Phenomenological description of anisotropy effects in some ferromagnetic superconductors

Physics Letters A 379 (2015) 1391–1396

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Physics Letters A www.elsevier.com/locate/pla

Phenomenological description of anisotropy effects in some ferromagnetic superconductors Diana V. Shopova a,∗ , Michail D. Todorov b a b

TCCM Research Group, Institute of Solid State Physics, Bulgarian Academy of Sciences, BG-1784 Sofia, Bulgaria Department of Applied Mathematics and Computer Science, Technical University of Sofia, 1000 Sofia, Bulgaria

a r t i c l e

i n f o

Article history: Received 13 November 2014 Received in revised form 12 February 2015 Accepted 23 February 2015 Available online 19 March 2015 Communicated by L. Ghivelder Keywords: Ginzburg–Landau theory Superconductivity Ferromagnetism Magnetization

a b s t r a c t We study phenomenologically the role of anisotropy in ferromagnetic superconductors UGe2 , URhGe, and UCoGe for the description of their phase diagrams. We use the Ginzburg–Landau free energy in its uniform form as we will consider only spatially independent solutions. This is an expansion of previously derived results where the effect of Cooper-pair and crystal anisotropies is not taken into account. The three compounds are separately discussed with the special stress on UGe2 . The main effect comes from the strong uniaxial anisotropy of magnetization while the anisotropy of Cooper pairs and crystal anisotropy only slightly change the phase diagram in the vicinity of Curie temperature. The limitations of this approach are also discussed. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The discovery of ferromagnetic superconductors UGe2 [1], URuGe [2], UCoGe [3], in which ferromagnetic ordering coexists with superconductivity, has given a new trend in understanding of unconventional superconductivity. The pressure–temperature phase diagrams of these compounds differ, but the common feature is that the superconductivity occurs in the domain of ferromagnetic phase and the superconducting transition temperature T s is lower than the Curie temperature, T c . UGe2 orders ferromagnetically at relatively high Curie temperature of 53 K and superconductivity appears upon the application of pressure of about 1 GPa, and at low temperature <1 K. The increase of pressure to the critical value P c = 1.5 GPa results in disappearance of both ferromagnetic and superconducting orders. URuGe and UCoGe are weaker ferromagnets with T c of 9.5 K and 3 K, respectively and the superconducting phase appears at ambient pressure as well. For URuGe the increase of pressure leads to the collapse of superconductivity at about 4 GPa, while for UCoGe the phase transition line gradually grows reaching maximum at 1.1 GPa, where the ferromagnetic order collapses and superconductivity persists also in the paramagnetic region. All three uranium compounds have orthorhombic crystal structure with highly anisotropic magnetic moment of Ising type. For de-

*

Corresponding author. E-mail address: [email protected] (D.V. Shopova).

http://dx.doi.org/10.1016/j.physleta.2015.02.045 0375-9601/© 2015 Elsevier B.V. All rights reserved.

tailed experimental presentation of ferromagnetic superconductors, see, for example, the recent review [4]. It is commonly accepted that 5f electrons of uranium atoms are responsible for both ferromagnetic and superconducting orders. In the presence of magnetization, the ferromagnetic exchange field is expected to rule out spin-singlet Cooper pairing and unconventional superconductivity of p-type, mediated through some magnetic mechanism is considered as the most likely. The experimental discovery of huge upper critical field in URhGe and UCoGe also confirms the triplet pairing because the Pauli paramagnetic effect characteristic of spin singlet pairing is absent there, see, for example [5,20] and the papers cited therein. The coexistence of itinerant ferromagnetism and superconductivity is theoretically proposed in [6] and the main idea is that the exchange of longitudinal spin fluctuations may lead to the triplet pairing in weak itinerant ferromagnets. There are theoretical considerations supported by experimental results that the scenario of spin-triplet superconductivity induced by longitudinal ferromagnetic spin fluctuations can be realized in UCoGe [7,8]. A recent theoretical trend generalizes the phenomenological Ginzburg–Landau approaches on the basis of lattice model for the description of both Meissner and inhomogeneous states of ferromagnetic superconductivity, see [9] and the review [10]. The anisotropic properties of superconductivity in ferromagnetic superconductors are vastly studied experimentally, especially the anisotropic properties of upper critical field [4]. Hattori et al. [7] claimed that the ferromagnetic fluctuations with Ising

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type anisotropy are responsible for the appearance of homogeneous ferromagnetic superconductivity in UCoGe. Here we will study phenomenologically the role of magnetic, crystal and Cooper-pair anisotropy on the phase diagram and possible phases using the previously derived Ginzburg–Landau free energy [11]. 2. Landau free energy We will consider only the Meissner phases – pure superconductors and phases of coexistence of ferromagnetism and superconductivity in the absence of external magnetic field. In earlier papers [11] we did not take in account the Cooper-pair and crystal anisotropies as the purpose there was to model phenomenologically the P –T phase diagram of UGe2 and ZnZr. Later it was experimentally proven that the superconductivity occurring in ZnZr is not volume but surface effect. Because the coexisting phase of UGe2 is totally within the domain of ferromagnetic phase we have assumed that it is the presence of ferromagnetism that triggers the appearance of superconductivity under external pressure. The pressure participates only through the linear dependence of Curie temperature on P , namely T c = T c0 (1 − P / P 0 ), where T c0 is the Curie temperature at zero (ambient) pressure and P 0 is the pressure close to the critical P c where ferromagnetism and superconductivity disappear. This free energy and the obtained results may be a good starting point for the description of ( P , T ) phase diagram in ferromagnetic superconductors. The general form of Ginzburg–Landau free energy we use in our considerations of ferromagnetic superconductors with p-pairing is [12]:





→ → − −

F[ψ, M ) =

− →

→ −

→ → − −

d3 x f s (ψ) + f F ( M ) + f I (ψ, M ) +

→ −2

B

8π

 → − − → − B .M , (1)

→ −

where the superconducting order parameter ψ = (ψ1 , ψ2 , ψ3 ) is a − → 3D complex vector, M is the magnetization and both depend on → − → − − → → − the spatial variable x . The magnetic induction is B = H + 4π M = → − → − → − ∇ × A with H – the external magnetic field and A – the magnetic vector potential. The free energy density that describes the superconductivity in → − the absence of magnetization and external magnetic field f s (ψ) is expanded up to the fourth order in superconducting order param→ − eter ψ , including the respective anisotropic terms. Here we suppose tetragonal symmetry for superconductors with triplet Cooper pairing. Although all three uranium compounds UGe2 , URhGe and UCoGe have orthorhombic symmetry and the structure of superconducting order parameter for orthorhombic symmetry has been derived by general group considerations [13,14], we shall not consider for the time being the anisotropy in (x, y ) plane, but only the uniaxial anisotropy, connected with the Ising-like anisotropy of magnetization: → −

→ −

→ −

fs (ψ) = f grad (ψ) + as |ψ|2 +

+

bs → us → − − |ψ|4 + |ψ 2 |2 2 2

v s −−→ 4 − − → (|ψ1 | + |ψ2 |4 ). 2

(2)

The above free energy density f s of superconducting subsystem is written following the classification of superconducting states with triplet pairing deduced by general group symmetry approach in [15,16]. The obtained expression f s possesses the symmetry of point group for tetragonal crystal symmetry, the symmetry of the spin rotations group, the time-reversal symmetry group and the gauge symmetry group by the way of its derivation. → − The term f grad (ψ) gives the spatial dependence of the superconducting order parameter in the form:

→ −

→ −

fgrad (ψ) = K 1 ( D i ψ j )∗ ( D i D j )



→ − → − → − → −  + K 2 ( D i ψ i )∗ ( D j ψ j ) + ( D i ψ j )∗ ( D j ψ i ) → −

→ −

+ K 3 ( D i ψ i )∗ ( D i ψ i ),

(3)

where the symbol D j stands for covariant differentiation D j = −ih¯ ∂/∂ xi + 2|e |/c A j , and over the indices (i , j ) summation is supposed, see [12]. The material parameters K j are related to the effective mass tensor of anisotropic Cooper pairs [15,16]. The Landau material parameters in (2) are given by as = αs [ T − T s ( P )], where T s ( P ) is the critical temperature for pure superconducting system and b s > 0. The parameters for Cooper-pair anisotropy u s and crystal lattice anisotropy v s within this approach are considered as undetermined material constants, characteristic of each particular substance and should be taken from the respective experimental data. All quantities αs , b s , u s , v s may be derived in principle from BCS-type microscopic models of superconductivity but vastly differ for weak and strong coupling limits. The ferromagnetic energy density up to the forth order in mag− → netization M is denoted by f F

fF = c f

3 

− →

|∇ j M j |2 + a f | M |2 +

j =1

bf − → | M |4 . 2

(4)

Here a f = α f [ T n − T nf ( P )] with a material parameter α f > 0, T is the temperature and T f is the Curie temperature for ferromagnetic subsystem. The experimentally found ferromagnetic superconductivity in UGe2 , UCoGe and URhGe strongly depends on pressure and we take this into account by the dependence of Curie temperature T f ( P ) of pure ferromagnetic system on pressure P [11]. For n = 1 the usual Landau form of a f is achieved; n = 2 describes the Stoner–Wohlfarth model [17]; b f > 0. In the calculations below we will consider the case n = 1 and n = 2 will be only shortly discussed in connection with the phase diagram of UGe2 . The interaction between the superconducting and magnetic order parameters is given by f I , see [18,12]: − → → −

− → → −

→ −

f I = i γ0 M .(ψ × ψ ∗ ) + δ M 2 |ψ|2 ,

(5)

with γ0 ∼ J , where J is the ferromagnetic exchange constant. Most experiments on UGe2 , UCoGe and URhGe (see the review paper [4]) show that the magnetization is of expressed Ising type. The strong uniaxial anisotropy of magnetic moment plays an important role for proposed magnetic mediated mechanisms for the appearance of triplet pairing favored by longitudinal magnetic fluctuations [7]. Recently it has been shown that in the fluctuation region UGe2 and URhGe does not belong to the 3D Ising universality class [19]. But in this paper we will use the Landau approach so the critical fluctuations will be not considered and the critical exponents are the mean-field ones. The uniaxial magnetic anisotropy means that magnetic mo− → ment in the above equations can be represented in the form M = (0, 0, M z = M ) by choosing z as the easy axis of magnetization. As we study only uniform phases in the absence of external magnetic field, H = 0, we will drop the dependence on the spatial variables − → → − of magnetization M and superconducting order parameter ψ and write for the free energy density f u = F / V , where by f u we have denoted the uniform part of free energy density Eq. (1) with V – the volume. Then the uniform magnetic and superconducting order parameters will depend only on T and P . To facilitate our considerations we make the uniform part of free energy f u dimensionless with the help of the relation [11]:

f =

fu b f M 04

(6)

, 

where M 0 = α f T f 0 / b f is the magnetic moment at T = 0, P = 0 with T f 0 the Curie temperature at zero pressure in the absence of

D.V. Shopova, M.D. Todorov / Physics Letters A 379 (2015) 1391–1396

superconducting order. The dimensionless order parameters then become:

m=

M M0

− →

;

ϕ i = φ i e θi ; φ i =

|ψ i | M 0 (b f /b)1/4

(7)

with b = b s + u s + v s , see (2). The complex 3-component superconducting order parameter ψi here is represented by its modulus φi and phase angle θi . The alternative representation of ψi by its real and imaginary parts is also possible [12]. The material parameter → − as in front of |ψ|2 in (2), which depends on temperature,in dimensionless form becomes r = β( T − T s )/ T f 0 , with β = αs b f /b/α f . For m = 0 and r ≤ 0 namely T ≤ T s the usual second order phase transition to pure superconducting phases will occur. The experiments show that the superconductivity for UGe2 and URhGe appears only in the domain of ferromagnetic ordering while for UCoGe the superconductivity exists also in the paramagnetic region. For all three ferromagnetic superconductors, the superconducting transition temperature is lower than the Curie temperature. This may be described by r > 0 for free energy (6), i.e., the superconductivity for m = 0 may appear above the critical temperature of the pure superconducting phase transition T > T s . The dimensionless parameters of Cooper-pair anisotropy w = u s /b and uniaxial crystal anisotropy v = v s /b, see (2), can take both positive and negative values, but their modulus remains smaller than unity by these definitions. For the pure ferromagnetic system, Eq. (4), the thermodynamic parameter a f in dimensionless form will be written as t = ( T n − T f ( P )n )/ T nf 0 with T f – the Curie temperature. Further in the calculations we will work in the standard Landau approach n = 1. The interaction between the pure superconducting and ferromagnetic subsystems, described by free energy density f I , Eq. (5), contains two parameters which in dimensionless form are: γ =

√



γ0 /( bb1f /4 ); γ1 = δ/ bb f . Using the definitions (7) and the above

explanations we can write down the dimensionless free energy density (6) in the form:

f = r (φ12 + φ22 + φ32 ) +

1 2

(φ12 + φ22 + φ32 )2

− 2w φ12 φ22 sin2 (θ2 − θ1 ) + φ12 φ32 sin2 (θ1 − θ3 )

+ φ22 φ32 sin2 (θ2 − θ3 )

− v φ12 φ22 + 2γ φ1 φ2 M sin(θ2 − θ1 ) + γ1 (φ12 + φ22 + φ32 ) M 2 + t M 2 +

1 2

M4.

(8)

The equilibrium phases for the free energy (8) are found from the equations of state

∂ f ( xi ) = 0, ∂ xi

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The calculations are accomplished in the space of thermodynamic parameters r ( T , P ), t ( T , P ); with T and P – the temperature and pressure, respectively. Here we do not make any assumptions for the particular dependence of r and t on the pressure, as we want to see the general trend and possibility to use the free energy (8) for qualitative estimates. In order to describe the ( P , T ) phase diagrams and make quantitative comparison with the available experimental data, UGe2 , URhGe, and UCoGe should be separately considered as discussed below. In this letter we do not aim at making a full analysis of all existing phases, stable and unstable, that are solutions of equations of state (9). Such analysis has been made in previous papers [12] for the same free energy Eq. (1) not taking into account the Cooper-pair and crystal anisotropies. We will discuss changes and show only those new results following from inclusion of respective anisotropies in free energy (8). 3. Results and discussion The crystal anisotropy and the Cooper pair anisotropy lift the degeneracy of the free energy (8) and change both the number and stability domains of the possible phases. Here we assume that γ1 > 0 as the negative sign means some redefinition of the free energy in order to ensure it is limited at infinity. We will not write down all seven equations of state following from (9) in explicit form, but will focus only on the stable phase that appears for T > T s in the domain of ferromagnetic ordering for φ1 = 0, φ2 = 0, φ3 = 0, m = 0. We call this phase the basic coexisting phase. Direct calculations show that the stable phase is given by φ1 = φ2 = φ . The equation describing the phase difference (θ1 − θ2 ) of the basic ferromagnetic superconducting phase is:



φ 2 cos(θ1 − θ2 ) 2w φ 2 sin(θ1 − θ2 ) + γ m = 0

(11)

with a solution cos(θ1 − θ2 ) = 0. The other solution sin(θ1 − θ2 ) = −γ m/(2w φ 2 ) is unstable for φ = 0. The solution φ1 = φ2 = φ = 0, cos(θ1 − θ2 ) = 0, φ3 = 0, m = 0 describes a two-domain phase with mutually perpendicular components of the superconducting order parameter with equal modulus in the complex plane. The domains differ in the sign of magnetic moment and sin(θ1 − θ2 ), but in this approximation the domains are undistinguishable with equal energies and the inclusion of Cooper-pair and crystal anisotropy terms does not lift this degeneracy. Here we consider only the positive sign of m. The equations of state for superconducting order parameter and magnetization of the basic coexisting phase are:



φ r + φ 2 (2 − 2w − v ) − γ m + γ1m2 = 0,

(12)

and

(9)

tm + m3 + 2γ1mφ 2 − γ φ 2 = 0.

(13)

with {xi } = (m, φ1 , φ2 , φ3 ; θ1 , θ2 , θ3 ). The stability of obtained phases and phase transition lines can be calculated from the stability matrix, with elements defined by:

Solving together the above equations we find the equilibrium value of r (m, t ) = r0 , for which the ferromagnetic superconductor exists. Detailed description of the way of calculation is given in [12]:

∂2 f Ai j = . ∂ xi ∂ x j

r0 =

(10)

The above expression (8) for the free energy is not simple and there is a great number of solutions for the equations of state (9). The stability in case of too many phases cannot be calculated only trough the stability matrix with elements (10) but in order to find the phases corresponding to a global minimum of the free energy, direct comparison of respective free energies should be done.

 −2m (1 − γ12 − w − v /2)m2 m0 − m 3

γ2

2

2

+ γ1 γ m + (1 − w − v /2)t +

 .

(14)

The value m0 = γ /(2γ1 ) of magnetic moment corresponds to the maximum of phase line r0 (t ) at the point

t X = −γ 2 /(4γ12 ),

r X = γ 2 /(4γ1 )

(15)

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Fig. 1. The (r , t ) phase diagram for ferromagnetic superconductor for γ > γ1 : P – disordered phase; F – ferromagnetic phase; FS – ferromagnetic superconductor.

for the transition from ferromagnetic to coexisting phase. This maximum is shown in the figures below by the point X . The expression for superconducting order parameter φ of coexisting phase becomes

φ2 =

m(m2 + t )

(m0 − m)

Fig. 2. The (r , t ) phase diagram for ferromagnetic superconductor for γ < γ1 : P – disordered phase; F – ferromagnetic phase; FS – ferromagnetic superconductor; S – superconducting phases.

(16)

.

The superconducting order √ √ parameter is positive for m < m0 , m > −t, and m > m0 , m < −t. So, there is a region of existence of φ also for t > 0 in the paramagnetic region. The superconducting order parameter φ is determined mainly by the uniaxial magnetization m and the interaction parameters γ1 and γ , the influence of crystal and Cooper-pair anisotropy parameters w, v within this approximation is manifested in the stability conditions. The uniaxial ferromagnetic phase exists for m2 = −t, t < 0 and its domain of stability is given by r ≥ re with

r e = γ1 t + γ

√ −t

(17)

√ and re = r0 (m = −t ).

We show in Fig. 1 the phase diagram for the basic coexisting phase, whose existence and stability domain is given in grey color. The line A B is a first order phase transition line: for t > 0 it describes the phase transition between the paramagnetic phase and the ferromagnetic superconductor, and for t < 0 – the transition between the ferromagnetic phase and coexisting phase. The interval t A B = (t A , t B ) of the first order phase transition lines is:

tA =

γ2 2(1 − w − v /2)

tB = −

4(γ1 +

√

γ2

, (18)

,

1 − w − v /2 )2  γ2 with r A = 0 and r B = 4 (γ1 + 2 1 − w −

v 2

 )/(γ1 + 1 − w −

v 2 ) . 2

The point B is tricritical one – to the left of it, the phase transition between the ferromagnet and coexisting phase is of second order. The second order transition line is given by (17) for t < t B . When Cooper-pair anisotropy and crystal anisotropy are not considered, the interval t A B is determined only by the interaction parameters γ and γ1 . The effect of anisotropy parameters w , v on the form of phase diagram is mainly on the first order phase transition lines in the vicinity of t = 0. The second order phase transition line to the left of (t B , r B ) is not affected by the inclusion of anisotropy and the dependence r (t ) there is described by (17). Depending on the sign of anisotropy parameters w , v the interval t A B becomes wider for w > 0 and more narrow for w < 0. The

Fig. 3. The (r , t ) phase diagram for ferromagnetic superconductor for w < 0, v < 0 including the stability lines of purely superconducting phases: P – disordered phase; F – ferromagnetic phase; FS – ferromagnetic superconductor.

sign of crystal anisotropy parameter v has the same effect as seen in Fig. 2 and Fig. 3; see (18). For w < 0 there is one more effect as the stability of the ferromagnetic superconductor for r < 0 is limited in the area enclosed by the curve numbered by 1 and the dashed vertical line. This dashed line marks the maximum point X on the phase line of transition between the coexisting and ferromagnetic phase. For t D < t < t X the coexisting phase is stable only for r > 0. The 2nd order phase line to the left of (t X , r X ) is determined only by interaction parameters γ1 and γ and the uniaxial symmetry of magnetization. The temperature parameter t by rendering the free energy (1) in the dimensionless uniform form (6) is limited from below and the numerical value t = −1 corresponds to the temperature T = 0 and pressure P = 0. The comparison of Fig. 1, Fig. 2 and Fig. 3 shows the different position of point D. Fig. 1 is drawn for γ > γ1 , i.e., the linear interaction is stronger than the quadratic one, see (5) and r D = 0 for t D = −1, which means that the coexisting phase appears also at zero pressure as

D.V. Shopova, M.D. Todorov / Physics Letters A 379 (2015) 1391–1396

is experimentally found in UCoGe and URhGe. In Fig. 2 and Fig. 3, r D = 0 at t D = −γ 2 /γ12 > −1 for γ < γ1 . This case applies for UGe2 as discussed in detail in [12] for w ≡ 0; v ≡ 0. In Fig. 2 the phase diagram is shown for w < 0, v > 0. The pure superconducting phase for these parameters defined by φ1 = φ2 , φ3 = 0, m = 0, sin (θ1 − θ2 ) = 0 exists for r < 0. For w = 0, v = 0 this phase shows marginal stability, but the inclusion of anisotropies stabilizes it for r < (2 − v )(2wt + γ 2 )/(4γ1 w ); in Fig. 2 its stability line is denoted by 2. The point t D > −1 for the parameters in Fig. 2, which means that for T = 0, P = 0 (t = −1) the only stable phase is the ferromagnetic one. The stability of coexisting phase for r < 0, w < 0 is limited, see the areas enclosed by line 1 and dashed line in Figs. 1 and 2, which is not the case for w ≥ 0 where this is not limited in any way. To draw the whole phase diagram together with the pure superconducting phases, it is necessary to work in the ( P , T ) space. In order to do so it is necessary to make some model assumptions for the dependence of Curie temperature on pressure T f ( P ), which have been already done within the Landau approach for linear dependence of T f on P [11]. Recently also model calculations are made for the pressure coefficients of Curie temperature [21] and the pressure coefficients of superconducting order parameter [22] for ferromagnetic superconductors. The pure superconducting phases that exist and are stable for r < 0 and w < 0, v < 0 are shown in Fig. 3 by their lines of stability numbered by 2, 3 and 4. The phase φ2 = φ3 , m = 0, φ1 = 0, sin (θ2 − θ3 ) = 0 exists for r < 0 and is stable for v < 0, w < 0 and r < r2 = t /γ1 with purely real collinear components, i.e., θ2 = kπ , θ3 = lπ . In Fig. 3, r2 is denoted by number 2. This phase may also have purely imaginary collinear components, namely, θ2 = (2k + 1)π /2, θ3 = (2l + 1)π /2; the stability is defined by w < 0, 4w + v < 0, and in Fig. 3 it is shown under number 4 with r < r4 = t /γ1 + 2γ 2 /γ1 (4w + v ). Both phases have equal free energies f s = −r 2 /2 but different domains of stability. Within this model with tetragonal crystal symmetry it is not possible to decide which one of these superconducting phases will be more stable. As in previous figures there only the phases which domains of existence and stability change under the effect of anisotropies, are shown. The line 3 describes a pure superconducting phase with φ2 = φ1 = 0, m = 0, φ3 = 0, which exists for r < 0 and is stable for θ3 = (2k + 1)π /2, w < 0, r < r3 = t /γ1 + γ 2 /(2w γ1 ). The parameters in all figures above are only guiding to explain the different possibilities. We should mention that the mean-field analysis here is made supposing that the Curie temperature is higher than the critical temperature of coexisting phase as experiments for UCoGe, URhGe and UGe2 show. Detailed limitations of the applied approach are given in [23] for w = 0, v = 0. The inclusion of anisotropies does not change in principle the conclusions given there. Direct comparison with experiment requires that assumptions should be made about the dependence of Curie temperature on the pressure for each particular ferromagnetic superconductor UGe2 , URhGe and UCoGe. Then different experimentally obtained phase diagrams and other experimental data should be used to find the material parameters of the Landau energy (8) for each compound. Here we discuss in detail the limitations for the use of free energy (8) to the description especially of UGe2 . The calculations show that including the Cooper-pair and crystal anisotropy terms for the description of P –T phase diagram using the approach described in [11] with linear dependence of Curie temperature on the pressure, does not rule out the main discrepancies between the experimentally found phase diagram and the one, calculated on the basis of Landau energy (8). First of all the transition from paramagnetic phase to ferromagnetic one around t = ( T − T f )/ T f 0 = 0 experimentally is of first order up to the tricritical point T TCP , where the transition

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Fig. 4. An illustration of T –P phase diagram of UGe2 : FM1 – high-pressure ferromagnetic phase, FM2 – low-pressure ferromagnetic phase, SC-phase of coexistence of ferromagnetism and superconductivity. T x ( P ) and T c ( P ) are the respective magnetic phase transition lines; T TCP is tricritical point; P c is the critical pressure, at which the ferromagnetic order and coexisting phase disappear.

becomes of second order down to P = 0. See Fig. 4 for the experimental phase diagram of UGe2 . The first order phase transition from paramagnetic to FM1 phase can be modeled by expanding the magnetic energy density (3) to M 6 for external magnetic field H = 0 and spatially uniform uniaxial magnetization. Then the parameter b f in front of M 4 -term should change sign with pressure from negative to positive at the tricritical point T TCP in Fig. 4. In [24] the first order transition from paramagnetic to low polarized ferromagnetic phase (FM1) is described in detail by including M 6 -term and the coupling of elastic and magnetic degrees of freedom in (4). The dependence of b f and T c on pressure is modeled according to the Stoner–Wohlfarth [17] theory of weak itinerant ferromagnets, namely T f = T f 0 (1 − P 2 / P 02 ), which corresponds to a f = α f ( T n − T nf ) in (4) with n = 2. The calculations show that when the ferromagnetic energy density is expanded up to M 6 to determine the total phase diagram with the coexisting phase becomes too complicated as the free energy density of the pure superconducting phase, (2), should be also expanded to sixth order in superconducting order parameter. Otherwise, there is no region of existence of ferromagnetic superconductor to the left of maximum on the phase transition line from ferromagnetic to coexisting phase within the numerical accuracy of calculation. Moreover, also assumptions should be made for the dependence of T s and b s ; see (2) and model calculations for the pressure coefficients of superconducting order parameter [22]. Taking into account the crystal and Cooper-pair anisotropy terms additionally complicates the problem as the Landau energy will depend on a bigger number of material parameters which are difficult to identify and compare with experimental data. Moreover, even the inclusion of sixth order term does not describe the transition from ferromagnetic low polarized (FM1) to high polarized (FM2) phase, line T x in Fig. 4, which is experimentally found to be of first order around the pressure P x . The maximum at the phase line between ferromagnetic phase and coexisting phase is near to this ferromagnetic transition. Within the Landau approach it will be necessary to include even M 8 terms in the expansion of ferromagnetic part of free energy (4), in order to describe the transition between FM1 and FM2. In principle, this can be done, but as aim is to have a tractable model for comparison with experimental ( P , T ) phase diagram, such an approach will be of little use. First of all the material parameters in front of higher order terms M 6 and M 8 will be difficult to determine from the available experimental data and, secondly, their dependence on pressure will be hard to define. As the appearance of superconductivity is triggered by the ferromagnetism in UGe2 this forces the inclusion of higher order terms also in the expansion of superconducting order parameter. The number of material parameters becomes too big in order to make any reasonable comparison with experiment.

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The experimental ( P , T ) phase diagram of URhGe is also peculiar as the ferromagnetic transition line with the increase of pressure goes upward up to the highest pressure measured of ∼13 GPa. The magnetic moment exhibits also strong uniaxial anisotropy. The coexisting phase is completely within the ferromagnetic domain and occurs at ambient pressure, too. When pressure grows the coexisting phase line decreases and vanishes at P = 4 GPa. The unusual experimental pressure dependence of Curie temperature cannot be easily modeled so to directly apply the Landau energy (1) to the description of this phase diagram as consistent microscopic models are not available to make reliable assumptions for the dependence of Landau free energy parameters on P . UCoGe is weak itinerant ferromagnet and is the only uranium compound found up to now, for which the superconductivity appears not only in the domain of ferromagnetic phase but also in the paramagnetic region up to 2.2 GPa, while the ferromagnetic order is suppressed for P ∼ 1.1 GPa. The ferromagnetic superconducting phase exists also at ambient pressure and magnetic moment shows uniaxial anisotropy of Ising type. In comparison with UGe2 and URhGe, the phase diagram of UCoGe may be described using the Landau energy (1), but the real orthorhombic crystal anisotropy should be considered together with the form of superconducting order parameter derived by general group considerations in [13,14] for this crystal symmetry. 4. Conclusion In this paper we have analyzed phenomenologically the role of magnetic, Cooper-pair and crystal anisotropies for the description of phase diagrams of some ferromagnetic superconductors within Landau free energy expansion to the forth order of magnetic and superconducting order parameters. The phase diagram is mainly determined by the uniaxial anisotropy of magnetization; the Cooper-pair and crystal anisotropies slightly change the behavior of phase transition lines in the vicinity of r = 0, t = 0. So the application of Ginzburg–Landau free energy up to the fourth order expansion of superconducting and magnetic order parameters may serve as an initial estimate. The results and analysis show that the crucial point for the proper phenomenological description of phase of coexistence between ferromagnetism and superconductivity in UGe2 , URhGe and UCoGe is related to the ferromagnetic

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