the zero temperature Stoner-type model employed in Ref. [38].

We redefine for convenience the free energy (6) in a dimensionless form by ˜ *f* = *f* /(*b <sup>f</sup> M*<sup>4</sup> 0), where *M*<sup>0</sup> = [*α<sup>f</sup> T<sup>n</sup> <sup>f</sup>* <sup>0</sup>/*b <sup>f</sup>* ] 1/2 > 0 is the value of the magnetization *M* corresponding to the pure magnetic subsystem (*ψ* ≡ 0) at *T* = *P* = 0 and *Tf* <sup>0</sup> = *Tf*(0). The order parameters assume the scaling *m* = *M*/*M*<sup>0</sup> and *ϕ* = *ψ*/[(*b <sup>f</sup>* /*b*)1/4*M*0], and as a result, the free energy becomes

$$f = r\phi^2 + \frac{\phi^4}{2} + tm^2 + \frac{m^4}{2} + 2\gamma m\phi\_1\phi\_2\sin\theta + \gamma\_1 m^2\phi^2. \tag{7}$$

where *φ<sup>j</sup>* = |*ϕj*|, *φ* = |*ϕ*|, and *θ* = (*θ*<sup>2</sup> − *θ*1) is the phase angle between the complex *ϕ*<sup>1</sup> = *φ*1*eiθ*<sup>1</sup> and *ϕ*<sup>2</sup> = *φ*2*eθ*<sup>2</sup> . Note that the phase angle *θ*3, corresponding to the third complex field component *ϕ*<sup>3</sup> = *φ*3*eiθ*<sup>3</sup> does not enter explicitly in the free energy ˜ *f* , given by Eq. (7), which is a natural result of the continuous space degeneration. The dimensionless parameters *t*, *r*, *γ* and *γ*<sup>1</sup> in Eq. (7) are given by

$$t = \tilde{T}^{\text{\textquotedblleft}} - \tilde{T}\_f^{\text{\textquotedblleft}}(P), \quad r = \kappa(\tilde{T} - \tilde{T}\_s), \tag{8}$$

and is well defined for any *P*˜. This allows for the consideration of pressures *P* > *P*<sup>0</sup> within the

Theory of Ferromagnetic Unconventional Superconductors with Spin-Triplet Electron Pairing 423

The model function *<sup>T</sup>*˜*f*(*P*) can be naturally generalized to *<sup>T</sup>*˜*f*(*P*)=(<sup>1</sup> <sup>−</sup> *<sup>P</sup>*˜*β*)1/*<sup>α</sup>* but the present needs of interpretation of experimental data do not require such a complex consideration (hereafter we use Eq. (9) which corresponds to *β* = 1 and *α* = *n*). Besides, other analytical forms of *T*˜*f*(*P*˜) can also be tested in the free energy (7), in particular, expansion in powers of *<sup>P</sup>*˜, or, alternatively, in (<sup>1</sup> <sup>−</sup> *<sup>P</sup>*˜) which satisfy the conditions *<sup>T</sup>*˜*f*(0) = 1 and *<sup>T</sup>*˜*f*(1) = 0. Note, that in URhGe the slope of *TF*(*P*) ∼ *Tf*(*P*) is positive from *P* = 0 up to high pressures [8] and for this compound the form (9) of *T*˜*f*(*P*) is inconvenient. Here we apply the simplest variants

In more general terms, all material parameters (*r*, *t*, *γ*, . . . ) may depend on the pressure. We suppose that a suitable choice of the dependence of *t* on *P* is enough for describing the main thermodynamic properties and this supposition is supported by the final results, presented in the remainder of this paper. But in some particular investigations one may need to introduce

The simplified model (7) is capable of describing the main thermodynamic properties of spin-triplet ferromagnetic superconductors. For *r* > 0, i.e., *T* > *Ts*, there are three stable phases [20]: (i) the normal (N-) phase, given by *φ* = *m* = 0 (stability conditions: *t* ≥ 0, *<sup>r</sup>* <sup>≥</sup> 0); (ii) the pure ferromagnetic phase (FM phase), given by *<sup>m</sup>* = (−*t*)1/2 <sup>&</sup>gt; 0, *<sup>φ</sup>* <sup>=</sup> 0,

already mentioned phase of coexistence of ferromagnetic order and superconductivity (FS

*<sup>T</sup>*˜ *<sup>n</sup>* <sup>+</sup> *κγ*1(*T*˜*<sup>s</sup>* <sup>−</sup> *<sup>T</sup>*˜) + *<sup>P</sup>*˜ <sup>−</sup> <sup>1</sup> <sup>−</sup> *<sup>γ</sup>*<sup>2</sup>

*<sup>c</sup>*<sup>2</sup> <sup>=</sup> <sup>3</sup>*γγ*1, *<sup>c</sup>*<sup>3</sup> <sup>=</sup> <sup>2</sup>(<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*<sup>2</sup>

The FS phase contains two thermodynamically equivalent phase domains that can be distinguished by the upper and lower signs (±) of some terms in Eqs. (11) and (12). The upper sign describes the domain (labelled bellow again by FS), where *m* > 0, sin*θ* = −1, whereas the lower sign describes the conjunct domain FS∗, where *m* < 0 and sin*θ* = 1 (for details, see, Ref. [20]). Here we consider one of the two thermodynamically equivalent phase domains, namely, the domain FS, which is stable for *m* > 0 (FS∗ is stable for *m* < 0). This "one-domain approximation" correctly presents the main thermodynamic properties

<sup>√</sup>2, where

*<sup>s</sup>* <sup>−</sup> *<sup>T</sup>*˜) <sup>±</sup> *<sup>γ</sup><sup>m</sup>* <sup>−</sup> *<sup>γ</sup>*1*m*<sup>2</sup> <sup>≥</sup> 0. (11)

*<sup>c</sup>*3*m*<sup>3</sup> <sup>±</sup> *<sup>c</sup>*2*m*<sup>2</sup> <sup>+</sup> *<sup>c</sup>*1*<sup>m</sup>* <sup>±</sup> *<sup>c</sup>*<sup>0</sup> <sup>=</sup> <sup>0</sup> (12)

2  1/2), and (iii) the

, (13)

<sup>1</sup>). (14)

which exists for *t* < 0 and is stable provided *r* ≥ 0 and *r* ≥ (*γ*1*t* + *γ*|*t*|

*φ*<sup>2</sup> = *κ*(*T*˜

*s*),

*c*<sup>1</sup> = 2 

free energy (7).

**4. Stable phases**

of *P*-dependence, namely, Eqs. (9) and (10).

a suitable pressure dependence of other parameters.

phase), given by sin*θ* = ∓1, *φ*<sup>3</sup> = 0, *φ*<sup>1</sup> = *φ*<sup>2</sup> = *φ*/

The magnetization *m* satisfies the equation

with coefficients *<sup>c</sup>*<sup>0</sup> <sup>=</sup> *γκ*(*T*˜ <sup>−</sup> *<sup>T</sup>*˜

where *κ* = *αsb*1/2 *<sup>f</sup>* /*α<sup>f</sup> <sup>b</sup>*1/2*Tn*−<sup>1</sup> *<sup>f</sup>* <sup>0</sup> , *<sup>γ</sup>* <sup>=</sup> *<sup>γ</sup>*0/[*α<sup>f</sup> <sup>T</sup><sup>n</sup> <sup>f</sup>* <sup>0</sup>*b*] 1/2, and *γ*<sup>1</sup> = *δ*0/(*bb <sup>f</sup>*)1/2. The reduced temperatures are *T*˜ = *T*/*Tf* 0, *T*˜*f*(*P*) = *Tf*(*P*)/*Tf* 0, and *T*˜ *<sup>s</sup>*(*P*) = *Ts*(*P*)/*Tf* 0.

The analysis involves making simple assumptions for the *P* dependence of the *t*, *r*, *γ*, and *γ*<sup>1</sup> parameters in Eq. (7). Specifically, we assume that only *Tf* has a significant *P* dependence, described by

*<sup>T</sup>*˜*f*(*P*)=(<sup>1</sup> <sup>−</sup> *<sup>P</sup>*˜) 1/*n*, (9)

where *P*˜ = *P*/*P*<sup>0</sup> and *P*<sup>0</sup> is a characteristic pressure deduced later. In ZrZn2 and UGe2 the *P*<sup>0</sup> values are very close to the critical pressure *Pc* at which both the ferromagnetic and superconducting orders vanish, but in other systems this is not necessarily the case. As we will discuss, the nonlinearity (*n* = 2) of *Tf*(*P*) in ZrZn2 and UGe2 is relevant at relatively high *P*, at which the N-FM transition temperature *TF*(*P*) may not coincide with *Tf*(*P*); *TF*(*P*) is the actual line of the N-FM phase transition, as shown in Fig. (1). The form (9) of the model function *T*˜*f*(*P*) is consistent with preceding experimental and theoretical investigations of the N-FM phase transition in ZrZn2 and UGe2 (see, e.g., Refs. [4, 24, 39]). Here we consider only non-negative values of the pressure *P* (for effects at *P* < 0, see, e.g., Ref. [44]).

The model function (9) is defined for *P* ≤ *P*0, in particular, for the case of *n* > 1, but we should have in mind that, in fact, the thermodynamic analysis of Eq. (7) includes the parameter *t* rather than *Tf*(*P*). This parameter is given by

$$t(T\_\prime P) = \tilde{T}^n - 1 + \tilde{P}\_\prime \tag{10}$$

and is well defined for any *P*˜. This allows for the consideration of pressures *P* > *P*<sup>0</sup> within the free energy (7).

The model function *<sup>T</sup>*˜*f*(*P*) can be naturally generalized to *<sup>T</sup>*˜*f*(*P*)=(<sup>1</sup> <sup>−</sup> *<sup>P</sup>*˜*β*)1/*<sup>α</sup>* but the present needs of interpretation of experimental data do not require such a complex consideration (hereafter we use Eq. (9) which corresponds to *β* = 1 and *α* = *n*). Besides, other analytical forms of *T*˜*f*(*P*˜) can also be tested in the free energy (7), in particular, expansion in powers of *<sup>P</sup>*˜, or, alternatively, in (<sup>1</sup> <sup>−</sup> *<sup>P</sup>*˜) which satisfy the conditions *<sup>T</sup>*˜*f*(0) = 1 and *<sup>T</sup>*˜*f*(1) = 0. Note, that in URhGe the slope of *TF*(*P*) ∼ *Tf*(*P*) is positive from *P* = 0 up to high pressures [8] and for this compound the form (9) of *T*˜*f*(*P*) is inconvenient. Here we apply the simplest variants of *P*-dependence, namely, Eqs. (9) and (10).

In more general terms, all material parameters (*r*, *t*, *γ*, . . . ) may depend on the pressure. We suppose that a suitable choice of the dependence of *t* on *P* is enough for describing the main thermodynamic properties and this supposition is supported by the final results, presented in the remainder of this paper. But in some particular investigations one may need to introduce a suitable pressure dependence of other parameters.

### **4. Stable phases**

8 Will-be-set-by-IN-TECH

According to the microscopic theory of band magnetism and superconductivity the macroscopic material parameters in Eq. (6) depend in a quite complex way on the density of states at the Fermi level and related microscopic quantities [37]. That is why we can hardly use the microscopic characteristics of these complex metallic compounds in order to elucidate their thermodynamic properties, in particular, in outlining their phase diagrams in some details. However, some microscopic simple microscopic models reveal useful results,

magnetic subsystem (*ψ* ≡ 0) at *T* = *P* = 0 and *Tf* <sup>0</sup> = *Tf*(0). The order parameters assume the scaling *m* = *M*/*M*<sup>0</sup> and *ϕ* = *ψ*/[(*b <sup>f</sup>* /*b*)1/4*M*0], and as a result, the free energy becomes

where *φ<sup>j</sup>* = |*ϕj*|, *φ* = |*ϕ*|, and *θ* = (*θ*<sup>2</sup> − *θ*1) is the phase angle between the complex *ϕ*<sup>1</sup> = *φ*1*eiθ*<sup>1</sup> and *ϕ*<sup>2</sup> = *φ*2*eθ*<sup>2</sup> . Note that the phase angle *θ*3, corresponding to the third complex field

is a natural result of the continuous space degeneration. The dimensionless parameters *t*, *r*, *γ*

*<sup>f</sup>* <sup>0</sup>*b*]

The analysis involves making simple assumptions for the *P* dependence of the *t*, *r*, *γ*, and *γ*<sup>1</sup> parameters in Eq. (7). Specifically, we assume that only *Tf* has a significant *P* dependence,

*<sup>T</sup>*˜*f*(*P*)=(<sup>1</sup> <sup>−</sup> *<sup>P</sup>*˜)

where *P*˜ = *P*/*P*<sup>0</sup> and *P*<sup>0</sup> is a characteristic pressure deduced later. In ZrZn2 and UGe2 the *P*<sup>0</sup> values are very close to the critical pressure *Pc* at which both the ferromagnetic and superconducting orders vanish, but in other systems this is not necessarily the case. As we will discuss, the nonlinearity (*n* = 2) of *Tf*(*P*) in ZrZn2 and UGe2 is relevant at relatively high *P*, at which the N-FM transition temperature *TF*(*P*) may not coincide with *Tf*(*P*); *TF*(*P*) is the actual line of the N-FM phase transition, as shown in Fig. (1). The form (9) of the model function *T*˜*f*(*P*) is consistent with preceding experimental and theoretical investigations of the N-FM phase transition in ZrZn2 and UGe2 (see, e.g., Refs. [4, 24, 39]). Here we consider only

The model function (9) is defined for *P* ≤ *P*0, in particular, for the case of *n* > 1, but we should have in mind that, in fact, the thermodynamic analysis of Eq. (7) includes the parameter *t*

*m*4

1/2 > 0 is the value of the magnetization *M* corresponding to the pure

<sup>2</sup> <sup>+</sup> <sup>2</sup>*γmφ*1*φ*2sin*<sup>θ</sup>* <sup>+</sup> *<sup>γ</sup>*1*m*2*φ*2, (7)

*<sup>f</sup>* (*P*), *<sup>r</sup>* <sup>=</sup> *<sup>κ</sup>*(*T*˜ <sup>−</sup> *<sup>T</sup>*˜*s*), (8)

*<sup>s</sup>*(*P*) = *Ts*(*P*)/*Tf* 0.

*<sup>t</sup>*(*T*, *<sup>P</sup>*) = *<sup>T</sup>*˜ *<sup>n</sup>* <sup>−</sup> <sup>1</sup> <sup>+</sup> *<sup>P</sup>*˜, (10)

1/2, and *γ*<sup>1</sup> = *δ*0/(*bb <sup>f</sup>*)1/2. The reduced

1/*n*, (9)

*f* = *f* /(*b <sup>f</sup> M*<sup>4</sup>

*f* , given by Eq. (7), which

0),

for example, the zero temperature Stoner-type model employed in Ref. [38]. We redefine for convenience the free energy (6) in a dimensionless form by ˜

<sup>2</sup> <sup>+</sup> *tm*<sup>2</sup> <sup>+</sup>

component *ϕ*<sup>3</sup> = *φ*3*eiθ*<sup>3</sup> does not enter explicitly in the free energy ˜

*<sup>t</sup>* <sup>=</sup> *<sup>T</sup>*˜ *<sup>n</sup>* <sup>−</sup> *<sup>T</sup>*˜ *<sup>n</sup>*

*<sup>f</sup>* <sup>0</sup> , *<sup>γ</sup>* <sup>=</sup> *<sup>γ</sup>*0/[*α<sup>f</sup> <sup>T</sup><sup>n</sup>*

non-negative values of the pressure *P* (for effects at *P* < 0, see, e.g., Ref. [44]).

where *M*<sup>0</sup> = [*α<sup>f</sup> T<sup>n</sup>*

*<sup>f</sup>* <sup>0</sup>/*b <sup>f</sup>* ]

˜

*<sup>f</sup>* /*α<sup>f</sup> <sup>b</sup>*1/2*Tn*−<sup>1</sup>

rather than *Tf*(*P*). This parameter is given by

temperatures are *T*˜ = *T*/*Tf* 0, *T*˜*f*(*P*) = *Tf*(*P*)/*Tf* 0, and *T*˜

and *γ*<sup>1</sup> in Eq. (7) are given by

where *κ* = *αsb*1/2

described by

*<sup>f</sup>* <sup>=</sup> *<sup>r</sup>φ*<sup>2</sup> <sup>+</sup> *<sup>φ</sup>*<sup>4</sup>

The simplified model (7) is capable of describing the main thermodynamic properties of spin-triplet ferromagnetic superconductors. For *r* > 0, i.e., *T* > *Ts*, there are three stable phases [20]: (i) the normal (N-) phase, given by *φ* = *m* = 0 (stability conditions: *t* ≥ 0, *<sup>r</sup>* <sup>≥</sup> 0); (ii) the pure ferromagnetic phase (FM phase), given by *<sup>m</sup>* = (−*t*)1/2 <sup>&</sup>gt; 0, *<sup>φ</sup>* <sup>=</sup> 0, which exists for *t* < 0 and is stable provided *r* ≥ 0 and *r* ≥ (*γ*1*t* + *γ*|*t*| 1/2), and (iii) the already mentioned phase of coexistence of ferromagnetic order and superconductivity (FS phase), given by sin*θ* = ∓1, *φ*<sup>3</sup> = 0, *φ*<sup>1</sup> = *φ*<sup>2</sup> = *φ*/ <sup>√</sup>2, where

$$
\phi^2 = \kappa (\tilde{T}\_s - \tilde{T}) \pm \gamma m - \gamma\_1 m^2 \ge 0. \tag{11}
$$

The magnetization *m* satisfies the equation

$$c\_3 m^3 \pm c\_2 m^2 + c\_1 m \pm c\_0 = 0\tag{12}$$

with coefficients *<sup>c</sup>*<sup>0</sup> <sup>=</sup> *γκ*(*T*˜ <sup>−</sup> *<sup>T</sup>*˜ *s*),

$$\mathcal{L}\_1 = 2\left[\mathcal{T}^n + \kappa \gamma\_1 (\mathcal{T}\_s - \mathcal{T}) + \mathcal{P} - 1 - \frac{\gamma^2}{2}\right],\tag{13}$$

$$\mathbf{c}\_2 = \mathbf{3}\gamma\gamma\_1, \quad \mathbf{c}\_3 = \mathbf{2}(1-\gamma\_1^2). \tag{14}$$

The FS phase contains two thermodynamically equivalent phase domains that can be distinguished by the upper and lower signs (±) of some terms in Eqs. (11) and (12). The upper sign describes the domain (labelled bellow again by FS), where *m* > 0, sin*θ* = −1, whereas the lower sign describes the conjunct domain FS∗, where *m* < 0 and sin*θ* = 1 (for details, see, Ref. [20]). Here we consider one of the two thermodynamically equivalent phase domains, namely, the domain FS, which is stable for *m* > 0 (FS∗ is stable for *m* < 0). This "one-domain approximation" correctly presents the main thermodynamic properties


**Table 1.** Theoretical results for the location [(*T*˜, *P*˜) - reduced coordinates] of the tricritical points A <sup>≡</sup> (*T*˜*A*, *<sup>P</sup>*˜ *<sup>A</sup>*) and B <sup>≡</sup> (*T*˜ *<sup>B</sup>*, *P*˜ *<sup>B</sup>*), the critical-end point C <sup>≡</sup> (*T*˜ *<sup>C</sup>*, *P*˜ *<sup>C</sup>*), and the point of temperature maximum, *max* =(*T*˜ *<sup>m</sup>*, *P*˜ *<sup>m</sup>*) on the curve *T*˜ *FS*(*P*˜) of the FM-FS phase transitions of first and second orders (for details, see Sec. 5). The first column shows *<sup>T</sup>*˜*<sup>N</sup>* <sup>≡</sup> *<sup>T</sup>*˜ (*A*,*B*,*C*,*m*). The second column stands for *tN* = *t*(*A*,*B*,*C*,*m*). The reduced pressure values *P*˜ (*A*,*B*,*C*,*m*) of points A, B, C, and *max* are denoted by *<sup>P</sup>*˜ *<sup>N</sup>*(*n*): *n* = 1 stands for the linear dependence *Tf*(*P*), and *n* = 2 stands for the nonlinear *Tf*(*P*) and *t*(*T*), corresponding to SFT.

described by the model (6), in particular, in the case of a lack of external symmetry breaking fields. The stability conditions for the FS phase domain given by Eqs.(11) and (12) are *γM* ≥ 0,

$$
\kappa (\tilde{T}\_s - \tilde{T}) \pm \gamma m - 2\gamma\_1 m^2 \ge 0,\tag{15}
$$

theory, namely, *T*˜

and the temperature *T*˜

*<sup>A</sup>* and *T*˜

*<sup>s</sup>*, as well as the dependence of *T*˜

derivation of the dependence of the multicritical temperatures *T*˜

*<sup>m</sup>* <sup>&</sup>gt; 0, i.e., when *<sup>γ</sup>*2/4*κγ*<sup>1</sup> <sup>&</sup>gt; <sup>|</sup>*T*˜*s*<sup>|</sup> (see Table 1).

where both superconductivity and ferromagnetic orders vanish.

where *tFS*(*P*˜) = *<sup>t</sup>*(*TFS*, *<sup>P</sup>*˜) <sup>≤</sup> 0, *<sup>γ</sup>*˜ <sup>=</sup> *<sup>γ</sup>*/*κ*, *<sup>γ</sup>*˜1 <sup>=</sup> *<sup>γ</sup>*1/*κ*, and 0 <sup>&</sup>lt; *<sup>P</sup>*˜ <sup>&</sup>lt; *<sup>P</sup>*˜

<sup>1</sup> <sup>−</sup> *nT*˜ *<sup>n</sup>*−<sup>1</sup> *FS* 

and 6). For certain ratios of *γ*˜, *γ*˜1, and values of *ρ*˜*s*, the curve *T*˜

figures. The second order phase transition line *T*˜

*T*˜

phase with respect to appearance of FS phase [20].

*FS*(*P*˜)/*∂P*˜, namely,

given by the solution of the equation

derivative *ρ*˜ = *∂T*˜

where *ρ*˜*<sup>s</sup>* = *∂T*˜

*T*˜ *<sup>m</sup>* = *T*˜

The shape of the line *T*˜

*FS*(*P*˜

above *Ts* owing to positive terms of order *γ*2. If *T*˜

**5. Temperature-pressure phase diagram**

temperatures *T*˜

provided *T*˜

*T*˜

*<sup>m</sup>* – the maximum of the curve *TFS*(*P*) (if available, see Sec. 5), the

Theory of Ferromagnetic Unconventional Superconductors with Spin-Triplet Electron Pairing 425

*<sup>A</sup>*, *P*˜

*<sup>A</sup>*, *T*˜

*m* on the same model parameters is outlined in Sec. 5. All

*<sup>C</sup>*, *P*˜

*<sup>s</sup>* < 0, the superconductivity appears,

*FS*(*P*) separating the FM and FS phases is

*<sup>γ</sup>*˜1 <sup>−</sup> *<sup>γ</sup>*˜/2[(−*tFS*)1/2 , (18)

*<sup>B</sup>*; *PB* is the pressure

*FS*(*P*˜) exhibits a maximum

*FS*(*P*˜) = *<sup>T</sup>*˜*<sup>s</sup>* <sup>+</sup> *<sup>γ</sup>*˜1*tFS*(*P*˜) + *<sup>γ</sup>*˜[−*tFS*(*P*˜)]1/2, (17)

*FS*(*P*) can vary depending on the theory parameters (see, e.g., Figs.3

*<sup>m</sup>*), given by *ρ*˜(*ρ*˜*s*, *Tm*, *Pm*) = 0. This maximum is clearly seen in Figs. 6 and 7. To

*<sup>B</sup>* and *T*˜

*<sup>A</sup>*) and B <sup>≡</sup> (*T*˜

*<sup>C</sup>*). The theoretical

*<sup>C</sup>* on *γ*, *γ*1, *κ*, and

*<sup>B</sup>*, *P*˜ *<sup>B</sup>*),

*<sup>B</sup>*, corresponding to the tricritical points A <sup>≡</sup> (*T*˜

*<sup>C</sup>*, corresponding to the critical-end point C <sup>≡</sup> (*T*˜

these temperatures as well as the whole phase transition line *TFS*(*P*) are considerably boosted

Although the structure of the FS phase is quite complicated, some of the results can be obtained in analytical form. A more detailed outline of the phase domains, for example, in *T* − *P* phase diagram, can be done by using suitable values of the material parameters in the free energy (7): *P*0, *Tf* 0, *Ts*, *κ*, *γ*, and *γ*1. Here we present some of the analytical results for the phase transition lines and the multi-critical points. Typical shapes of phase diagrams derived directly from Eq. (7) are given in Figs. 2–7. Figure 2 shows the phase diagram calculated from Eq. (7) for parameters, corresponding to the experimental data [13] for ZrZn2. Figures 3 and 4 show the low-temperature and the high-pressure parts of the same phase diagram (see Sec. 7 for details). Figures 5–7 show the phase diagram calculated for the experimental data [2, 4] of UGe2 (see Sec. 8). In ZrZn2, UGe2, as well as in UCoGe and UIr, critical pressure *Pc* exists,

As in experiments, we find out from our calculation that in the vicinity of *P*<sup>0</sup> ∼ *Pc* the FM-FS phase transition is of fist order, denoted by the solid line BC in Figs. 3, 4, 6, and 7. At lower pressure the same phase transition is of second orderq shown by the dotted lines in the same

corresponding to the multi-critical point B, where the line *TFS*(*P*) terminates, as clearly shown in Figs. 4 and 7). Note, that Eq. (17) strictly coincides with the stability condition for the FM

Additional information for the shape of this phase transition line can be obtained by the

*<sup>ρ</sup>*˜ <sup>=</sup> *<sup>ρ</sup>*˜*<sup>s</sup>* <sup>+</sup> *<sup>γ</sup>*˜1 <sup>−</sup> *<sup>γ</sup>*˜/2(−*tFS*)1/2

*<sup>s</sup>*(*P*˜)/*∂P*˜. Note, that Eq. (18) is obtained from Eqs. (10) and (17).

locate the maximum we need to know *ρ*˜*s*. We have already assumed *Ts* does not depend on *P*, as explained above, which from the physical point of view means that the function *Ts*(*P*) is

and

$$3(1 - \gamma\_1^2)m^2 + 3\gamma\gamma\_1 m + \tilde{T}^n - 1 + \tilde{P} + \kappa\gamma\_1(\tilde{T}\_s - \tilde{T}) - \frac{\gamma^2}{2} \ge 0. \tag{16}$$

These results are valid whenever *Tf*(*P*) > *Ts*(*P*), which excludes any pure superconducting phase (*ψ* �= 0, *m* = 0) in accord with the available experimental data.

For *r* < 0, and *t* > 0 the models (6) and (7) exhibit a stable pure superconducting phase (*φ*<sup>1</sup> = *φ*<sup>2</sup> = *m* = 0, *φ*<sup>2</sup> <sup>3</sup> = −*r*) [20]. This phase may occur in the temperature domain *Tf*(*P*) < *T* < *Ts*. For systems, where *Tf*(0) � *Ts*, this is a domain of pressure in a very close vicinity of *P*<sup>0</sup> ∼ *Pc*, where *TF*(*P*) ∼ *Tf*(*P*) decreases up to values lower than *Ts*. Of course, such a situation is described by the model (7) only if *Ts* > 0. This case is interesting from the experimental point of view only when *Ts* > 0 is enough above zero to enter in the scope of experimentally measurable temperatures. Up to date a pure superconducting phase has not been observed within the accuracy of experiments on the mentioned metallic compounds. For this reason, in the reminder of this paper we shall often assume that the critical temperature *Ts* of the generic superconducting phase transition is either non-positive (*Ts* ≤ 0), or, has a small positive value which can be neglected in the analysis of the available experimental data.

The negative values of the critical temperature *Ts* of the generic superconducting phase transition are generally possible and produce a variety of phase diagram topologies (Sec. 5). Note, that the value of *Ts* depends on the strength of the interaction mediating the formation of the spin-triplet Cooper pairs of electrons. Therefore, for the sensitiveness of such electron couplings to the crystal lattice properties, the generic critical temperature *Ts* depends on the pressure. This is an effect which might be included in our theoretical scheme by introducing some convenient temperature dependence of *Ts*. To do this we need information either from experimental data or from a comprehensive microscopic theory.

Usually, *Ts* ≤ 0 is interpreted as a lack of any superconductivity but here the same non-positive values of *Ts* are effectively enhanced to positive values by the interaction parameter *γ* which triggers the superconductivity up to superconducting phase-transition temperatures *TFS*(*P*) > 0. This is readily seen from Table 1, where we present the reduced critical temperatures on the FM-FS phase transition line *T*˜ *FS*(*P*˜), calculated from the present theory, namely, *T*˜ *<sup>m</sup>* – the maximum of the curve *TFS*(*P*) (if available, see Sec. 5), the temperatures *T*˜ *<sup>A</sup>* and *T*˜ *<sup>B</sup>*, corresponding to the tricritical points A <sup>≡</sup> (*T*˜ *<sup>A</sup>*, *P*˜ *<sup>A</sup>*) and B <sup>≡</sup> (*T*˜ *<sup>B</sup>*, *P*˜ *<sup>B</sup>*), and the temperature *T*˜ *<sup>C</sup>*, corresponding to the critical-end point C <sup>≡</sup> (*T*˜ *<sup>C</sup>*, *P*˜ *<sup>C</sup>*). The theoretical derivation of the dependence of the multicritical temperatures *T*˜ *<sup>A</sup>*, *T*˜ *<sup>B</sup>* and *T*˜ *<sup>C</sup>* on *γ*, *γ*1, *κ*, and *T*˜ *<sup>s</sup>*, as well as the dependence of *T*˜ *m* on the same model parameters is outlined in Sec. 5. All these temperatures as well as the whole phase transition line *TFS*(*P*) are considerably boosted above *Ts* owing to positive terms of order *γ*2. If *T*˜ *<sup>s</sup>* < 0, the superconductivity appears, provided *T*˜ *<sup>m</sup>* <sup>&</sup>gt; 0, i.e., when *<sup>γ</sup>*2/4*κγ*<sup>1</sup> <sup>&</sup>gt; <sup>|</sup>*T*˜*s*<sup>|</sup> (see Table 1).

## **5. Temperature-pressure phase diagram**

10 Will-be-set-by-IN-TECH

*<sup>N</sup>*(*n*)

(*A*,*B*,*C*,*m*). The second column stands for *tN* = *t*(*A*,*B*,*C*,*m*). The

*<sup>κ</sup>*(*T*˜*<sup>s</sup>* <sup>−</sup> *<sup>T</sup>*˜) <sup>±</sup> *<sup>γ</sup><sup>m</sup>* <sup>−</sup> <sup>2</sup>*γ*1*m*<sup>2</sup> <sup>≥</sup> 0, (15)

*<sup>s</sup>* <sup>−</sup> *<sup>T</sup>*˜) <sup>−</sup> *<sup>γ</sup>*<sup>2</sup>

<sup>1</sup> <sup>1</sup> <sup>−</sup> *<sup>T</sup>*˜ *<sup>n</sup>*

*FS*(*P*˜) of the FM-FS phase transitions of first and second orders (for details,

*<sup>C</sup>*, *P*˜

*C*

*<sup>s</sup>* + *γ*2/2

*<sup>m</sup>* <sup>−</sup> *<sup>γ</sup>*2/4*γ*<sup>2</sup>

*<sup>C</sup>*), and the point of temperature maximum,

*<sup>B</sup>* <sup>−</sup> *<sup>γ</sup>*2/4(<sup>1</sup> <sup>+</sup> *<sup>γ</sup>*1)<sup>2</sup>

1

*<sup>N</sup>*(*n*): *n* = 1 stands for the

<sup>2</sup> <sup>≥</sup> 0. (16)

*FS*(*P*˜), calculated from the present

*<sup>N</sup> tN P*˜

*<sup>s</sup>* <sup>+</sup> *<sup>γ</sup>*2(<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*1)/4*κ*(<sup>1</sup> <sup>+</sup> *<sup>γ</sup>*1)<sup>2</sup> <sup>−</sup>*γ*2/4(<sup>1</sup> <sup>+</sup> *<sup>γ</sup>*1)<sup>2</sup> <sup>1</sup> <sup>−</sup> *<sup>T</sup>*˜ *<sup>n</sup>*

*<sup>s</sup>* <sup>+</sup> *<sup>γ</sup>*2/4*κ*(<sup>1</sup> <sup>+</sup> *<sup>γ</sup>*1) <sup>0</sup> <sup>1</sup> <sup>−</sup> *<sup>T</sup>*˜ *<sup>n</sup>*

**Table 1.** Theoretical results for the location [(*T*˜, *P*˜) - reduced coordinates] of the tricritical points A

(*A*,*B*,*C*,*m*) of points A, B, C, and *max* are denoted by *<sup>P</sup>*˜

described by the model (6), in particular, in the case of a lack of external symmetry breaking fields. The stability conditions for the FS phase domain given by Eqs.(11) and (12) are *γM* ≥ 0,

These results are valid whenever *Tf*(*P*) > *Ts*(*P*), which excludes any pure superconducting

For *r* < 0, and *t* > 0 the models (6) and (7) exhibit a stable pure superconducting phase (*φ*<sup>1</sup> =

For systems, where *Tf*(0) � *Ts*, this is a domain of pressure in a very close vicinity of *P*<sup>0</sup> ∼ *Pc*, where *TF*(*P*) ∼ *Tf*(*P*) decreases up to values lower than *Ts*. Of course, such a situation is described by the model (7) only if *Ts* > 0. This case is interesting from the experimental point of view only when *Ts* > 0 is enough above zero to enter in the scope of experimentally measurable temperatures. Up to date a pure superconducting phase has not been observed within the accuracy of experiments on the mentioned metallic compounds. For this reason, in the reminder of this paper we shall often assume that the critical temperature *Ts* of the generic superconducting phase transition is either non-positive (*Ts* ≤ 0), or, has a small positive value

The negative values of the critical temperature *Ts* of the generic superconducting phase transition are generally possible and produce a variety of phase diagram topologies (Sec. 5). Note, that the value of *Ts* depends on the strength of the interaction mediating the formation of the spin-triplet Cooper pairs of electrons. Therefore, for the sensitiveness of such electron couplings to the crystal lattice properties, the generic critical temperature *Ts* depends on the pressure. This is an effect which might be included in our theoretical scheme by introducing some convenient temperature dependence of *Ts*. To do this we need information either from

Usually, *Ts* ≤ 0 is interpreted as a lack of any superconductivity but here the same non-positive values of *Ts* are effectively enhanced to positive values by the interaction parameter *γ* which triggers the superconductivity up to superconducting phase-transition temperatures *TFS*(*P*) > 0. This is readily seen from Table 1, where we present the reduced

<sup>3</sup> = −*r*) [20]. This phase may occur in the temperature domain *Tf*(*P*) < *T* < *Ts*.

linear dependence *Tf*(*P*), and *n* = 2 stands for the nonlinear *Tf*(*P*) and *t*(*T*), corresponding to SFT.

<sup>1</sup>)*m*<sup>2</sup> <sup>+</sup> <sup>3</sup>*γγ*1*<sup>m</sup>* <sup>+</sup> *<sup>T</sup>*˜ *<sup>n</sup>* <sup>−</sup> <sup>1</sup> <sup>+</sup> *<sup>P</sup>*˜ <sup>+</sup> *κγ*1(*T*˜

phase (*ψ* �= 0, *m* = 0) in accord with the available experimental data.

which can be neglected in the analysis of the available experimental data.

experimental data or from a comprehensive microscopic theory.

critical temperatures on the FM-FS phase transition line *T*˜

<sup>A</sup> *<sup>T</sup>*˜*<sup>s</sup> <sup>γ</sup>*2/2 <sup>1</sup> <sup>−</sup> *<sup>T</sup>*˜ *<sup>n</sup>*

*<sup>B</sup>*), the critical-end point C <sup>≡</sup> (*T*˜

*max <sup>T</sup>*˜*<sup>s</sup>* <sup>+</sup> *<sup>γ</sup>*2/4*κγ*<sup>1</sup> <sup>−</sup>*γ*2/4*γ*<sup>2</sup>

*N T*˜

B *T*˜

C *T*˜

*<sup>A</sup>*) and B <sup>≡</sup> (*T*˜

reduced pressure values *P*˜

*<sup>B</sup>*, *P*˜

*<sup>m</sup>*) on the curve *T*˜

see Sec. 5). The first column shows *<sup>T</sup>*˜*<sup>N</sup>* <sup>≡</sup> *<sup>T</sup>*˜

<sup>3</sup>(<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*<sup>2</sup>

<sup>≡</sup> (*T*˜*A*, *<sup>P</sup>*˜

and

*φ*<sup>2</sup> = *m* = 0, *φ*<sup>2</sup>

*max* =(*T*˜*m*, *P*˜

Although the structure of the FS phase is quite complicated, some of the results can be obtained in analytical form. A more detailed outline of the phase domains, for example, in *T* − *P* phase diagram, can be done by using suitable values of the material parameters in the free energy (7): *P*0, *Tf* 0, *Ts*, *κ*, *γ*, and *γ*1. Here we present some of the analytical results for the phase transition lines and the multi-critical points. Typical shapes of phase diagrams derived directly from Eq. (7) are given in Figs. 2–7. Figure 2 shows the phase diagram calculated from Eq. (7) for parameters, corresponding to the experimental data [13] for ZrZn2. Figures 3 and 4 show the low-temperature and the high-pressure parts of the same phase diagram (see Sec. 7 for details). Figures 5–7 show the phase diagram calculated for the experimental data [2, 4] of UGe2 (see Sec. 8). In ZrZn2, UGe2, as well as in UCoGe and UIr, critical pressure *Pc* exists, where both superconductivity and ferromagnetic orders vanish.

As in experiments, we find out from our calculation that in the vicinity of *P*<sup>0</sup> ∼ *Pc* the FM-FS phase transition is of fist order, denoted by the solid line BC in Figs. 3, 4, 6, and 7. At lower pressure the same phase transition is of second orderq shown by the dotted lines in the same figures. The second order phase transition line *T*˜ *FS*(*P*) separating the FM and FS phases is given by the solution of the equation

$$
\tilde{T}\_{FS}(\tilde{P}) = \tilde{T}\_s + \gamma\_1 t\_{FS}(\tilde{P}) + \tilde{\gamma} [-t\_{FS}(\tilde{P})]^{1/2},\tag{17}
$$

where *tFS*(*P*˜) = *<sup>t</sup>*(*TFS*, *<sup>P</sup>*˜) <sup>≤</sup> 0, *<sup>γ</sup>*˜ <sup>=</sup> *<sup>γ</sup>*/*κ*, *<sup>γ</sup>*˜1 <sup>=</sup> *<sup>γ</sup>*1/*κ*, and 0 <sup>&</sup>lt; *<sup>P</sup>*˜ <sup>&</sup>lt; *<sup>P</sup>*˜ *<sup>B</sup>*; *PB* is the pressure corresponding to the multi-critical point B, where the line *TFS*(*P*) terminates, as clearly shown in Figs. 4 and 7). Note, that Eq. (17) strictly coincides with the stability condition for the FM phase with respect to appearance of FS phase [20].

Additional information for the shape of this phase transition line can be obtained by the derivative *ρ*˜ = *∂T*˜ *FS*(*P*˜)/*∂P*˜, namely,

$$\tilde{\rho} = \frac{\tilde{\rho}\_s + \tilde{\gamma}\_1 - \tilde{\gamma}/2(-t\_{\rm FS})^{1/2}}{1 - n\tilde{T}\_{\rm FS}^{\rm n-1}\left[\tilde{\gamma}\_1 - \tilde{\gamma}/2\left[(-t\_{\rm FS})^{1/2}\right]\right]},\tag{18}$$

where *ρ*˜*<sup>s</sup>* = *∂T*˜ *<sup>s</sup>*(*P*˜)/*∂P*˜. Note, that Eq. (18) is obtained from Eqs. (10) and (17).

The shape of the line *T*˜ *FS*(*P*) can vary depending on the theory parameters (see, e.g., Figs.3 and 6). For certain ratios of *γ*˜, *γ*˜1, and values of *ρ*˜*s*, the curve *T*˜ *FS*(*P*˜) exhibits a maximum *T*˜ *<sup>m</sup>* = *T*˜ *FS*(*P*˜ *<sup>m</sup>*), given by *ρ*˜(*ρ*˜*s*, *Tm*, *Pm*) = 0. This maximum is clearly seen in Figs. 6 and 7. To locate the maximum we need to know *ρ*˜*s*. We have already assumed *Ts* does not depend on *P*, as explained above, which from the physical point of view means that the function *Ts*(*P*) is flat enough to allow the approximation *T*˜ *<sup>s</sup>* ≈ 0 without a substantial error in the results. From our choice of *P*-dependence of the free energy [Eq. (7)] parameters, it follow that *ρ*˜*<sup>s</sup>* = 0.

Setting *ρ*˜*<sup>s</sup>* = *ρ*˜ = 0 in Eq. (18) we obtain

$$t(T\_{m\nu}P\_m) = -\frac{\tilde{\gamma}^2}{4\tilde{\gamma}\_1^2},\tag{19}$$

**Figure 3.** Details of Fig. 2 with expanded temperature scale. The points A, B, C are located in the high-pressure part (*P* ∼ *Pc* ∼ 21 kbar). The *max* point is at *P* ≈ 0 kbar. The FS phase domain is shaded. The dotted line shows the second order FM-FS phase transition with *Pm* ≈ 0. The solid straight line BC shows the fist-order FM-FS transition for *P* > *PB*. The quite flat solid line AC shows the first order N-FS transition (the lines BC and AC are more clearly seen in Fig. 4. The dashed line stands for the second

At pressure *P* > *PB* the FM-FS phase transition is of first order up to the critical-end point C. For *PB* < *P* < *PC* the FM-FS phase transition is given by the straight line BC (see, e.g., Figs. 4 and 7). The lines of all three phase transitions, N-FM, N-FS, and FM-FS, terminate at point C. For *P* > *PC* the FM-FS phase transition occurs on a rather flat smooth line of equilibrium transition of first order up to a second tricritical point A with *PA* ∼ *P*<sup>0</sup> and *TA* ∼ 0. Finally, the third transition line terminating at the point C describes the second order phase

three multi-critical points (A, B, and C), and the maximum *Tm*(*Pm*) are given in Table 1. Note that, for any set of material parameters, *TA* < *TC* < *TB* < *Tm* and *Pm* < *PB* < *PC* < *PA*.

There are other types of phase diagrams, resulting from model (7). For negative values of the generic superconducting temperature *Ts*, several other topologies of the *T* − *P* diagram can be outlined. The results for the multicritical points, presented in Table 1, shows that, when *Ts* lowers below *T* = 0, *TC* also decreases, first to zero, and then to negative values. When *TC* = 0 the direct N-FS phase transition of first order disappears and point C becomes a very special zero-temperature multicritical point. As seen from Table 1, this happens for *Ts* <sup>=</sup> <sup>−</sup>*γ*<sup>2</sup>*Tf*(0)/4*κ*(<sup>1</sup> <sup>+</sup> *<sup>γ</sup>*1). The further decrease of *Ts* causes point C to fall below the zero temperature and then the zero-temperature phase transition of first order near *Pc* splits into two zero-temperature phase transitions: a second order N-FM transition and a first order

At lower *Ts* also point B falls below *T* = 0 and the FM-FS phase transition becomes entirely of second order. For very extreme negative values of *Ts*, a very large pressure interval below *Pc* may occur where the FM phase is stable up to *T* = 0. Then the line *TFS*(*P*) will exist only for relatively small pressure values (*P* � *Pc*). This shape of the stability domain of the FS phase

*<sup>N</sup>* and pressures *P*˜

Theory of Ferromagnetic Unconventional Superconductors with Spin-Triplet Electron Pairing 427

*<sup>N</sup>*, *N* = (A, B, C, *max*) at the

order N-FM transition.

transition N-FM. The reduced temperatures *T*˜

FM-FS transition, provided *TB* still remains positive.

is also possible in real systems.

namely, the value *tm*(*T*, *P*) = *t*(*Tm*, *Pm*) at the maximum *Tm*(*Pm*) of the curve *TFS*(*P*). Substituting *tm* back in Eq. (17) we obtain *Tm*, and with its help we also obtain the pressure *Pm*, both given in Table 1, respectively.

We want to draw the attention to a particular feature of the present theory that the coordinates *Tm* and *Pm* of the maximum (point *max*) at the curve *TFS*(*P*) as well as the results from various calculations with the help of Eqs. (17) and (18) are expressed in terms of the reduced interaction parameters *γ*˜ and *γ*˜1. Thus, using certain experimental data for *Tm*, *Pm*, as well as Eqs. (17) and (18) for *TFS*, *Ts*, and the derivative *ρ* at particular values of the pressure *P*, *γ*˜ and *γ*˜1 can be calculated without any additional information, for example, for the parameter *κ*. This property of the model (7) is quite useful in the practical work with the experimental data.

**Figure 2.** *T* − *P* diagram of ZrZn2 calculated for *Ts* = 0, *Tf* <sup>0</sup> = 28.5 K, *P*<sup>0</sup> = 21 kbar, *κ* = 10, *γ*˜ = 2*γ*˜1 ≈ 0.2, and *n* = 1. The dotted line represents the FM-FS transition and the dashed line stands for the second order N-FM transition. The dotted line has a zero slope at *P* = 0. The low-temperature and high-pressure domains of the FS phase are seen more clearly in the following Figs. 3 and 4.

The conditions for existence of a maximum on the curve *TFS*(*P*) can be determined by requiring *P*˜ *<sup>m</sup>* > 0, and *T*˜ *<sup>m</sup>* > 0 and using the respective formulae for these quantities, shown in Table 1. This *max* always occurs in systems where *TFS*(0) ≤ 0 and the low-pressure part of the curve *TFS*(*P*) terminates at *T* = 0 for some non-negative critical pressure *P*0*<sup>c</sup>* (see Sec. 6). But the *max* may occur also for some sets of material parameters, when *TFS*(0) > 0 (see Fig. 3, where *Pm* = 0). All these shapes of the line *TFS*(*P*) are described by the model (7). Irrespectively of the particular shape, the curve *TFS*(*P*) given by Eq. (17) always terminates at the tricritical point (labeled B), with coordinates (*PB*, *TB*) (see, e.g., Figs. 4 and 7).

12 Will-be-set-by-IN-TECH

our choice of *P*-dependence of the free energy [Eq. (7)] parameters, it follow that *ρ*˜*<sup>s</sup>* = 0.

*<sup>t</sup>*(*Tm*, *Pm*) = <sup>−</sup> *<sup>γ</sup>*˜ <sup>2</sup>

namely, the value *tm*(*T*, *P*) = *t*(*Tm*, *Pm*) at the maximum *Tm*(*Pm*) of the curve *TFS*(*P*). Substituting *tm* back in Eq. (17) we obtain *Tm*, and with its help we also obtain the pressure

We want to draw the attention to a particular feature of the present theory that the coordinates *Tm* and *Pm* of the maximum (point *max*) at the curve *TFS*(*P*) as well as the results from various calculations with the help of Eqs. (17) and (18) are expressed in terms of the reduced interaction parameters *γ*˜ and *γ*˜1. Thus, using certain experimental data for *Tm*, *Pm*, as well as Eqs. (17) and (18) for *TFS*, *Ts*, and the derivative *ρ* at particular values of the pressure *P*, *γ*˜ and *γ*˜1 can be calculated without any additional information, for example, for the parameter *κ*. This property of the model (7) is quite useful in the practical work with the experimental

**Figure 2.** *T* − *P* diagram of ZrZn2 calculated for *Ts* = 0, *Tf* <sup>0</sup> = 28.5 K, *P*<sup>0</sup> = 21 kbar, *κ* = 10,

high-pressure domains of the FS phase are seen more clearly in the following Figs. 3 and 4.

the tricritical point (labeled B), with coordinates (*PB*, *TB*) (see, e.g., Figs. 4 and 7).

*γ*˜ = 2*γ*˜1 ≈ 0.2, and *n* = 1. The dotted line represents the FM-FS transition and the dashed line stands for the second order N-FM transition. The dotted line has a zero slope at *P* = 0. The low-temperature and

The conditions for existence of a maximum on the curve *TFS*(*P*) can be determined by

Table 1. This *max* always occurs in systems where *TFS*(0) ≤ 0 and the low-pressure part of the curve *TFS*(*P*) terminates at *T* = 0 for some non-negative critical pressure *P*0*<sup>c</sup>* (see Sec. 6). But the *max* may occur also for some sets of material parameters, when *TFS*(0) > 0 (see Fig. 3, where *Pm* = 0). All these shapes of the line *TFS*(*P*) are described by the model (7). Irrespectively of the particular shape, the curve *TFS*(*P*) given by Eq. (17) always terminates at

*<sup>m</sup>* > 0 and using the respective formulae for these quantities, shown in

4*γ*˜ <sup>2</sup> 1

*<sup>s</sup>* ≈ 0 without a substantial error in the results. From

, (19)

flat enough to allow the approximation *T*˜

Setting *ρ*˜*<sup>s</sup>* = *ρ*˜ = 0 in Eq. (18) we obtain

*Pm*, both given in Table 1, respectively.

data.

requiring *P*˜

*<sup>m</sup>* > 0, and *T*˜

**Figure 3.** Details of Fig. 2 with expanded temperature scale. The points A, B, C are located in the high-pressure part (*P* ∼ *Pc* ∼ 21 kbar). The *max* point is at *P* ≈ 0 kbar. The FS phase domain is shaded. The dotted line shows the second order FM-FS phase transition with *Pm* ≈ 0. The solid straight line BC shows the fist-order FM-FS transition for *P* > *PB*. The quite flat solid line AC shows the first order N-FS transition (the lines BC and AC are more clearly seen in Fig. 4. The dashed line stands for the second order N-FM transition.

At pressure *P* > *PB* the FM-FS phase transition is of first order up to the critical-end point C. For *PB* < *P* < *PC* the FM-FS phase transition is given by the straight line BC (see, e.g., Figs. 4 and 7). The lines of all three phase transitions, N-FM, N-FS, and FM-FS, terminate at point C. For *P* > *PC* the FM-FS phase transition occurs on a rather flat smooth line of equilibrium transition of first order up to a second tricritical point A with *PA* ∼ *P*<sup>0</sup> and *TA* ∼ 0. Finally, the third transition line terminating at the point C describes the second order phase transition N-FM. The reduced temperatures *T*˜ *<sup>N</sup>* and pressures *P*˜ *<sup>N</sup>*, *N* = (A, B, C, *max*) at the three multi-critical points (A, B, and C), and the maximum *Tm*(*Pm*) are given in Table 1. Note that, for any set of material parameters, *TA* < *TC* < *TB* < *Tm* and *Pm* < *PB* < *PC* < *PA*.

There are other types of phase diagrams, resulting from model (7). For negative values of the generic superconducting temperature *Ts*, several other topologies of the *T* − *P* diagram can be outlined. The results for the multicritical points, presented in Table 1, shows that, when *Ts* lowers below *T* = 0, *TC* also decreases, first to zero, and then to negative values. When *TC* = 0 the direct N-FS phase transition of first order disappears and point C becomes a very special zero-temperature multicritical point. As seen from Table 1, this happens for *Ts* <sup>=</sup> <sup>−</sup>*γ*<sup>2</sup>*Tf*(0)/4*κ*(<sup>1</sup> <sup>+</sup> *<sup>γ</sup>*1). The further decrease of *Ts* causes point C to fall below the zero temperature and then the zero-temperature phase transition of first order near *Pc* splits into two zero-temperature phase transitions: a second order N-FM transition and a first order FM-FS transition, provided *TB* still remains positive.

At lower *Ts* also point B falls below *T* = 0 and the FM-FS phase transition becomes entirely of second order. For very extreme negative values of *Ts*, a very large pressure interval below *Pc* may occur where the FM phase is stable up to *T* = 0. Then the line *TFS*(*P*) will exist only for relatively small pressure values (*P* � *Pc*). This shape of the stability domain of the FS phase is also possible in real systems.

the low-temperature and zero-temperature phase transitions of first order are favored by both

Theory of Ferromagnetic Unconventional Superconductors with Spin-Triplet Electron Pairing 429

There is a number of experimental [9, 40] and theoretical [17, 41, 42] investigations of the problem for quantum phase transitions in unconventional ferromagnetic superconductors, including the mentioned intermetallic compounds. Some of them are based on different theoretical schemes and do not refer to the model (6). Others, for example, those in Ref. [41] reported results about the thermal and quantum fluctuations described by the model (6) before the comprehensive knowledge for the results from the basic treatment reported in the present investigation. In such cases one could not be sure about the correct interpretation of the results from the RG and the possibilities for their application to particular zero-temperature phase transitions. Here we present basic results for the zero-temperature

**Figure 5.** *T* − *P* diagram of UGe2 calculated taking *Ts* = 0, *Tf* <sup>0</sup> = 52 K, *P*<sup>0</sup> = 1.6 GPa, *κ* = 4, *γ*˜ = 0.0984, *γ*˜1 = 0.1678, and *n* = 1. The dotted line represents the FM-FS transition and the dashed line stands for the N-FM transition. The low-temperature and high-pressure domains of the FS phase are seen more

The RG investigation [41] has demonstrated up to two loop order of the theory that the thermal fluctuations of the order parameter fields rescale the model (6) in a way which corresponds to first order phase transitions in magnetically anisotropic systems. This result is important for the metallic compounds we consider here because in all of them magnetic anisotropy is present. The uniaxial magnetic anisotropy in ZrZn2 is much weaker than in UGe2 but cannot be neglected when fluctuation effects are accounted for. Owing to the particular symmetry of model (6), for the case of magnetic isotropy (Heisenberg symmetry), the RG study reveals an entirely different class of (classical) critical behavior. Besides, the different spatial dimensions of the superconducting and magnetic quantum fluctuations imply a lack of stable quantum critical behavior even when the system is completely magnetically isotropic. The pointed arguments and preceding results lead to the reliable conclusion that the phase transitions, which have already been proven to be first order in the lowest-order approximation, where thermal and quantum fluctuations are neglected, will not

symmetry arguments and detailed thermodynamic analysis.

phase transitions described by the model (6).

clearly in the following Figs. 6 and 7.

**Figure 4.** High-pressure part of the phase diagram of ZrZn2, shown in Fig. 1. The thick solid lines AC and BC show the first-order transitions N-FS, and FM-FS, respectively. Other notations are explained in Figs. 2 and 3.

## **6. Quantum phase transitions**

We have shown that the free energy (6) describes zero temperature phase transitions. Usually, the properties of these phase transitions essentially depend on the quantum fluctuations of the order parameters. For this reason the phase transitions at ultralow and zero temperature are called quantum phase transitions [31, 32]. The time-dependent quantum fluctuations (correlations) which describe the intrinsic quantum dynamics of spin-triplet ferromagnetic superconductors at ultralow temperatures are not included in our consideration but some basic properties of the quantum phase transitions can be outlines within the classical limit described by the free energy models (6) and (7). Let we briefly clarify this point.

The classical fluctuations are entirely included in the general GL functional (1)–(5) but the quantum fluctuations should be added in a further generalization of the theory. Generally, both classical (thermal) and quantum fluctuations are investigated by the method of the renormalization group (RG) [31], which is specially intended to treat the generalized action of system, where the order parameter fields (*ϕ* and *M*) fluctuate in time *t* and space *x* [31, 32]. These effects, which are beyond the scope of the paper, lead either to a precise treatment of the narrow critical region in a very close vicinity of second order phase transition lines or to a fluctuation-driven change in the phase-transition order. But the thermal fluctuations and quantum correlation effects on the thermodynamics of a given system can be unambiguously estimated only after the results from counterpart simpler theory, where these phenomena are not present, are known and, hence, the distinction in the thermodynamic properties predicted by the respective variants of the theory can be established. Here we show that the basic low-temperature and ultralow-temperature properties of the spin-triplet ferromagnetic superconductors, as given by the preceding experiments, are derived from the model (6) without any account of fluctuation phenomena and quantum correlations. The latter might be of use in a more detailed consideration of the close vicinity of quantum critical points in the phase diagrams of ferromagnetic spin-triplet superconductors. Here we show that the theory predicts quantum critical phenomena only for quite particular physical conditions whereas the low-temperature and zero-temperature phase transitions of first order are favored by both symmetry arguments and detailed thermodynamic analysis.

14 Will-be-set-by-IN-TECH

**Figure 4.** High-pressure part of the phase diagram of ZrZn2, shown in Fig. 1. The thick solid lines AC and BC show the first-order transitions N-FS, and FM-FS, respectively. Other notations are explained in

We have shown that the free energy (6) describes zero temperature phase transitions. Usually, the properties of these phase transitions essentially depend on the quantum fluctuations of the order parameters. For this reason the phase transitions at ultralow and zero temperature are called quantum phase transitions [31, 32]. The time-dependent quantum fluctuations (correlations) which describe the intrinsic quantum dynamics of spin-triplet ferromagnetic superconductors at ultralow temperatures are not included in our consideration but some basic properties of the quantum phase transitions can be outlines within the classical limit

The classical fluctuations are entirely included in the general GL functional (1)–(5) but the quantum fluctuations should be added in a further generalization of the theory. Generally, both classical (thermal) and quantum fluctuations are investigated by the method of the renormalization group (RG) [31], which is specially intended to treat the generalized action of system, where the order parameter fields (*ϕ* and *M*) fluctuate in time *t* and space *x* [31, 32]. These effects, which are beyond the scope of the paper, lead either to a precise treatment of the narrow critical region in a very close vicinity of second order phase transition lines or to a fluctuation-driven change in the phase-transition order. But the thermal fluctuations and quantum correlation effects on the thermodynamics of a given system can be unambiguously estimated only after the results from counterpart simpler theory, where these phenomena are not present, are known and, hence, the distinction in the thermodynamic properties predicted by the respective variants of the theory can be established. Here we show that the basic low-temperature and ultralow-temperature properties of the spin-triplet ferromagnetic superconductors, as given by the preceding experiments, are derived from the model (6) without any account of fluctuation phenomena and quantum correlations. The latter might be of use in a more detailed consideration of the close vicinity of quantum critical points in the phase diagrams of ferromagnetic spin-triplet superconductors. Here we show that the theory predicts quantum critical phenomena only for quite particular physical conditions whereas

described by the free energy models (6) and (7). Let we briefly clarify this point.

Figs. 2 and 3.

**6. Quantum phase transitions**

There is a number of experimental [9, 40] and theoretical [17, 41, 42] investigations of the problem for quantum phase transitions in unconventional ferromagnetic superconductors, including the mentioned intermetallic compounds. Some of them are based on different theoretical schemes and do not refer to the model (6). Others, for example, those in Ref. [41] reported results about the thermal and quantum fluctuations described by the model (6) before the comprehensive knowledge for the results from the basic treatment reported in the present investigation. In such cases one could not be sure about the correct interpretation of the results from the RG and the possibilities for their application to particular zero-temperature phase transitions. Here we present basic results for the zero-temperature phase transitions described by the model (6).

**Figure 5.** *T* − *P* diagram of UGe2 calculated taking *Ts* = 0, *Tf* <sup>0</sup> = 52 K, *P*<sup>0</sup> = 1.6 GPa, *κ* = 4, *γ*˜ = 0.0984, *γ*˜1 = 0.1678, and *n* = 1. The dotted line represents the FM-FS transition and the dashed line stands for the N-FM transition. The low-temperature and high-pressure domains of the FS phase are seen more clearly in the following Figs. 6 and 7.

The RG investigation [41] has demonstrated up to two loop order of the theory that the thermal fluctuations of the order parameter fields rescale the model (6) in a way which corresponds to first order phase transitions in magnetically anisotropic systems. This result is important for the metallic compounds we consider here because in all of them magnetic anisotropy is present. The uniaxial magnetic anisotropy in ZrZn2 is much weaker than in UGe2 but cannot be neglected when fluctuation effects are accounted for. Owing to the particular symmetry of model (6), for the case of magnetic isotropy (Heisenberg symmetry), the RG study reveals an entirely different class of (classical) critical behavior. Besides, the different spatial dimensions of the superconducting and magnetic quantum fluctuations imply a lack of stable quantum critical behavior even when the system is completely magnetically isotropic. The pointed arguments and preceding results lead to the reliable conclusion that the phase transitions, which have already been proven to be first order in the lowest-order approximation, where thermal and quantum fluctuations are neglected, will not undergo a fluctuation-driven change in the phase transition order from first to second. Such picture is described below, in Sec. 8, and it corresponds to the behavior of real compounds.

in the relatively narrow interval (*Pm*, *PB*) and can appear for some special sets of material

Theory of Ferromagnetic Unconventional Superconductors with Spin-Triplet Electron Pairing 431

treatment of the case *Ts* �= 0 can be performed within the entire two-domain picture for the phase FS (see, also, Ref. [20]). The complete study of this case is beyond our aims but here we will illustrate our arguments by investigation of the conditions, under which the critical pressure *Poc* occurs in systems with *Ts* ≈ 0. Moreover, we will present the general result for

**Figure 7.** High-pressure part of the phase diagram of UGe2, shown in Fig. 4. Notations are explained in

*oc*)]1/2 = (<sup>1</sup> <sup>−</sup> *<sup>P</sup>*˜

<sup>0</sup>*c*. For *Ts* �<sup>=</sup> 0, Eqs. (20) and (21) have two solutions with respect to *<sup>P</sup>*˜

some sets of material parameters these solutions satisfy the physical requirements for *P*0*<sup>c</sup>* and

<sup>0</sup>*<sup>c</sup>* and can be identified with the critical pressures. The conditions for existence of *P*0*<sup>c</sup>* and *P*�

can be obtained either by analytical calculations or by numerical analysis for particular values

<sup>0</sup>*<sup>c</sup>* <sup>−</sup> *<sup>γ</sup>*˜*m*0*<sup>c</sup>* <sup>−</sup> *<sup>T</sup>*˜*<sup>s</sup>* <sup>=</sup> 0, (20)

1/2 (21)

<sup>0</sup>*c*. For

0*c*

<sup>0</sup>*c*)

Setting *TFS*(*P*0*c*) = 0 in Eq. (17) we obtain the following quadratic equation,

*γ*˜1*m*<sup>2</sup>

*<sup>m</sup>*0*<sup>c</sup>* = [−*t*(0, *<sup>P</sup>*˜

<sup>0</sup>*<sup>c</sup>* do not exists for *Ts* ≥ 0.

<sup>0</sup>*<sup>c</sup>* for the general case of *Ts* �= 0

<sup>0</sup>*c*. The correct

parameters (*r*, *t*, *γ*, *γ*1). In particular, as our calculations show, *P*�

The analytical calculation of the critical pressures *P*0*<sup>c</sup>* and *P*�

<sup>0</sup>*<sup>c</sup>* ≥ 0 in systems where *Ts* �= 0.

leads to quite complex conditions for appearance of the second critical field *P*�

**7. Criteria for type I and type II spin-triplet**

**ferromagnetic superconductors**

*P*0*<sup>c</sup>* ≥ 0 and *P*�

Figs. 2, 3, 5, and 6.

and, hence, for *P*˜

*P*�

for the reduced magnetization,

of the material parameters.

Our results definitely show that the quantum phase transition near *Pc* is of first order. This is valid for the whole N-FS phase transition below the critical-end point C, as well as the straight line BC. The simultaneous effect of thermal and quantum fluctuations do not change the order of the N-FS transition, and it is quite unlikely to suppose that thermal fluctuations of the superconductivity field *ψ* can ensure a fluctuation-driven change in the order of the FM-FS transition along the line BC. Usually, the fluctuations of *ψ* in low temperature superconductors are small and slightly influence the phase transition in a very narrow critical region in the vicinity of the phase-transition point. This effect is very weak and can hardly be observed in any experiment on low-temperature superconductors. Besides, the fluctuations of the magnetic induction *B* always tend to a fluctuation-induced first-order phase transition rather than to the opposite effect - the generation of magnetic fluctuations with infinite correlation length at the equilibrium phase-transition point and, hence, a second order phase transition [31, 43]. Thus we can quire reliably conclude that the first-order phase transitions at low-temperatures, represented by the lines BC and AC in vicinity of *Pc* do not change their order as a result of thermal and quantum fluctuation fluctuations.

**Figure 6.** Low-temperature part of the *T* − *P* phase diagram of UGe2, shown in Fig. 5. The points A, B, C are located in the high-pressure part (*P* ∼ *Pc* ∼ 1.6 GPa). The FS phase domain is shaded. The thick solid lines AC and BC show the first-order transitions N-FS, and FM-FS, respectively. Other notations are explained in Figs. 2 and 3.

Quantum critical behavior for continuous phase transitions in spin-triplet ferromagnetic superconductors with magnetic anisotropy can therefore be observed at other zero-temperature transitions, which may occur in these systems far from the critical pressure *Pc*. This is possible when *TFS*(0) = 0 and the *TFS*(*P*) curve terminates at *T* = 0 at one or two quantum (zero-temperature) critical points: *P*0*<sup>c</sup>* < *Pm* - "lower critical pressure", and *P*� <sup>0</sup>*<sup>c</sup>* > *Pm* – "upper critical pressure." In order to obtain these critical pressures one should solve Eq. (17) with respect to *P*, provided *TFS*(*P*) = 0, *Tm* > 0 and *Pm* > 0, namely, when the continuous function *TFS*(*P*) exhibits a maximum. The critical pressure *P*� <sup>0</sup>*<sup>c</sup>* is bounded in the relatively narrow interval (*Pm*, *PB*) and can appear for some special sets of material parameters (*r*, *t*, *γ*, *γ*1). In particular, as our calculations show, *P*� <sup>0</sup>*<sup>c</sup>* do not exists for *Ts* ≥ 0.
