Can we explain the coexistence of ferromagnetism and

superconductivity based on two parameters model?

Yuan Zhoua,*, Jun Lia, Chang-De Gonga,b

aNational Key Laboratory of Solid State Microstructures, Department of Physics, Nanjing University, Nanjing 210093, ChinabDepartment of Physics, The Chinese University of Hongkong, Hongkong, China

Received 11 May 2004; received in revised form 1 September 2004; accepted 7 September 2004 by P. Sheng

Available online 18 September 2004

Abstract

A possible explanation about the coexistence of ferromagnetism (FM) and superconductivity (SC) based on a two parameters

mean field model in a two-dimensional system is discussed. The key feature of this model is that there are two independent

parameters which are responsible for ferromagnetism and superconductivity, respectively. We point out that the coexisting FM

and s-wave pairing SC state is energetically not favorable among all possible state. We generalize the two parameter model to

include the coexistence of FM with p-wave SC. We find that the phase diagram is not consistent with what experimentally

discovered in UGe2.

q 2004 Elsevier Ltd. All rights reserved.

PACS: 74.10.Cv; 75.50.Cc; 75.10.Lp

Keywords: A. Superconductors; D. Electronic band structure

1. Introduction

It is a long-time period for people to believe that the

singlet s-wave pairing superconductivity (SC) and ferro-

magnetism (FM) are mutually exclusive in the case of spin

singlet pairing because of Pauli paramagnetic limit. But

interests in this aspect are revived due to a recent discovery

of the coexistence of itinerant FM and SC in UGe2 [1,2] and

other compounds, such as URhGe [3] and ZnZr2 [4]. These

experiments show that the same electrons are responsible for

both the FM and SC. UGe2 is a heavy-fermion compound

with highly anisotropic structure, however, its 5f electrons

are more itinerant than that in many other heavy-fermion

systems. Below the Curie temperature, Both UGe2 and

URhGe contain zigzag chains of nearest-neighbour uranium

0038-1098/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ssc.2004.09.012

* Corresponding author. Tel.: C86 2583594678; fax: C86

2583595535.

E-mail address: cdgongsc@nju.edu.cn (Y. Zhou).

ions [1,3]. An ab initial calculation pointed out that the

quasi-two-dimensional Fermi surface has weak dispersion

along ky [5]. It means that the structure of these compounds

are anisotropic two-dimensional plane system [6], the inter-

plane distance is much larger than the lattice constants (a,b)

within the plane, or more precisely it is equivalent to a 2Ce-

dimension system, (where 0!e!1), in which the thermal

fluctuation has been suppressed. So far as the conduction

problems are concerned, including the superconductivity,

the system can be described by a 2-D mean field model with

appropriately selected parameters for considering the 2Ce

dimensionality effect. In general, high pressure would favor

the choice of a more symmetric cell [2]. Another important

experimental result about the superconducting critical

temperature is its non-monotonic pressure dependence in

coexistent phase [1]. Recently, Karchev et al. [7] proposed

the first three-dimensional mean field model where s-wave

SC and FM are caused by two independent coupling

constants. Though it seems to be questionable to derive this

model from the original 4-fermion Hamiltonian [Eq. (1) in

Solid State Communications 132 (2004) 507–511

www.elsevier.com/locate/ssc

Y. Zhou et al. / Solid State Communications 132 (2004) 507–511508

[7]] by means of the saddle point approximation [8].

Furthermore, the stability of their coexistent solution in

comparison with other solutions is doubtful [9]. On the other

hand, there are experimental evidences from FM exchange

band splitting and specific heat to preclude the spin singlet

pairing and support the spin-triplet p-wave pairing in UGe2.

Theories based on spin-fluctuation mediated p-wave pairing

[10] are applied to explain the UGe2 compound [6,11–15].

Motivated by these progresses in the field of unconven-

tional ferromagnetic superconductivity, we believe more

works within the two parameters model should be

performed. But the question is that what the symmetry of

the SC pairing can be tolerated in the ferromagnetic SC in

the frame of mean field theory and two parameters model.

We will present numerical study with more realistic two-

dimensional (2D) band structure instead of the three-

dimensional free electron energy. We show that the

coexisting FM and s-wave pairing SC state has highest

ground state energy among all possible solutions. Further-

more, we generalize the two parameters model to the case of

spin triplet pairing. We find that the coexisting FM and p-

wave paring SC phase is the most stable one. Unfortunately,

the parameter phase diagram of this coexistence phase is not

consistent with the experimental finding in UGe2 system.

This might be due to the unrealistic isotropic dispersion to

be used. By using a more realistic tight-binding anisotropic

2D band structure, we will find that the magnetization

dependence of the SC energy gap is non-monotonic, which

may explain the experimental result discovered in UGe2[16].

Fig. 1. The total energy vs. g with s-wave pairing. Solid line for the

FM phase; dot line for the coexistence phase and dash line for pure

SC phase at half-filling case. The total energy of the coexistence

phase is always highest among three possible phase, while the total

energy of the FM phase is smaller than that of the pure SC phase

unless g is large enough.

2. Theory and results for s-wave pairing

The mean-field Hamiltonian with s-wave pairing and

itinerant FM order is [7]

Hseff Z

Xks

ekscCkscks K

Xk

ðD�cKkYck[CDcCk[cCKkYÞ

C1

2JM2 C

jDj2

g; (1)

where eksZekGðJM=2Þ (sZG1 for spin-up and spin-

down, respectively), 2kZK2tðcos kxCcos kyÞKm for 2D

system with the nearest-neighbor hopping t. The SC energy

gap is DZghcKkYck[i. The ferromagnetic exchange J and

SC coupling g are independent to each other in this model.

The magnetization of the system defined by MZKhSZ iZ1=2ðhnYiK hn[iÞ and the chemical potential m determined by

the total particle number per site NZhnYiChn[i should be

solved jointly with the gap equation

1

gZ

1

2

XEkO

JM2

1

Ek

: (2)

The particle distribution functions at zero temperature are

found to be

hnk[iZ1

21K

ek

Ek

� �q Ek K

JM

2

� �; (3)

and

hnkYiZ 1K1

21C

ek

Ek

� �q Ek K

JM

2

� �; (4)

where EkZffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2kC jDj2

q: The total energy is

hHiZXEk!

JM2

ek CXEkO

JM2

ek 1Kek

Ek

� �K

JM2

2K

D2

g

CmN: (5)

Solutions of the above three self-consistent equations are

corresponding to the free electron state (DZ0, MZ0), the

ferromagnetic state (DZ0, Ms0), the BCS state (Ds0,

MZ0) and the coexisting SC and FM state (Ds0, Ms0).

From the point of view of the total energy hHi, the free

electron state is eliminated. The relative stabilities of the

remaining three phases could be obtained by comparing

the total energy as plotted in Fig. 1. The total energy of the

coexistence phase is higher than that of the two single

phases for all values of J and g. On the other hand, the

normal FM phase and the BCS phase compete with each

other. In mean-field theory, non-zero magnetization is

available only when JOJc. The FM phase is stable for

small enough g at JOJc, otherwise, the BCS phase is the

ground state. The (J, g) phase diagrams at three different

electron fillings are plotted in Fig. 2. So that it is clear that

the two parameters model with tight-binding electron band

in the s-wave pairing channel cannot explain the coexistence

of SC and FM. The possibility of other pairing symmetry

Fig. 2. Phase diagram in the condition of s-wave pairing, solid line

for NZ1.0 case; dotted line for NZ0.95 case and dash line for NZ0.9 case. The system prefer to FM phase unless the SC coupling

constant g is large enough.

Y. Zhou et al. / Solid State Communications 132 (2004) 507–511 509

should be considered to understand the coexistence

phenomena within the two parameters model.

3. Theory and results for p-wave pairing

We generalize the above two-parameter model to the

case of triplet p-wave pairing. Due to a large spin-splitting

of Fermi surfaces, the possibility of pairing opposite

momentum states with anti-parallel spins is removed [3],

and only equal-spin pairing (ESP) should be considered.

The effective Hamiltonian with ESP is

Hpeff Z

Xks

eksCCksCks K

1

2

Xks

ðD�s ðkÞcKkscks

CDðkÞcCkscCKksÞ:

The p-wave SC energy gap is defined as DsðkÞZP

k 0 Vðk;

k 0ÞhcKk 0sck 0si; where the interaction is assumed as V(k,k 0)ZggkKk 0 with gkKk 0ZcosðkxKk 0xÞCcosðkyKk 0yÞ: The anti-

symmetric p-wave energy gap is taken as DsðkÞZDsðsin

kxC i sin kyÞ; a form analogous to that in the A1 phase of

He3[17]. Hence, the four self-consistent equations for the

magnetization M, chemical potential m and the SC order

parameters (D[, DY) are derived as

M ZK1

4

Xks

seksffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

e2ks C jDsj2ðsin2 kx Csin2 kyÞ

q0B@

1CA; (6)

N ZXk

1K1

2

Xks

eksffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2ks C jDsj

2ðsin2 kx Csin2 kyÞ

q ; (7)

1

gZ

1

2

Xk

sin2 kxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2ks C jDsj

2ðsin2 kx Csin2 kyÞ

q ; (8)

where sZG1 for spin-up and spin-down channel, respect-

ively. The total energy with p-wave pairing is

hHiZXks

ekhnksiK1

4

!Xks

jDsj2ðsin2 kx Csin2 kyÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

e2ks C jDsj2ðsin2 kx Csin2 kyÞ

q0B@

1CA

KJM2

2CmN: (9)

If the magnetization equal to zero, it refers to the pure SC

phase.

There exist two types of equal spin pairing in the FM

phase due to spin rotation symmetry breaking. The ESP SC

gap in spin-down (which is taken as the majority spin

polarization direction) is generally larger than that in the

spin-up channel except the half-filled band. The chemical

potential is zero in this special case, so that ek[ZKekC~QY

with ~QZ ðp;pÞ: Thus the gap equation in the two ESP

channels are identical. When FM disappears, the two type of

the SC will be coincident. The results are shown in Fig. 3.

The SC energy gap increases with the increasing g, while the

magnetization decreases, which means that the SC pairing

will suppress the magnetization. On the other hand, the

increasing of J, i.e. the strong magnetization will also

depress the SC energy gap as shown in Fig. 4, the SC energy

gap will disappear if magnetization is large enough. The FM

wishes the spin to be in same direction, while SC prefers to

the equal amount of spin-up or spin-down pairing for

lowering the total energy. The competition between SC and

FM is the reason of the depression of each other. Is the

coexistence phase stable in the p-wave pairing case? We can

calculate the total energy of the coexistence phase and

compare with that of the FM phase and the pure SC phase

with the same parameters. The Fig. 5 shows that the total

energy of the p-wave type coexistence phase is the lowest

one among the pure SC phase, FM phase and the

coexistence phase, which means the coexistence phase

could be established. The two parameter (J, g) p-wave

pairing model tolerate the coexistence phase, but shows only

monotonous behavior of SC energy gap vs. g unless g/J have

some anomalous behaviors under the pressure. If instead of

the isotropic 2D tight-binding hand, we use more realistic

anisotropic dispersion given by Sandeman et al. [6]: ekZKa cos kxKb cos kx cos kyKg cosð2kxÞKd cosð3kxÞ with

aZ1, bZ0.7, gZ0.03, dZK0.03. Through the same

calculations as given in (6)–(9), the SC energy gap versus

Fig. 3. The magnetization and SC energy gap vs. g with p-wave pairing. (a) NZ1.0, JZ8.0 case, DYZD[; (b) NZ0.9, JZ8.8 case, DYOD[

until MZ0. The SC energy gap increases with the increasing g, while magnetization decreases down to zero.

Fig. 4. The SC energy gap with p-wave pairing vs. the

magnetization with different g at half-filling case.

Fig. 5. The total energy vs. g with p-wave pairing at NZ1.0 case. It

shows that the total energy of coexistence phase is always smaller

than that of the pure SC phase with p-wave pairing in 2D case.

Y. Zhou et al. / Solid State Communications 132 (2004) 507–511510

Fig. 6. The SC energy gap with p-wave spin-down pairing vs.

magnetization with anisotropic band structure given in the text. NZ0.72, solid line for gZ1.2, dash line for gZ1.0, and dotted line for

gZ0.8 [16].

Y. Zhou et al. / Solid State Communications 132 (2004) 507–511 511

magnetization plot shows non-monotonic behavior as

shown in Fig. 6, i.e. a two parameters model in an

anisotropic 2D system with p-wave pairing is a possible

choice to describe the coexistence of FM and SC in UGe2within the mean field theory. Other choice beyond the two

parameters model, for example the mean field one parameter

model may also give reasonable coexistence phase with

p-wave pairing.

4. Conclusion

In conclusion, the coexistence of FM and SC in the two-

dimensional system was investigated in the case of s-wave

or p-wave pairing based on the two parameters mean field

model. We find that the order parameters of the FM and the

SC depress each other in both cases. By comparing the total

free energy, we show that a s-wave type coexistence is

always unstable, while a p-wave type coexistence is

possible. Our results on the p-wave pairing fail to show

the non-monotonous behaviors of the SC energy gap as

experimental data shown, which means that the two-

parameters model with isotropic two-dimensional band

structure is not adequate to describe the coexistence of FM

and SC discovered in UGe2.

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