Phenomenological study of spin-triplet ferromagnetic superconductors
Unconventional superconductivity with spin-triplet Cooper pairing is reviewed based on the quasi-phenomenological Ginzburg-Landau theory. The superconductivity, in particular, the mixed phase of coexistence of ferromagnetizm and unconventional superconductivity is triggered by the spontaneous ma...
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irk-123456789-1194822017-06-08T03:06:08Z Phenomenological study of spin-triplet ferromagnetic superconductors Shopova, D.V. Tsvetkov, T.E. Uzunov, D.I. Unconventional superconductivity with spin-triplet Cooper pairing is reviewed based on the quasi-phenomenological Ginzburg-Landau theory. The superconductivity, in particular, the mixed phase of coexistence of ferromagnetizm and unconventional superconductivity is triggered by the spontaneous magnetization. The mixed phase is stable whereas the other superconducting phases that usually exist in unconventional superconductors are either unstable, or, for particular values of the parameters of the theory, some of these phases are metastable at relatively low temperatures in a quite narrow domain of the phase diagram. The phase transitions from the normal phase to the phase of coexistence is of first order while the phase transition from the ferromagnetic phase to the coexistence phase can be either of first or second order depending on the concrete substance. The Cooper pair and crystal anisotropy are relevant to a more precise outline of the phase diagram shape and reduce the degeneration of the ground states of the system but they do not drastically effect the phase stability domains and the thermodynamic properties of the respective phases. Оглянуто нетрадиційну надпровідність із спін-триплетним куперівським спарюванням, базуючись на квазі-феноменологічній теорії Гінзбурга-Ландау. Надпровідність, а зокрема змішана фаза співіснування феромагнетизму і незвичної надпровідності викликається спонтанною намагніченістю. Змішана фаза є стійкою, в той час як інші надпровідні фази, що зазвичай існують у нетрадиційних надпровідниках, є або нестійкими, або, при певних значеннях параметрів теорії, метастабільними при відносно низьких температурах у вкрай вузькій області фазової діаграми. Фазовий перехід із нормальної фази у фазу співіснування є переходом першого роду, в той час як фазовий перехід із феромагнітної фази у фазу співіснування може бути або першого або другого роду залежно від типу речовини. Куперівське спарювання і анізотропія кристалу є суттєвими для більш точного опису форми фазової діаграми і знімають виродження основного стану системи, проте вони не чинять значного впливу на області фазової стійкості і термодинамічні властивості відповідних фаз. 2005 Article Phenomenological study of spin-triplet ferromagnetic superconductors / D.V. Shopova, T.E. Tsvetkov, D.I. Uzunov // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 181–192. — Бібліогр.: 53 назв. — англ. 1607-324X PACS: 74.20.De, 74.20.Rp DOI:10.5488/CMP.8.1.181 http://dspace.nbuv.gov.ua/handle/123456789/119482 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
Unconventional superconductivity with spin-triplet Cooper pairing is reviewed
based on the quasi-phenomenological Ginzburg-Landau theory. The
superconductivity, in particular, the mixed phase of coexistence of ferromagnetizm
and unconventional superconductivity is triggered by the spontaneous
magnetization. The mixed phase is stable whereas the other superconducting
phases that usually exist in unconventional superconductors
are either unstable, or, for particular values of the parameters of the theory,
some of these phases are metastable at relatively low temperatures in a
quite narrow domain of the phase diagram. The phase transitions from the
normal phase to the phase of coexistence is of first order while the phase
transition from the ferromagnetic phase to the coexistence phase can be
either of first or second order depending on the concrete substance. The
Cooper pair and crystal anisotropy are relevant to a more precise outline
of the phase diagram shape and reduce the degeneration of the ground
states of the system but they do not drastically effect the phase stability
domains and the thermodynamic properties of the respective phases. |
| format |
Article |
| author |
Shopova, D.V. Tsvetkov, T.E. Uzunov, D.I. |
| spellingShingle |
Shopova, D.V. Tsvetkov, T.E. Uzunov, D.I. Phenomenological study of spin-triplet ferromagnetic superconductors Condensed Matter Physics |
| author_facet |
Shopova, D.V. Tsvetkov, T.E. Uzunov, D.I. |
| author_sort |
Shopova, D.V. |
| title |
Phenomenological study of spin-triplet ferromagnetic superconductors |
| title_short |
Phenomenological study of spin-triplet ferromagnetic superconductors |
| title_full |
Phenomenological study of spin-triplet ferromagnetic superconductors |
| title_fullStr |
Phenomenological study of spin-triplet ferromagnetic superconductors |
| title_full_unstemmed |
Phenomenological study of spin-triplet ferromagnetic superconductors |
| title_sort |
phenomenological study of spin-triplet ferromagnetic superconductors |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119482 |
| citation_txt |
Phenomenological study of spin-triplet ferromagnetic superconductors / D.V. Shopova, T.E. Tsvetkov, D.I. Uzunov // Condensed Matter Physics. — 2005. — Т. 8, № 1(41). — С. 181–192. — Бібліогр.: 53 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT shopovadv phenomenologicalstudyofspintripletferromagneticsuperconductors AT tsvetkovte phenomenologicalstudyofspintripletferromagneticsuperconductors AT uzunovdi phenomenologicalstudyofspintripletferromagneticsuperconductors |
| first_indexed |
2025-07-08T15:57:06Z |
| last_indexed |
2025-07-08T15:57:06Z |
| _version_ |
1837094905537626112 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 1(41), pp. 181–192
Phenomenological study of spin-triplet
ferromagnetic superconductors
D.V.Shopova, T.E.Tsvetkov, D.I.Uzunov
CP Laboratory, Institute of Solid State Physics,
Bulgarian Academy of Sciences,
BG–1784 Sofia, Bulgaria
Received November 15, 2004
Unconventional superconductivity with spin-triplet Cooper pairing is revi-
ewed based on the quasi-phenomenological Ginzburg-Landau theory. The
superconductivity, in particular, the mixed phase of coexistence of ferro-
magnetizm and unconventional superconductivity is triggered by the spon-
taneous magnetization. The mixed phase is stable whereas the other su-
perconducting phases that usually exist in unconventional superconductors
are either unstable, or, for particular values of the parameters of the theory,
some of these phases are metastable at relatively low temperatures in a
quite narrow domain of the phase diagram. The phase transitions from the
normal phase to the phase of coexistence is of first order while the phase
transition from the ferromagnetic phase to the coexistence phase can be
either of first or second order depending on the concrete substance. The
Cooper pair and crystal anisotropy are relevant to a more precise outline
of the phase diagram shape and reduce the degeneration of the ground
states of the system but they do not drastically effect the phase stability
domains and the thermodynamic properties of the respective phases.
Key words: superconductivity, ferromagnetizm, phase diagram, order
parameter profile, anisotropy
PACS: 74.20.De, 74.20.Rp
1. Inroduction
The formation of Cooper pairs with a nonzero angular momentum was theoreti-
cally predicted [1] in 1959 as a mechanism of superfluidity of Fermi liquids. In 1972
the same phenomenon – unconventional superfluidity due to a p-wave (spin triplet)
Cooper pairing of 3He atoms, was experimentally discovered in the mK range of
temperatures; for details and theoretical description, see [2–4]. In contrast to the
standard s-wave pairing in usual (conventional) superconductors, where the elec-
tron pairs are formed by an attractive electron-electron interaction due to a virtual
c© D.V.Shopova, T.E.Tsvetkov, D.I.Uzunov 181
D.V.Shopova, T.E.Tsvetkov, D.I.Uzunov
phonon exchange, the widely accepted mechanism of the Cooper pairing in super-
fluid 3He is based on an attractive interaction between the fermions (3He atoms)
due to a virtual exchange of spin fluctuations. Certain spin fluctuation mechanisms
of unconventional Cooper pairing of electrons were also proposed for the depiction
of heavy fermion superconductors discovered in 1979 (see, e. g., [5–7]) as well as for
some classes of high-temperature superconductors (see, e. g., [8–16]).
The possible superconducting phases in unconventional superconductors are de-
scribed in the framework of the general Ginzburg-Landau (GL) effective free energy
functional [13] using the symmetry groups theory. A variety of possible supercon-
ducting orderings were predicted for different crystal structures [17–22]. A detailed
thermodynamic analysis [11,18] of the homogeneous (Meissner) phases and a renor-
malization group investigation [11] of the superconducting phase transition up to the
two-loop approximation have been also performed (for a three-loop renormalization
group analysis, see [23]; for effects of magnetic fluctuations and disorder, see [24,25]).
In 2000, experiments [26] at low temperatures (T ≈ 1 K) and high pressure
(T ≈ 1 GPa) demonstrated the existence of spin triplet superconducting states in
the metallic compound UGe2. This superconductivity is triggered by spontaneous
magnetization of the ferromagnetic phase which exists at much higher temperatures
and coexists with the superconducting phase in the whole domain of existence of
the latter below T ≈ 1 K; see also experiments published in [27,28], and the dis-
cussion in [29]. Moreover, the same phenomenon of existence of superconductivity
at low temperatures and high pressure in the domain of the (T, P ) phase diagram
where the ferromagnetic order is present has been observed in other ferromagnetic
metallic compounds (ZrZn2 [30] and URhGe [31]) soon after the discovery [26] of
superconductivity in UGe2.
In contrast to other superconducting materials, for example, ternary and Chevrel
phase compounds, where the effects of magnetic order on superconductivity are
also substantial (see, e. g., [32–35]), 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
ferromagnetizm and superconductivity. For example, in UGe2 TFS is 0.8 K whereas
the critical temperature of the phase transition from paramagnetic to ferromagnetic
state in the same material is Tf = 35 K [26,27]. It can be supposed that in such
substances the material parameter Ts defined as the (usual) critical temperature of
the second order phase transition from normal to uniform (Meissner) supercondicting
state in a zero external magnetic field is much lower than the phase transition
temperature TFS. Note, that the mentioned experiments with the compounds UGe2,
URhGe, and ZrZn2 do not give any evidence for the existence of a standard normal-
to-superconducting phase transition in a zero external magnetic field.
Moreover, it seems that the superconductivity in the metallic compounds, men-
tioned above, always coexists with the ferromagnetic order and is enhanced by it.
As claimed in [26] in these systems the superconductivity seems to arise from the
same electrons that create the band magnetism, and is most naturally understood as
a triplet rather than spin-singlet pairing phenomenon. Note, that all three metallic
182
Phenomenological study of spin-triplet ferromagnetic superconductors
compounds, mentioned so far, are itinerant ferromagnets. A similar type of uncon-
ventional superconductivity has been suggested [36] as a possible outcome of recent
experiments in Fe [37], where a superconducting phase is discovered at tempera-
tures below 2 K for pressures between 15 and 30 GPa. Note, that both vortex and
Meissner superconductivity phases [37] have been found in the high-pressure crys-
tal modification of Fe which has a hexagonal close-packed crystal structure. In this
hexagonal lattice the strong ferromagnetizm of the usual bcc iron crystal probably
disappears [36]. Thus one can hardly claim that there is a coexistence of ferromag-
netizm and superconductivity in Fe but the clear evidence of a superconductivity is
also a remarkable achievement.
The important point in all discussions of the interplay between superconductivi-
ty and ferromagnetizm is that a small amount of magnetic impurities can destroy
superconductivity in conventional (s-wave) superconductors by breaking up the (s-
wave) electron pairs with opposite spins (paramagnetic impurity effect [38]). In this
aspect the phenomenological arguments [39] and the conclusions based on the mi-
croscopic theory of magnetic impurities in s-wave superconductors [38] are in a
complete agreement with each other; see, e. g., [32–35]. In fact, a total suppressi-
on of conventional (s-wave) superconductivity should occur in the presence of an
uniform spontaneous magnetization M , i.e., in a standard ferromagnetic phase [39].
The physical reason for this suppression is the same as in the case of magnetic impu-
rities, namely, the opposite electron spins in the s-wave Cooper pair turn along the
vector M in order to lower their Zeeman energy and, hence, the pairs break down.
Therefore, the ferromagnetic order can hardly coexist with conventional supercon-
ducting states. Especially this is so for the coexistence of uniform superconducting
and ferromagnetic states when the superconducting order parameter ψ(x) and the
magnetization M(x) do not depend on the spatial vector x.
However, a coexistence of s-wave superconductivity and ferromagnetizm may ap-
pear in uncommon materials and under quite special circumstances. Furthermore,
let us emphasize that the conditions for the coexistence of nonuniform (“vertex”,
“spiral”, “spin-sinosoidal” or “helical”) superconducting and ferromagnetic states
are less restrictive than those for the coexistence of uniform superconducting and
ferromagnetic orders. Coexistence of nonuniform phases has been discussed in de-
tail, in particular, in the experiment and theory of ternary and Chevrel-phase com-
pounds, where such coexistence seems quite likely; for a comprehensive review, see,
for example, [32–35,40].
In fact, the only two superconducting systems for which the experimental data
allow assumptions in favor of a coexistence of superconductivity and ferromagne-
tizm are the rare earth ternary boride compound ErRh4B4 and the Chervel phase
compound HoMo6S8; for a more extended review, see [33,41]. In these compounds
the phase of coexistence most likely appears in a very narrow temperature region
just below the Curie temperature Tf of the ferromagnetic phase transition. At lower
temperatures the magnetic moments of the rare earth 4f electrons become better
aligned, the magnetization increases and the s-wave superconductivity pairs formed
by the conduction electrons disintegrate.
183
D.V.Shopova, T.E.Tsvetkov, D.I.Uzunov
We shall not extend our consideration over all important aspects of the long
standing problem of coexistence of superconductivity and ferromagnetizm. We shall
rather concentrate on the description of the newly discovered coexistence of fer-
romagnetizm and unconventional (spin-triplet) superconductivity in the itinerant
ferromagnets UGe2, ZrZn2, and URhGe. Here we wish to emphasize that the main
object of our discussion is the superconductivity of these compounds and, in a sec-
ond place in the rate of importance we put the problem of coexistence. The reason
is that the existence of superconductivity in such itinerant ferromagnets is a highly
nontrivial phenomenon. As noted in [42] the superconductivity in these materials
seems difficult to be explained in terms of previous theories [32,33,35] and requires
new concepts for the interpretation of experimental data.
We have already mentioned that in ternary compounds the ferromagnetizm
comes from the localized 4f electrons whereas the s-wave Cooper pairs are formed
by conduction electrons. In UGe2 and URhGe the 5f electrons of U atoms form both
superconducting and ferromagnetic order [26,31]. In ZrZn2 the same twofold role is
played by the 4d electrons of Zr. Therefore the task is to describe this behavior of the
band electrons at a microscopic level. One may speculate about a spin-fluctuation
mediated unconventional Cooper pairing as is in case of 3He and heavy fermion
superconductors. These important issues do not have yet a reliable answer and for
this reason we shall confine our consideration to a phenomenological level.
Reliable experimental data, for example, the data on the coherence length and
the superconducting gap [26,27,31,30], are in favor of the conclusion about a spin-
triplet Cooper pairing in these metallic compounds, although the mechanism of the
pairing remains unclear. We shall essentially use this reliable conclusion. Besides,
this point of view is consistent with the experimental observation of coexistence of
superconductivity only in the low temperature part of the ferromagnetic domain
of the phase diagram (T, P ) which means that a pure (non ferromagnetic) super-
conducting phase has not been observed. This circumstance is also in favor of the
assumption of a spin-triplet superconductivity. Our investigation leads to the results
which confirm this general picture.
Based on the experimental data and on the conclusions presented for the first
time in [26,29] and shortly afterwards confirmed in [27,28,30,31] it can be accepted
that the superconductivity in these magnetic compounds is considerably enhanced
by the ferromagnetic order parameter M and, perhaps, it could not exist without
this “mechanism of ferromagnetic trigger”, or, in short, “M -trigger”. Such a phe-
nomenon is possible for spin-triplet Cooper pairs, where the electron spins point
parallel to each other and their turning along the vector of spontaneous magneti-
zation M does not cause a breakdown of the spin-triplet Cooper pairs but rather
stabilizes them and, perhaps, stimulates their creation. We shall describe this phe-
nomenon at a phenomenological level.
Recently, the phenomenological theory that explains the coexistence of ferro-
magnetizm and unconventional spin-triplet superconductivity of GL type was de-
veloped [42,43]. The possible low-order couplings between superconducting and
ferromagnetic order parameters were derived by means of general symmetry group
184
Phenomenological study of spin-triplet ferromagnetic superconductors
arguments. Thus, several important features of the superconducting vortex state
in the ferromagnetic phase of unconventional ferromagnetic superconductors were
established [42,43].
In this paper we shall use the approach presented in [42,43] to investigate the
conditions for the occurrence of the Meissner phase and to demonstrate that the
presence of ferromagnetic order enhances the p-wave superconductivity. Our consid-
eration is focused on the ground state, namely, we are interested in uniform phases,
where the order parameters (the superconducting order parameter ψ and the mag-
netization vector M = {Mj, j = 1, 2, 3}), do not depend on the spatial vector ~x ∈ V ,
where V is the volume of the system. Recent results regarding the phase diagram and
the phase transitions [44,45], and thermodynamic quantities [46] will be essentially
used in our investigation.
Our study is based on the mean-field approximation [13] as well as on familiar
results for the possible phases in nonmagnetic superconductors with triplet (p-wave)
Cooper pairs [11,12,18]. Results from [44–47] will be reviewed and extended. In our
proceeding investigation [44–46] both Cooper pair anisotropy and crystal anisotropy
have been neglected in order to clarify the main effect of the coupling between the
ferromagnetic and superconducting order parameters. The phenomenological GL
free energy is quite complex and the inclusion of these anisotropies is related with
lengthy formulae and a multivariant analysis which obscures the final results. Here
we shall pay our attention to the effect of the Cooper pairs anisotropy.
There exists a formal similarity between the phase diagram we have obtained
and the phase diagram of certain improper ferroelectrics [48–53]. The variants of
the theory of improper ferroelectrics, known before 1980, were criticized in [53] for
their oversimplification and inconsistency with the experimental results. But the
further development of the theory has no such disadvantages (see, e. g., [51,52]). We
should emphasize that the symmetry of the GL model of spin-triplet ferromagnetic
superconductors is quite different from the symmetry of the known models in fer-
roelectrics and, hence, the results for ferroelectric systems can hardly be applied to
superconductors without additional investigations.
2. Ginzburg-Landau free energy
Consider the GL free energy F (ψ,M) = V f(ψ,M) , where the free energy den-
sity f(ψ,M) (for short hereafter called “free energy”) of a spin-triplet ferromagnetic
superconductor is given by
f(ψ,M) = as|ψ|2 +
bs
2
|ψ|4 +
us
2
|ψ2|2 +
vs
2
3
∑
j=1
|ψj |4 + afM
2 +
bf
2
M
4
+ iγ0M .(ψ × ψ∗) + δM 2|ψ|2 . (1)
In (1), ψ = {ψj ; j = 1, 2, 3} is a three-dimensional complex vector (ψj = ψ′
j+iψ′′
j )
describing the unconventional (spin-triplet) superconducting order and B = (H +
4πM) = ∇ × A is the magnetic induction; H = {Hj; j = 1, 2, 3} is the external
185
D.V.Shopova, T.E.Tsvetkov, D.I.Uzunov
magnetic field, A = {Aj ; j = 1, 2, 3} is the magnetic vector potential (∇. A = 0). In
(1), bs > 0, bf > 0, af = αf(T − Tf) is given by the positive material parameter αf
and the ferromagnetic critical temperature Tf corresponding to a simple ferromagnet
(ψ ≡ 0), and as = αs(T − Ts), where αs is another positive material parameter and
Ts is the critical temperature of a standard second order phase transition which may
occur at |H| = M = 0; M = |M |. The parameter us describes the anisotropy
of the spin-triplet Cooper pair whereas the crystal anisotropy is described by the
parameter vs [11,18].
The two orders – the magnetization vector M = {Mj} and ψ = {Aj}, interact
through the last two terms in (1). The γ0−term [43] ensures the triggering of the
superconductivity by the ferromagnetic order (γ0 > 0) whereas the δ−term makes
the model more realistic in the strong coupling limit [42]. Both ψM-interaction terms
included in (1) are important for a correct description of the temperature-pressure
(T, P ) phase diagram of the ferromagnetic superconductor [44,45]. In general, the
parameter δ for ferromagnetic superconductors may take both positive and negative
values.
As we are interested in the ground state properties, we set the external magnetic
field equal to zero (H = 0). Besides, we emphasize that the magnetization vector
M may produce vortex superconducting phase in case of type II superconductivi-
ty. The investigation of nonuniform (vortex) states can be made with the help of
gradient terms in the free energy which take into account the spatial variations of
the order parameter field ψ. This task is beyond our present consideration. Rather
we investigate the basic problem about the possible stable uniform (Meissner) su-
perconducting phases which may coexist with uniform ferromagnetic order. For this
aim the free energy (1) is quite convenient.
In case of a strong easy axis type of magnetic anisotropy, as is in UGe2 [26], the
overall complexity of mean-field analysis of the free energy f(ψ,M) can be avoided
by performing an “Ising-like” description: M = (0, 0,M). Further, because of the
equivalence of the “up” and “down” physical states (±M) the thermodynamic anal-
ysis can be done within the “gauge” M > 0. But this stage of consideration can also
be achieved without the help of crystal anisotropy arguments. When the magnetic
order has a continuous symmetry one may take advantage of the symmetry of the
total free energy f(ψ,M) and avoid the study of equivalent thermodynamic states
that occur as a result of the respective symmetry breaking at the phase transition
point but they have no effect on thermodynamics of the system. In the isotropic
system one may again choose a gauge, in which the magnetization vector has the
same direction as z-axis (|M | = Mz = M) and this will not influence the gen-
erality of thermodynamic analysis. Here we shall prefer the alternative description
within which the ferromagnetic state may occur as two equivalent “up” and “down”
domains with magnetizations M and −M, respectively.
We shall use adequate notations which reduce the number of parameters. With
the help of
b = (bs + us + vs) > 0 (2)
186
Phenomenological study of spin-triplet ferromagnetic superconductors
we redefine the order parameters and the other parameters in the following way:
ϕj = b1/4ψj = φje
θj , M = b
1/4
f
M ,
r =
as√
b
, t =
af√
bf
, w =
us
b
, v =
vs
b
,
γ =
γ0
b1/2b
1/4
f
, γ1 =
δ
(bbf)1/2
. (3)
Having in mind our approximation of uniform ψ and M and the notations (2)–
(3), the free energy density f(ψ,M) can be written in the form
f(ψ,M) = rφ2 +
1
2
φ4 + 2γφ1φ2Msin(θ2 − θ1) + γ1φ
2M2 + tM2 +
1
2
M4
− 2w
[
φ2
1φ
2
2sin
2(θ2 − θ1) + φ2
1φ
2
3sin
2 (θ1 − θ3) + φ2
2φ
2
3sin
2(θ2 − θ3)
]
− v
[
φ2
1φ
2
2 + φ2
1φ
2
3 + φ2
2φ
2
3
]
. (4)
In the above free energy the order parameters ψ and M are defined per unit volume.
We assume that Tf > Ts. This is the case when the superconductivity is triggered
by the magnetic order. We shall discuss the stable phases in the temperature region
T > Ts. The case Tf < Ts may be also important for neutron stars so it needs a special
investigation. When Ts ∼ Tf a quite simple analytical treatment is possible. All
these cases may be of interest to the description of ferromagnetic superconductivity
in stellar objects whereas in condensed matter only the case of Tf � Ts has been
observed so far.
We work in the framework of the standard mean-field analysis [13]. The stable
phases correspond to global minima of the GL energy (1). The equilibrium phase
transition line separating two phases is defined by thermodynamic states, where the
respective GL free energies are equal.
3. Phases
We shall not enumerate and discuss all phases described by (1). Rather we shall
focus our attention on the stable phases at relatively high temperatures (T > Ts).
The calculations show that for temperatures T > Ts, i.e., for r > 0, there are three
stable phases. Two of them are quite simple: the normal phase (ψ = M = 0) with
existence and stability domains given by t > 0 and r > 0, and the ferromagnetic
phase (FM) given by ψ = 0 and M2 = −t whose existence condition is t < 0, there
is a stability domain defined by the inequalities r > γ1t and
r > γ1t+ γ
√
−t . (5)
The third stable phase is the phase of coexistence of superconductivity and fer-
romagnetizm (hereafter referred to as FS). This phase is the main object of our
187
D.V.Shopova, T.E.Tsvetkov, D.I.Uzunov
consideration. It is given by the following equations:
φ1 = φ2 =
φ√
2
, φ3 = 0 , (6)
φ2 =
(
±γM − r − γ1M
2
)
, (7)
(
1 − γ2
1
)
M3 ± 3
2
γγ1M
2 +
(
t− γ2
2
− γ1r
)
M ± γr
2
= 0 , (8)
and
(θ2 − θ1) = ∓π
2
+ 2πk , (9)
(k = 0,±1, . . .). The upper sign in (7)–(9) corresponds to a domain where sin(θ2 −
θ1) = −1 and the lower sign corresponds to a second domain which may be referred
to as FS∗; there sin(θ2 − θ1) = 1. These two domains are equivalent and describe
the same ordering. We shall focus on the upper sign in (7)–(9), i.e. on FS. The
phase diagram (t, r) has qualitatively the same shape as the phase transition lines
corresponding to w = 0 but there are essential quantitative differences between
them. We shall discuss them in the next section. Note, that the system exhibits
both first and second order phase transitions and complex phase transition points:
triple and tricritical points [44,47].
4. Anisotropy effects
Our analysis demonstrates that when the anisotropy of the Cooper pairs is taken
into account, there will not be drastic changes in the shape of the phase diagram for
r > 0 and the order of the respective phase transitions. Of course, there will be some
changes in the size of the phase domains and the formulae for the thermodynamic
quantities. Besides, and this seems to be the main anisotropy effect, the w- and
v-terms in the free energy lead to a stabilization of the order along the main crystal
directions which, in other words, means that the degeneration of the possible ground
states is considerably reduced. So there will be a smaller number of marginally stable
states.
Let us neglect the crystal anisotropy by setting vs = 0 in (1) and concentrate our
attention on the Cooper pair anisotropy described by the us-term in the GL model.
The dimensionless anisotropy parameter w ∼ us given by (3) can be either positive
or negative depending on the sign of us. Obviously when us > 0, the parameter
w will be positive too (0 < w < 1). We shall illustrate the effect of Cooper-pair
anisotropy in this case. The order parameters (M , φj, θj) are given by (6), (9),
φ2 =
±γM − r − γ1M
2
(1 − w)
> 0 , (10)
and
(
1 − w − γ2
1
)
M3 ± 3
2
γγ1M
2 +
[
t(1 − w) − γ2
2
− γ1r
]
M ± γr
2
= 0 , (11)
188
Phenomenological study of spin-triplet ferromagnetic superconductors
where the meaning of the upper and lower sign is the same as explained just below
(9). We consider the FS domain corresponding to the upper sign in the (10) and
(11). The stability conditions for FS read,
(2 − w)γM − r − γ1M
2
1 − w
> 0 , (12)
γM − wr − wγ1M
2
> 0 , (13)
and
1
1 − w
[
3(1 − w − γ2
1)M
2 + 3γγ1M + t(1 − w) − γ2
2
− γ1r
]
> 0 . (14)
For M 6= (γ/2γ1) we can express the function r(M, t) defined by (11), substitute the
obtained expression for r(M, t) in the existence (10) and stability conditions (12)-
(14) and do the analysis in the same way as for w = 0 [47]. The most substantial
qualitative difference between the cases w > 0 and w < 0 is that for w < 0 the
stability of FS is bounded for r < 0.
5. Conclusion
We have done an investigation of the M-trigger effect in unconventional ferro-
magnetic superconductors. This effect due to the Mψ1ψ2-coupling term in the GL
free energy consists in bringing into existence the superconductivity in a domain
of the phase diagram of the system that is entirely in the region of existence of
the ferromagnetic phase. This form of coexistence of unconventional superconduc-
tivity and ferromagnetic order is possible for temperatures above and below the
critical temperature Ts, which corresponds to the standard phase transition of sec-
ond order from normal to Meissner phase, i. e., a usual uniform superconductivity
in a zero external magnetic field, which appears outside the domain of existence of
the ferromagnetic order. Our investigation has been mainly intended to clarify the
thermodynamic behaviour at temperatures Ts < T < Tf , where the superconduc-
tivity cannot appear without the mechanism of M-triggering. We have described the
possible ordered phases (FM and FS) in this most interesting temperature interval.
The Cooper pair and crystal anisotropies have also been investigated and their
main effects on the thermodynamics of the triggered phase of coexistence are es-
tablished. In discussions of concrete real materials one should take into account
the respective crystal symmetry but the variation of the essential thermodynamic
properties with the change of the type of symmetry is not substantial when the low
symmetry and low order (in both M and ψ) γ-term is present in the free energy.
Below the superconducting critical temperature Ts a variety of pure supercon-
ducting and mixed phases of coexistence of superconductivity and ferromagnetizm
exists and the thermodynamic behavior at these relatively low temperatures is more
complex than in the known cases of improper ferroelectrics. The case Tf < Ts also
needs a special investigation. Our results are referred to the possible uniform super-
conducting and ferromagnetic states. Vortex and other nonuniform phases need a
separate study.
189
D.V.Shopova, T.E.Tsvetkov, D.I.Uzunov
The relation of the present investigation to the properties of real ferromagnetic
compounds, such as UGe2, URhGe, and ZrZn2, has been discussed throughout the
text. In these real compounds the ferromagnetic critical temperature is much larg-
er than the superconducting critical temperature (Tf � Ts) and that is why the
M-triggering of the spin-triplet superconductivity is very strong. Moreover, the γ1-
term is important to stabilize the FM order up to the absolute zero (0 K), as is in
the known spin-triplet ferromagnetic superconductors. The neglect [43] of the sym-
metry conserving γ1-term hinders the proper description of real substances of this
type. More experimental information about the values of the material parameters
(as, af , . . .) is required in order to outline the thermodynamic behavior and the phase
diagram in terms of thermodynamic parameters T and P . In particular, a reliable
knowledge about the dependence of the parameters as and af on the pressure P ,
on the value of the characteristic temperature Ts and on the ratio as/af at zero
temperature are of primary interest.
6. Acknowledgements
The authors warmly greet Reinhard Folk with his 60th anniversary! D.V.S. and
D.I.U. thank Reinhard Folk for the valuable collaboration, discussions, and profes-
sional support over the last fifteen years.
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Феноменологічне дослідження спін-триплетних
феромагнітних надпровідників
Д.В.Шопова, Т.Є.Цвєтков, Д.І.Узунов
CP лабораторія, Інститут фізики твердого стану,
Болгарська академія наук,
BG–1784 Софія, Болгарія
Отримано 15 листопада 2004 р.
Оглянуто нетрадиційну надпровідність із спін-триплетним купе-
рівським спарюванням, базуючись на квазі-феноменологічній тео-
рії Гінзбурга-Ландау. Надпровідність, а зокрема змішана фаза спів-
існування феромагнетизму і незвичної надпровідності викликаєть-
ся спонтанною намагніченістю. Змішана фаза є стійкою, в той
час як інші надпровідні фази, що зазвичай існують у нетради-
ційних надпровідниках, є або нестійкими, або, при певних зна-
ченнях параметрів теорії, метастабільними при відносно низьких
температурах у вкрай вузькій області фазової діаграми. Фазовий
перехід із нормальної фази у фазу співіснування є переходом пер-
шого роду, в той час як фазовий перехід із феромагнітної фази
у фазу співіснування може бути або першого або другого роду
залежно від типу речовини. Куперівське спарювання і анізотропія
кристалу є суттєвими для більш точного опису форми фазової
діаграми і знімають виродження основного стану системи, проте
вони не чинять значного впливу на області фазової стійкості і
термодинамічні властивості відповідних фаз.
Ключові слова: надпровідність, феромагнетизм, фазова діаграма,
профіль параметра впорядкуваня, анізотропія
PACS: 74.20.De, 74.20.Rp
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