Coexistence of superconductivity and magnetism

ADVANCES IN PHYSICS, 1985, VOL. 34, NO. 2, 175-261

Coexistence of superconductivity and magnetism Theoretical predictions and experimental results By L. N. BULAEVSKII P.N. Lebedev Physical Institute, Moscow, U.S.S.R. A. I. BUZDIN MOSCOW State University Physics Department, M o s c o w , U.S.S.R. M. L. KULIt~ Institute o f Physics, Belgrade, Yugoslavia and S. V. PANJUKOV P.N. Lebedev Physical Institute, Moscow, U.S.S.R. [Received 6 September 1984] Abstract Superconductivity and ferromagnetic ordering are two antagonistic types of ordering, and their mutual influence leads to many interesting phenomena which have been studied recently in ternary compounds. Theoretical analysis of ferromagnetic materials which are type II superconductors near the superconducting transition point Tc~ shows that they become type I near the magnetic transition point TM. The proposed theory constructed for the case TM ,~ Tcl predicts the formation of a transverse domain-like (DS phase) magnetic structure below TM. The electronic spectrum appears to be gapless in the DS phase of clean compounds with a re-entrant transition. The change from type II to type I behaviour as the sample is cooled to TM has been observed in ErRh4B4. Experimental data for HoM06S8, HoMo6Se 8 and ErRh4B 4 give evidence for the coexistence of superconductivity and non-uniform magnetic ordering below TM. Mutual influence of superconducting and magnetic orderings is also studied. Contents

PAGE

List of symbols and abbreviations. 1. Introduction.

Interaction between conducting electrons and localized moments in magnetic superconductors. 2.1. Structure of magnetic superconductors and hamiltonian of magnetic and superconducting systems. 2.2. Crystal-field effects. 2.3. Indirect exchange interaction in the normal state--RKKY interaction. 2.4. Electromagnetic (dipole~lipole) interaction in normal state. 2.5. Magnetic functional of normal ferromagnetic state. 3. Properties of ferromagnetic conductors (FS) in non-magnetic superconducting phase--above TM. 3.1. Role of magnetic scattering.

176 178

187 187 189 190 191 192 193 193

L. B. Bulaevskii, A. L Buzdin, M. L. Kulik and S. V. Panjukov

176

3.2. Upper critical field Hc2. 3.3. Lower critical field Hal. 3.4. Experimental data for critical fields in ErRh4B4. 4. Structure of coexistence phase in ferromagnetic superconductors. 4.1. General method for calculation of free-energy functional. 4.2. Coexistence phase near TM and critical fluctuations. 4.2.1. The temperature region very close to TM. 4.2.2. The temperature region slightly below TM. 4.3. Coexistence phase with strong exchange fields. 4.4. Transition from DS phase to F N phase. 4.5. Superconducting properties of DS phase. 4.6. Role of spin-orbit scattering and phase diagram of ferromagnetic superconductors. 4.7. Effect of magnetic structure defects on coexistence phase.

205 205 207 207 208 210 214 215

5. The effect of magnetic field on coexistence phase. 5.1. Change of parameters of DS phase in magnetic field H. 5.2. (H, T) phase diagram for T < TM.

221 221 225

6. Theoretical predictions for FS and comparison with experimental results. 6.1. The main conclusions of theory. 6.2. Properties of HoMo6Ss, and HoMo6Se 8. 6.3. Experimental data for ErRh4B4. 6.4. Re-entrant superconductivity in ternary silicides, stannides and pseudoternary compounds.

217 219

226 226 227 234 239

7. Does superconductivity exist in the domain walls of the F N phase? 7.1. Formulation of problem. 7.2. Localized superconducting solution. 7.3. Type of transition to S - D W state and region of its existence.

240 240 241 243

8. Antiferromagnetic superconductors. 8.1. Interaction of superconductivity and antiferromagnetic ordering. 8.2. Upper critical magnetic field He2 in A F superconductors. 8.3. Coexistence phases of superconducting systems with weak ferromagnetism.

245 245 246

9. Conclusions.

249

Appendices. A. Superconducting functional of dirty systems. B. Superconducting functional of clean systems. C. Phase transition near tricritical point. D. Surface energy of domain wall. References.

a

A B(O) d D

F~, Fm Fint

194 200 201

247

251 253 254 255 256

List of symbols and abbreviations Magnetic correlation length at T = 0 Vector potential Magnetic induction in ferromagnetic state at T = 0 Domain thickness in coexistence phase Parameter of magnetic anisotropy Magnetic free energy Interacting part of free-energy between superconducting electrons and localized moments Land6 factor

Coex&tence of superconductivity and magnet&m G

ho H Ha

Ho, n~2 H~ J J(r) J(q) l I M n

N(0) QM

S(T) T~o To, T~, T~2

TM, T~ 'OF

A0

0

0~,, 0~m 2L

~o ~(E) T ~'s

Z~ Z~, Zn

AF DS EM, EX F FN IS LOFF MS S S-DW VS WF

177

Vector of reciprocal magnetic lattice Exchange field (in energetic units) due to localized moments, acting on conduction electrons Magnetic field Applied magnetic field Lower critical field Upper critical field Upper orbital critical field Crystal field hamiltonian Magnitude of localized moment Parameter of exchange interaction between electrons and localized moments Fourier transform of J(r) Mean free path of electrons Antiferromagnetic vector Magnetization Concentration of localized moments Density of states of Fermi level Wave-vector of non-uniform magnetic structure in coexistence phase Normalized magnetic moment of rare earth atom Superconducting transition temperature in absence of localized magnetic moments Superconducting critical temperatures on cooling Temperature at superconducting-normal magnetic state transition Supercooling temperature of superconducting (or coexistence) phase in ferromagnetic region Ferromagnetic overheating temperature in region of superconducting phase Magnetic transition temperature in ferromagnetic and antiferromagnetic superconductors respectively Fermi velocity Superconducting gap at T = 0 in absence of magnetic moments Surface energy of domain wall Ferromagnetic ground-state energy (per ion) in absence of superconductivity Contribution to 0 from long-range part of exchange and electromagnetic interactions respectively London penetration depth Magnetic moment at T = 0 Superconducting correlation length at T = 0 in absence of magnetic moments Energy dependence of density of electronic states Mean free time of electrons Mean free time of exchange scattering of electrons on localized moments Electronic paramagnetic susceptibility Paramagnetic susceptibility of localized moments Electronic paramagnetic susceptibility in superconducting and normal state respectively Antiferromagnetic state Coexistence phase with magnetic domain structure Electromagnetic and exchange interaction (mechanism) respectively between electrons and localized moments Ferromagnetic state Normal ferromagnetic phase Intermediate state Larkin-Ovchinikov-Fuld~Ferrel state Meissner state Superconducting phase Magnetic domain wall in superconducting state Vortex state State with weak ferromagnetism

178

L. B. Bulaevskii, A. L Buzdin, M. L. KuliO and S. V. Panjukov 1.

Introduction

Superconductivity and magnetic ordering are two very different cooperative phenomena, and the question regarding their coexistence is important because of their antagonistic character. There are two mechanisms of mutual influence between magnetic moments and superconducting electrons---exchange and electromagnetic interactions. The electromagnetic mechanism (EM), which suppresses superconductivity in ferromagnetic systems, was considered by Ginzburg [1] in 1956, before the invention of the Bardeen-Coope~Schrieffer (BCS) theory. He studied ferromagnetic Meissner state of a type I superconductor as a coexistence phase, where the superconducting order parameter and magnetization were supposed to be constant throughout the sample, as in figure 1. In this case, magnetic induction is zero inside the sample, due to screening effects of superconducting currents flowing in the surface region with the thickness 2L, the London penetration depth. Assuming the magnetic induction of the ferromagnetic system to be large in comparison with critical field He, Ginzburg and Zharkov [2] came to the conclusion that coexistence is possible only in very exceptional cases, where the effect of magnetic induction is suppressed for some reason, for instance in thin films, or when magnetization is in the opposite direction to the applied field. They concluded that, in massive samples, competition between ferromagnetic and superconducting ordering practically prevents their coexistence. The antagonistic character of superconducting and ferromagnetic orderings, in the framework of the EM mechanism, has been analyzed by Blount and Varma [3], Ferrel et al. [4], and by Matsumoto et al. [5]. They concluded that since the long-range part of the dipole~lipole interaction (between localized moments, LM) is strongly screened by Meissner currents, the ferromagnetic transition temperature is drastically decreased in the superconducting phase. In 1958 Matthias et al. [6] considered another mechanism of interaction between superconducting electrons and magnetic moments--the exchange (EX) interaction. The exchange field, in the magnetically ordered state, tends to align spins of Cooper pairs in the same direction, thus preventing pairing effects. This is called the paramagnetic effect [7]. Besides this effect, superconducting pairing is suppressed by the exchange scattering of electrons on LMs, because the exchange scattering has a detrimental effect on singlet pairing. This has been considered first by Abrikosov and Gor'kov [8]. We point out that the magnetic scattering is present in both the paramagnetic and ferromagnetic phases.

f I I I

/

J

Figure 1. The Meissner superconducting ferromagnetic phase. The arrows show the direction of screening currents J~ and magnetization M inside the sample.

Coexistence of superconductivity and magnetism

179

Anderson and Suhl [9] have shown that ferromagnetic ordering is less likely in the superconducting phase, than in the normal phase. The main reason for this is a suppression of the zero wave-vector component (q = 0) of the electronic paramagnetic susceptibility gs (q) in the presence of superconductivity. It should be stressed that the ferromagnetic transition temperature is proportional to Zs(0) in the case of the R K K Y interaction between LMs, due to the direct exchange interaction between conducting electrons and LMs. From the preceding discussion it should be clear that a strong mutual influence exists only between superconductivity and ferromagnetism. In the case of antiferromagnetism the average values of magnetic induction B and exchange field h are practically zero on the scale of the superconducting correlation length, which means that the mutual influence of superconductivity and antiferromagnetism is small unless the exchange scattering is large. On the basis of these facts, Baitensperger and Str/issler [10] concluded that the two phenomenon were not incompatible, which has been confirmed by subsequent experiments. In the ferromagnetic case, the common effect of the EX and EM mechanisms excludes a simple coexistence of ferromagnetism and superconductivity, giving rise either to the first-order transition S - F N between nonmagnetic superconducting (S) and normal ferromagnetic (FN) phases [11], or to coexistence of superconductivity and non-uniform magnetic ordering. The latter possibility was predicted by Anderson and Suhi [9] in 1959, in the framework of the EX mechanism, and in subsequent papers [3-5] in the framework of the EM mechanism. In the above discussion we said that the electronic paramagnetic susceptibility Z~(0) is decreased in the superconducting state, leading to the depression of the ferromagnetic transition temperature as shown in figure 2. However, the effect of Cooper pairing on the zs(q) is reduced at large wave-vectors ]q[. Assuming that the magnetic critical temperature is much smaller than the superconducting critical

0M

2KF

q

Figure 2. Dependence of electronic susceptibility g~ on wave-vector q in the superconducting state at T ~ Tc (solid line). Zo reaches a maximum at QM ~ (a2~0) ~. Normal state susceptibility L,(q) is represented by the broken line.

180


temperature To,, i.e. TM '~ Tc~, Anderson and Suhl calculated xs(q) in the isotropic model and in the limit of small exchange field. It has the following form in the limit of pure superconductor: Zs(q) =

ZR(0)(1 -- (n/2~0q) -- a2q2),

(1.1)

for T = 0 and q~0 >3> 1. Here, a means the magnetic stiffness, and is of the order of atomic radius, and ~0 is the superconducting coherence length. The curve Zs(q) tends to zero as q ---) 0 and approaches the corresponding one for the normal metal at large q >> Go ) . It has a maximum at QM ~ (a2¢0) -1/3 where ~o I ,~ QM '~ a-~ because G0 >> a. Thus, a non-uniform magnetic structure with the wave-vector QM appears in the superconducting state. It has been called cryptoferromagnetism by Anderson and Suhl. The magnetic energy of the cryptoferromagnetic state is very close to the ferromagnetic energy, because QM '~ a -l , and hence the effect of magnetic ordering on superconductivity is reduced, because QM >> ~o ~. This situation is possible when a ,~ G, which leads to the modification of the ordering with smaller stiffness; in this case it is the magnetic which is modified. The situation is similar in the framework of the EM mechanism. As a spontaneous magnetization appears the superconductor responds with screening currents (the Meissner effect) to compensate for the average magnetization. In order to minimize the energy of this process, the magnetic moments tilt slightly, so as to form a longwavelength oscillatory state. If this wavelength (.-. QM)) is shorter than the superconducting penetration depth 2L no Meissner effect will result. The wave-vector QM minimizes the non-uniform part of the energy a2q21M, I2 + ~

1

IA, I2,

(1.2)

where Mq and A, are the Fourier transforms of the magnetization and the vector potential. The first term in (1.2) is due to the non-uniformity of the magnetization, and the second term describes the EM interaction between superconducting electrons and the magnetic field by LMs. Assuming that q)'L ~ 1 has i[qAq] = 4nMq. Putting this into (1.2) and minimizing with respect to q one gets the wave-vector QM ~ (a2L) -I/2 [3--5]. In the paper by Anderson and Suhl, as well as in [3-5, 12, 13], the wave-vector QM of the non-uniform magnetic structure was calculated at the transition point TM assuming a second-order phase transition. Many of the theoretical studies have been directed toward the explanation of the type of magnetic structure, as well as of superconducting properties in the coexistence phase. However, the answer depends on which type of interaction (EX or EM) is dominant. The helical (simple spiral) magnetic structure (figure 3) was predicted within the framework of the isotropic EX mechanism [14]. This was done in the limit of small and large exchange fields. It was also predicted that far away from the transition point TM, and in clean systems, superconductivity has a gapless character. By lowering the temperature from TM, there are two possibilities for the further fate of the coexistence phase; either it disappears at some point To2 of first order, or it survives down to zero temperature, depending on parameters of the system. However, it was clear that the spiral structure is practically realizable only in an isotropic model. The real magnetic anisotropy transforms the spiral structure into the domain-like struc-


181

Figure 3. Spiral magnetic structure in the coexistence phase of an isotropic ferromagnetic superconductor. ture shown in figure 4. The first authors to point out the domain-like magnetic structure in the coexistence phase appear to have been Fulde and Ferrell [15], who found the period of this structure for a small exchange field, i.e., near TM. The spiral structure also comes out naturally within the framework of the isotropic EM mechanism [4, 5], while the real magnetic anisotropy favours domain-like structure [16, 17]. As the temperature is lowered, there are two possible transitions; either into the phase with a spontaneous flux line lattice (figure 5) when 47rM > Hcl, (Hcl is lower critical field), or directly to the FN phase, depending on the value of the Ginzburg-Landau parameter x. In the state with a spontaneous vortex lattice (VS) the magnetic induction B is non-zero inside the sample (see figure 5). In the framework of the EM mechanism such a state is more favourable than the domain-like structure if the magnetization [MI is sufficiently large. In the last case the increase of magnetic energy cannot be compensated by superconducting condensation energy. At the same time, in the VS state, B ( = 4rcM) should be sufficiently small to preserve superconductivity, i.e., B < H~2. All known magnetic superconductors are type II with respect to the parameter x = 2/~, and a more precise statement of this will be given in §3. Overviews of this approach have been given by Tachiki [18] and Ishikawa [19], and in reviews [20, 21].

/// 1

I

1

1

t

I

/

d

Figure 4. One-dimensional transverse domain-like magnetic structure in the DS phase. The arrows indicate directions of the moments and d is the domain width.

182

L. B. Bulaevskii, A. L Buzdin, M. L. Kufi? and S. V. Panjukov

@@ @

@

@ @ @ ® / @ @ @ @ / @

@

/

o ooo I I I I I Figure 5. The spontaneous vortex phase. Straight arrows show the magnetization inside the sample, circles show supercurrents and the shaded regions are normal kernels of the vortices. The next step towards the understanding of the coexistence phase in ferromagnetic superconductors was made when both mechansims (EX and EM) were treated simultaneously, taking into account magnetic anisotropy [22-24], using the BCS model and Gor'kov's equations or the corresponding quasiclassical Eilenberger equations. They make possible a quantitative analysis of the real and most interesting magnetic superconductors with TM "~ Tc~. In [22-24], and in some previous works [12, 13, 25, 26], it was shown that the E X mechanism plays the dominant role in the formation of the coexistence phase in real systems. This is also true when the characteristic parameter (0ex) of the EX mechanism is smaller than the parameter (0em) which characterizes the EM mechanism; 0ex is the contribution to the ferromagnetic energy due to the EX interaction, while 0em is the corresponding contribution from the EM interaction (per LM at T = 0). This is connected with the important fact that the effect of the non-uniform magnetic field on superconductivity is reduced more than the effect of a non-uniform exchange field by an increase in the wave-vector Q. Their relative effect on superconductivity is characterized by the parameter 0em/0ex(•L Q)2. Later in the text we shall demonstrate that 0em and 0~x are of the same order in real ternary compounds like ErRh4B4, HoMo6X 8 (X = S, Se). Therefore, due to the large factor (2LQ) 2 ,~ 2~/a~o , the magnetic structure of the coexistence phase is basically determined by the EX mechanism and by the magnetic anisotropy. The role of the EM mechanism is to make the structure transverse, and to suppress magnetic fluctuations and their effect on superconductivity near the point TM. The spin-orbit scattering of superconducting electrons decreases the effect of the exchange field on superconductivity by increasing Xe(0) [27]. Theoretical estimates show that complete suppression of the EX mechanism by the spin-orbit scattering could be realized only in very dirty superconductors, when the electronic mean-free path is of the order of the interatomic distance. Thus, the theory proposed in [22-24]

Coexistence o f superconductivity and magnetism

183

predicts one-dimensional transverse domain-like magnetic structure with the wavevector Q ,~ (a~0)-~/2, in the case of not very dirty superconductors, and the gapless character of superconductivity in clean systems, far from TM (the strong exchange field regime). Less predictive conclusions regarding the properties of the coexistence phase, as well as conditions for its appearance, can be drawn from experimental results for magnetic superconductors. The initial experiments in this field were carried out by Matthias and co-workers in 1958 [6] on superconductors with rare earth (RE) magnetic ions as impurities. The aim of their study was to find such materials where the EX mechanism is particularly weak. They found several systems that appeared to satisfy this criterion, such as Cet xGdxRu2 and Y~ xGdxOS2. However, the magnetic order observed, in these relatively dilute alloy systems, was a short range or spin glass type, rather than true long-range magnetic order (see the reviews by Maple [28], and Ishikawa [19]). In addition, it often proved very difficult to avoid problems associated with chemical clustering. Thus, although considerable progress was made in understanding pairbreaking effects in twenty years from 1959, the questions relating directly to the mutual effect of long-range magnetic order and superconductivity were left more or less unresolved. This situation changed completely with the discovery in 1976 of the ternary rare earth compounds, RE RhaB 4 and RE Mo6X 8 (X = S, Se) and other similar materials. These compounds differ from the earlier pseudo-binary compounds in that they contain an ordered sublattice of magnetic RE ions, but in spite of that most of them are superconducting. Neutron scattering measurements have shown that ternary rare earth compounds are magnetically ordered at low temperatures and below TM(< Tc~) (tables 1, 2). It was also shown experimentally that many of the RERh4B4 and REMo6X8 compounds exhibit coexistence of superconductivity and long-range antiferromagnetic order [48-50]. The appearance of long-range ferromagnetic order in ErRh4B4 and HoMo6S 8 is accompanied by the destruction of superconductivity at the point To2(< TM) [33-35], which is re-entrant behaviour. The neutron scattering measurements, carried out by Lynn et al. [51-53] on polycrystalline samples of HoMo6Ss, by Moncton et al. [36] on polycrystalline ErRh4B4, and by Sinha et al. [37] on a monocrystalline sample of ErRhaB 4, has given evidence for the coexistence of a spatially inhomogeneous magnetic state and superconductivity in a narrow temperature interval (TM -- To2). Recently Lynn et al. [54] observed, in HoMo6Ses, the same coexistence phase at all temperatures below TM. So, HoMo6Se8 is the first ferromagnetic superconductor without re-entrant behaviour. This is in agreement with the Anderson and Suhl prediction. The magnitude of the wave-vector Q of the structure, as well as the data for T~, TM and T~2 are presented in table 3. Later on, it was shown that the coexistence phase is not always realized. So, in ErRh H Sn36 [59], Tm2Fe2Si5 [30, 60] and Ho0.6EroaRh4B 4 [61, 62], the coexistence phase is absent, i.e. there is a first-order phase transition from a non-magnetic superconducting (S) phase directly to the FN phase at some transition point T~2. The theory of the coexistence phase with the domain-like magnetic structure--the DS phase--is basically in agreement with experimental data for HoM06S 8 and HoMo6Ses. However, many of the principal conclusions of the theory cannot be confirmed, because only polycrystalline samples of H o M o 6S8 are presently available. This means that it is not possible to determine the magnetic structure of the coexistence phase precisely. They are all dirty systems, and the gapless behaviour, predicted

4.3 1.8

AF, complex AF F F F F F AF, complex

magnetic structure

[29, 30] [29, 30, 31, 32] [29, 31, 32] [29, 30] [29, 30] [29, 30] [29, 30, 32] [29, 30, 32~.3] [29-32, 44, 45] [29, 301

references

RE

Nd Gd Tb Dy Ho Er Tm Lu Gd Tb Er Lu Ho

X

S S S S S S S S Se Se Se Se Se

3.7 1.4 2.04 2.05 1.8 2.2 2.3 2.2 5-6 5.7 6.0 6-2 5-5

Tel [K]

0.65

T~2 [K]

F

AF AF AF F AF

magnetic structure

0-53

0.6 0.9 1.2 1.3 0-2

0em [K]

AF, complex

0.5 0.32 0.17 0-15 0.06

0ex [K]

0-75 0-85 1-1

0.8 0.84 1.05 0.4 0.70 0.2

TM [K]

[54]

[56] [32, [32, [32, [32, [32, [32] [321 [32] [32] [32]

56] 56] 56] 52-53, 55, 56] 56]

references

Table 2. Basic parameters and type of magnetic ordering in ternary chalcogenides RE MO6(X)8. For X = S, N(0) ~ 3 [states/eV spin RE], ~oD ~ 220K [56] and VF ~ 1"8 × 107 cms ! [ 5 7 ] . For X = Se, N(0) ~ 6.5 [states/eV spin RE], o90 ~ 200K [56], and v v ~ 1.8 x 107cms l [57].

0.7

!.5 2-4 4.4 2.5 1.6 1.1 0.5 0.3

0em [K]

6.02 5.13 5.5

1.31, 0.89 0.87 5.8 7.4 10.7 6.7 0.8 0.4

0~x [K]

8.7 9.8 11.5

1/2 7/2 1/2 1/2 1/2

TM [K]

5.5

J~ff

10.8 5.3 2.7

~,EJ

T~2 [K]

Y Nd Sm Gd Tb Dy Ho Er Tm Lu

eV ~

]

T~t [K]

RE

N(0) States

Basic parameters and type of magnetic ordering in ternary borides (RE) Rh4B4, where 0em = 2np2n, # is the moment of ion at T = 0, n is the concentration of RE, o90 ~ 200 K, and 0~ = - 0.2 dT¢(x)/dx for (RE)~Lul ~RH4B4.

Table 1.

4.

v,~v

e~

t~

ErRh4B4 HoMo6S8

40 ~ 24g

1-8 1-3

0~m [K]

8.7 1.8

1.0k 0.741

TM [K]

18' 4'

Ao [K]

~0 [A]

l [58]

S~ )2

0.56"

10~ 4.8f

H*(C) [KOe]

0.62 t

[K]

~)

1022 4 x 1021

n [cm -3]

m from Q and ~o n [53]

0.35"

S~2

15.5 3.2

A0 [K]

~-~ = 2~0ox

i [56], 0ex ,,~ 0e~

j

0.8 k 0-7l

k [37]

1m 2.5 m

g from 0ex h 0ex = hZoN(O)

45-50 / 100"

T~2 [K]

a

>>~o 60 ~

l [/~]

[/k]

210a(a), 160"(e) 1500

d = z~/Q [AI

900 ~ 1200a

2L(0) [A]

d from N(0) and vF e from H*(0) [35] and 40 f [35]

Y 0.9~

r~ ~ [K]

1.3 a 1.8 b

Tot [K]

Basic parameters or re-entrant superconductors ErRh4B4 and HoMo6S 8

Vl.-× 107 [cm s 1]

a [41] b [57] c Ao = 1"76 T~o

0.8 h 0.15 i

0ex [K]

1850 3600

ErRh4B4 HoMo6S ~

h0 [K]

N 1(0) [K" spin RE]

Compound

Table 3.

6.5 4.8

[KOe]

B(O)

5.6 9.1

# ( T = 0) [,uB]

~ 3"

¢2

e~

~"

e~

186

L. B. Bulaevskii, A. L Buzdin, M. L. KuliO and S. V. Panjukov

for clean systems, is absent. Therefore, special attention is paid to the re-entrant superconductor ErRh4B4, which was obtained very recently as a monocrystal [37], and as evaporated films [47]. The neutron scattering measurements by Sinha et al. [37, 67], carried out on a monocrystal, as well as earlier work by Moncton et al. [36] on polycrystals of ErRh4B4, show that a non-uniform magnetic structure develops only in a very small part of the sample. This fact cannot be described by the theories outlined above. The cause of this 'anomalous' behaviour in real monocrystalline samples of ErRh4B4 has not yet been explained. The difference between the results of neutron and Mrssbauer measurements of magnetic moment gives evidence for an irregular asperomagnetic ordering in real samples [65, 66]. However, all theoretical investigations of the coexistence phase [3-5, 14, 17, 18, 22-24, 63, 64] were based on the assumption that the magnetic subsystem is completely regular, which means that their predictions are inapplicable to system with sufficiently strong magnetic disorder, which suppresses the formation of a coherent non-uniform magnetic structure [66]. The experimental results of Crabtree et al. [41], and Behroozi et al. [68], for ErRh4Ba monocrystals, display very interesting behaviour in the temperature region near and above TM. Specific behaviour is exhibited in the presence of an external magnetic field, when, near TM, the polarization of LMs is increased. This leads to an increase in the effect of the exchange field, which together with the magnetic induction crucially determines the nucleation of superconductivity. As a result, the upper critical field Ho2 falls off faster than the thermodynamic critical field, H~, giving rise to the change in type of superconductivity. So, type II behaviour near Td is replaced by type I behaviour near T~. The alternative case of clean superconductors gives the non-uniform Larkin-Ovchinikov-Fulde-Ferrel (LOFF) state. It is very important that the monocrystalline ErRh, B4 is a unique system where the LOFF state can be realized and studied. The theory of critical magnetic fields, in the temperature region above TM, including the effects of the exchange field, is developed in [170], where it is shown that its results are in agreement with existing experimental data [41, 68]. Finally, there is still one very interesting aspect regarding the coexistence of magnetism and superconductivity. In 1960, it was proposed by Matthias and Suhl that superconductivity may exist in the region of a magnetic domain wall, where the destructive effect of the exchange field is weakened. This question has been studied theoretically in [69], and re-examined in subsequent papers [70, 71] in connection with anomalous resistivity behaviour of the FN phase below T~zin ErRh 4B4 and H o M o 6 S 8 [171]. However, the analysis given very recently in [72] shows that this possibility is probably not realized in systems like ErRh4B4 and HoMo6Ss, while it might be realized in pseudo-ternary systems like Ho~ xYxRh4B4, and Ho~ xYxMo6S8 in some region of x. In §2 we analyze basic structural properties, as well as magnetic interactions of ternary compounds and §3 deals with the behaviour of ferromagnetic superconductors above the point TM. The estimates of characteristic parameters of the EX and EM interactions in real compounds are given also. Methods for studying the coexistence phase are outlined in ~ , where the role of the EX and EM mechanisms in the formation of the coexistence phase is explained and the main characteristics of the coexistence phase are determined. The very important effect of defects on the proposed domain-like magnetic structure is discussed in §4.7. The effect of an applied magnetic field on the coexistence phase is studied in §5, and a comparison of theoretical predictions and experimental data for ferromagnetic superconductors is given in §6. §7 deals with the question regarding the existence of

Coex&tence of superconductivity and magnet&m

187

superconducting domain walls, discussed in the light of experimental realization. Particular properties of antiferromagnetic superconductors are considered briefly in §8. The most interesting properties of magnetic superconductors and further intriguing problems are summarized in the conclusion. 2. Interaction of conducting electrons and localized magnetic moments in magnetic superconductors 2.1. Structure of magnetic superconductors and hamiltonian of magnetic and superconducting subsystems The synthesis of ternary compounds has been very successful; 15 classes of ternary compounds have been discovered to date (see [29, 73-75]). In five of them both superconductivity and magnetism have been found with certainty. These five are either borides of the type RETaB 4 (T = Rh, Ir), chalcogenides REMo6X 8 (X = S, Se), silicides of transition metals (RE)2T3Si 5 (T = Fe, Co), or stannides RETxSny (T = Rh, Os). In ternary borides both ferromagnetic and antiferromagnetic orderings have been observed (see table 1), as has the famous re-entrant superconductivity in ErRh4B4 (see the review by Maple et al. [30]). A very interesting antiferromagnetic superconductor with the same composition as ErRh4B4, but with another crystallographic structure [46, 47] has also been synthesized. The second re-entrant ferromagnetic superconductor with a regular sublattice of magnetic ions, HoMo6Ss, belongs to the class of chalcogenides (see the review by Ichikawa et al. [56]). The third is a silicide, Tm2Fe2Sis, with Tel ~ 1.3K and Tcz = TM ~ 1.1 K [30, 60]. In the stannides ErRhv~Sn36, ErOsxSny and TmOsxSny, the re-entrant superconductivity is also observed with Tc~= 1.2K, TM = T~2 0.6K [30, 59, 76], but without long-range magnetic order down to the very low temperatures T ,~ To2. This is shown by neutron and specific heat measurements [30, 77]. Perhaps this is due to the chaotic and incomplete filling of equivalent lattice sites by RE and Sn atoms, so that there is irregularity in the positions of the magnetic atoms [30, 78]. A large number of pseudoternaries are also known; among them there are re-entrant ferromagnetic superconductors Er~ xHoxRh4B4 [61,62], Ho(IrxRh~ x)4B4 with x < 0.25 [79, 80], GdxYJ_xRh4B4 with x > 0"3 [79], antiferromagnetic superconductors Ho(IrxRhl x)4B4 with x > 0.4, and Tb(IrxRhj_x)4B4 with x > 0.2 [80, 81]. As in stannides, the distribution of LMs in pseudoternary compounds is irregular. In this review we shall study only those ternary superconductors with regular sublattices of magnetic ions as these are the only systems presently described by adequate theory. For further understanding of the physics of magnetic superconductors we need to describe the crystallographic structures or RE Rh4B4 and RE Mo6X8 systems, as shown in figure 6. The Rh and B atoms (or Mo and X, respectively) form clusters; which are well connected, but separated from the RE ions by a rather large distance. Magnetism in these systems is due to the 4f RE electrons, while the d electrons of transition elements are responsible for superconductivity. The distance between the nearest RE atoms is large, 6-5/k in chalcogenides and 5.3 A in borides, and so the direct exchange interaction of LMs is negligible. For better understanding of ternaries, the band structure calculations, which have been performed by Freeman and Jarlborg [57], are of great importance, as they show that there is a significant transfer of electrons from the RE atoms to the clusters. This produces a small density of electronic states on the RE ions; in chalcogenides it is 2 3 % of the

188

L. B. Bulaevskii, A. L Buzdin, M. L. Kuli( and S. V. Panjukov

0 Er • Rh

o B

(b)

(c)

(~

RE

•

Mo

•

S

Figure 6. Crystal structures of (a) primitive tetragonal ferromagnetic ErRh4B4 phase, (b) tetragonal body centred antiferromagnetic ErRh4 B4phase and (c) RE MO6S8,where RE = Ho.


189

corresponding density on the Mo atoms, while in borides it is about 15% of the density on the Rh atoms, and this is the main reason why the EX interaction between 4f RE electrons and conduction electrons is small. It also causes a weak indirect R K K Y interaction and low magnetic transition temperature, ~ 1 K. This makes possible the coexistence of magnetism and superconductivity in these systems. In what follows we shall be interested only in such systems where f electronic levels are far below the Fermi level, so that the magnetism of 4f electrons can be described by the localized moments model. Thus we shall not study systems with intermediate valence, nor compounds of the type Y4Co3 [21, 82], RE Rh2Si 2 [74], nor quasi-onedimensional compounds (TMTSF)2PF 6 [83] with itinerant magnetism. Theoretical descriptions of these systems are given in [84-86]. The systems we study here can be divided into magnetic subsystems with regularly distributed localized moments J in a crystal field, and electronic subsystems with usual Cooper pairing in the presence of nonmagnetic impurities. The interaction between these two subsystems is realized through the EX and EM interactions between LMs and conduction electrons. In the framework of the BCS model, the total hamiltonian of the system has the form I-I

=

f

d3r

(l/2m)W+(r)

P - ce A

W(r)

+ A(r)q~+(r)ia,,W+(r) - A*(r)W(r)ia, qU(r) + W+(r)[~J(r

-

r/)(g-1)3i+

geltsB]aq/

(r) + lqsc + IA(r)12 +

+ ~, [B(r;)g/tB3i + I2Ic~(3,)],

(2.1)

i

B =

rot A.

Here, A(r) is the superconducting order parameter for singlet pairing, W(r) is the spinor operator, a = {ax, o-~., az} are Pauli matrices, A is the vector potential, 2 is the electron-phonon coupling constant, J(r) is the exchange integral, g is the Lande factor, and 3~ is the operator of the moment at ith site. The term IZlcv(]~) describes the crystal-field effect on LMs, and the term I?tsc describes the electronic scattering on nonmagnetic impurities. The electronic part of (2.1) describes the Cooper pairing in the presence of the EX field (of LMs), and of magnetic induction B. Both these fields act on electronic spins - - t h e so called paramagnetic effect--but B also affects the orbital motion of the electrons. If B is fixed the Gibbs potential can be found, in principle. Minimizing it with respect to A gives Maxwell's equations for B, where the sources of B are g/~B(J~> and the superconducting screening currents. 2.2. Crystal-field effects The crystal field partially lifts the (2J + l)-fold degeneracy of the Hund's rule state of the RE ions, which leads to a distribution of energy levels responsible for magnetic behaviour [87, 88].

190

L. B. Bulaevskii, A. I. Buzdin, M. L. Kulik and S. V. Panjukov

From the measurements of specific heat and from M6ssbauer studies [30, 43, 56, 89] it is known that the crystal-field effects are essential, and that usually energy splitting of ionic levels in these systems, is much higher than the magnetic ordering energy. Therefore, the magnetic properties of most ternary compounds are determined by the lowest ionic level in the crystal field. If this level is doubly degenerate, then the effect of HCFis to replace, in (2.1), 3 by ,leerwith Jeer = ½, and to introduce an effective g factor and energy of anisotropy. Although the complete arrangement of Ho 3+ ionic levels in HoMo6Ss is not established yet, it is widely believed that the Jeer = } here, with the easymagnetization axis along the [111] direction [53]. The situation in ErRh4 B4 is more complicated. The distribution of energy levels is known from specific heat measurements carried out by Dunlap et al. [65, 90], and the lowest levels are two doublets with a small energy separation of 1.2 K, consistent with M6ssbauer studies [91], which is of the same order of magnitude as the magnetic transition temperature. All other excited energy levels lie much higher--at more than 12 K, and play an unimportant role in the temperature region T < Tc~ (see also [38, 73, 89]). As is known, the system with two close doublets cannot be described by an effective moment. Nevertheless, according to [66], the magnetic behaviour of the ErRh4B4 ideal crystal is similar to usual ferromagnetic behaviour with the secondorder phase transition at the Curie point 0 and the easy-magnetization plane is the basal plane of the ErRh4B 4 crystal. Neutron measurements on the monocrystalline ErRh4B4 sample, carried out by Sinha et al. [37, 41, 42], show that in the FN phase magnetic moments are oriented along the preferred direction in the basal plane. Probably this is due to the presence of internal stresses. 2.3. Indirect exchange interaction in normal s t a t e - - R K K Y interaction Magnetic moments polarize conduction electrons because of the exchange interaction, and this effect leads to the indirect R K K Y interaction of LMs. In the second order perturbation produces the effective R K K Y hamiltonian [29, 92], I2iex _

N2

1)2 [ j2(q + G)Ze(q + G ) -

n2(g_

~/1~ j2 (k) Z~(k)] ~]q,~_q

q,G

(2.2) 3.

1

= ~y

J, exp iqr,,

J(q)

=

I d 3 r f ( r ) exp iqr,

"7 where N is the total number of LMs, ri are the coordinates of LMs, ze(q) is the electronic paramagnetic susceptibility in units (g2 #2) and per LM, G are the reciprocal vectors of the magnetic lattice, and n is the concentration of LMs. The sum over G takes care of the discreteness of the magnetic lattice. The last term in the squarebracket excludes the self-interaction of LMs. In the sum over q in the first Brillouin zone, terms with G = 0 and Iql "~ IGI represent the long-wavelength part of the R K K Y interaction. The sum gives the contribution ( - 0 , x ) (per LM) to the ferromagnetic ground state energy, where O~x = xdO)h~, ho = J(O)(g - 1)n(Jz>, and (Jz> is the average moment in the FN phase. In the absence of crystal-field effects, (Jz> = J. The z axis is an easymagnetization axis. In the framework of the hamiltonian (2.1) one obtains the well-known result Zs = N(0), where N(0) is electronic density of states at the Fermi level (per LM).


191

The parameter h0 represents the electronic eneregy in the EX field at T = 0, and it is responsible for the energy splitting of electrons in a Cooper pair--the so-called paramagnetic effect [7]. The quantity 0ex = N(O)h2 > 0 favours ferromagnetic ordering and it is only part of the Curie temperature 0c (see below). Superconductivity decreases the electronic paramagnetic susceptibility Zs(q) at q < ~o ~. This depression of the long-wavelength part of the R K K Y interaction is shown in figure 2 (see also [93, 94]). The decrease of zs(q) in ternary borides, at temperatures below Tot, was observed by Kumagai and Fradin [95]. These authors measured the nuclear magnetic relaxation time of the B H nuclear spin in Y~_xErxRh4B4 with x = 0.0002. They observed, on cooling below Tc, a reduction in the electronic spin relaxation time, which is mainly determined by z~(q) (see [175]). Since superconductivity suppressed the long-wavelength part of the R K K Y interaction, the parameter 0ex is an energy measure of the EX interaction between superconducting electrons and LMs. In fact 0~x is the increase of energy (per LM) of the ferromagnetic state due to superconducting screening of the R K K Y interaction at q -- 0 and T = 0. The terms with G ~- 0, in (2.2), describe the short-wavelength part of the R K K Y interaction, which is practically unaltered by superconductivity. The contribution of the short-wavelength part of the FN phase energy at T = 0, is denoted by ( - 0"x). Its magnitude and sign (10~xl ~ 0ox) depend essentially on the electronic band structure given by ze(q + G), G ¢: 0. We stress here that in antiferromagnetic systems this procedure of separation into short and long wavelength parts has no meaning, because the wavelength of magnetic structure is G/2. In that case h0 characterizes the magnitude of the EX field, and 0~x the contribution of the R K K Y interaction to the Nrel temperature TN. Later, we shall estimate h0 and 0~x using experimental data for superconducting compounds. 2.4. Electromagnetic (dipole-dipole) interaction in normal state The localized moments generate a magnetic field, which leads to the effective dipole~lipole interaction of LMs, which has the form HE~

=

1 ~

I'

3

.~ ~ j,j,_ r~(roj,)(rujj)

],

(2.3)

where r~ = ri - rj. The contribution of (2.3) to the ferromagnetic ground state may be characterized by the parameter 0era = 2nnlt 2, where/~ = g/~B(J~)T=0. The contribution of HEM to the ground state energy essentially depends on the type of magnetic lattice. According to the calculation by Redi and Anderson [96] the magnetic~lipole interaction favours antiferromagnetic ordering in ternary chalcogenides, contrary to the ferromagnetic ground state realized in HoMo6S 8. The dipole~tipole interaction in ferromagnetic superconductors can be divided into long and short wavelength parts. Here it is better to do this in real space, rather than in momentum space [97]. Let us place the ith dipole inside the sphere of radius R so that G- ~ ,~ R ,~ d ,~ J-L- Here d is the characteristic length of magnetic nonuniformity and in the coexistence phase 2dis the period of the magnetic structure. The interaction of the ith dipole with all the dipoles inside this sphere represents the short wavelength part of the EM interaction. This part is not affected by superconductivity because R ~ 2nQ-~ ~ 2L. The contribution of this part to the ground state energy is denoted by ( - 0~m), where the sign of 0~m depends on the type of magnetic lattice

192

L. B. Bulaevskii, A. L Buzdin, M. L. Kufi? and S. V. Panjukov

(for a cubic lattice, 0~m = 0). The interaction of the ith dipole with dipoles outside the sphere gives the long wavelength part of the EM interaction. As a result, the contribution of the long wavelength part of the ground state energy is ( - 0,m + 20em/3) = - 0era/3, where the term 20em/3 originates from the Lorentz field. The long wavelength part of the EM and EX interactions always favour ferromagnetic ordering. On the other hand, in a superconductor there is screening of the field generated by LMs, because of the Meissner effect, which leads to an increase in the energy of the ferromagnetic state by an amount of 0era. However, the magnetic field B(O) = 4nn#, generated by LMs, suppresses superconductivity due to the orbital effect [7]. In that way the mutual influence of superconductivity and magnetic order (in the framework of the EM mechanism) is characterized by the parameters 0era and B(0), which play the same role as 0ex and h0 in the EX case. Values for 0eraand B(0) for some compounds are given in tables 1-3. So the total contribution of all mechanisms to the energy of the FN state at T = 0 is 00 (per LM), where 00 = 0ex + (0,m/3) + 0~x + 0~m. AS we explained earlier, (0ex -~ 0em/3 ) > 0, while the sign of (0~x + 0~m) depends on the type of magnetic lattice and band structure of the compounds under consideration.

2.5. Magnetic functional of the normal ferromagnetic state For further consideration we need the form of the magnetic free-energy functional FM, which describes non-uniform structure close to the ferromagnetic one. This closeness means that wave-vectors characterizing a non-uniform state are much smaller than the reciprocal wave-vector IGI. FM is easily obtained using the mean-field approximation (MFA), the applicability of which in magnetic superconductors will be discussed in subsequent sections. The magnetic subsystem, in the MFA, is described by quantities Si = (3i)/(J~)r=o replacing 3~ by (3~) in (2.2). To get the complete magnetic functional we add the term which describes the crystal-field effect on non-interacting LMs. We separate the interaction energy between LMs into short and long wavelength parts, then the EX interaction of LMs as well as the short wavelength part of the EM interaction can be written in a form which is quadratic in Sq (the Fourier transform of S3 with coefficients analytic in q when q ~ 0. The long wavelength part of the EM interaction can be obtained, from (2.1), immediately introducing an average magnetization S(r), over the sphere with the radius R. Then, the total magnetic function can be written in the following form: FM(S,, T) = ~ Fo(S,, T) + ~ [(-00 + O,~aZq2)SqS_q i

q

+ ½0:mq-2(qSq) (qS_q)]

+

[-#HS,

+

(2.4)

i

where F0 is the isotropic part of functional for an isolated ion, H is the internal magnetic field in the sample, D is the anisotropy parameter. In the case of an ion characterized by an effective moment J~e = ½, one has Fo(S, T) = T~Sbl(x)dx, where b~(x) is the inverse Brillouin function of the moment ½. In the case of the two lowest doublets the functional for an isolated ion could be determined numerically if /tcv(-]) is known (see [66]). Let us now discuss the question of the applicability of the M F A to the study of ternary ferromagnetic superconductors. In the usual ferromagnetic systems with large


193

exchange interaction we have 0 >> 0em , and the fluctuation region is broad, due to the short-range character of the direct exchange interaction. However, in those systems where 0 ~ 0em, and which have an easy-magnetization axis, fluctuations are strongly suppressed due to the long-range part of the EM interaction [22, 98]. As a result, fluctuations have four-dimensional character [99], because of the easy-axis anisotropy and the long-range part of the EM interaction. Since the fluctuations have logarithmic character near 0 ( ~ 0em), the M F A can be applied in normal ferromagnetic systems. This M F A type behaviour has been observed in HoMo6S 8 [100, 101]. Therefore, we use the MFA for the study of ferromagnetic superconductors, bearing in mind that the easy-magnetization axis is realized in HoMo6S8, and in real samples of ErRh4B4. However, M F A neglects the effects of spin waves on superconductivity, but the approximation is probably justified in systems with strong magnetic anisotropy when a gap in the spin wave spectrum exists.

3. Properties of ferromagnetic superconductors in non-magnetic superconducting phase, above TM 3.1. Role of magnetic scattering Before studying the coexistence phase, we must analyze the effect of magnetic (exchange) scattering of electrons in the non-magnetic superconducting phase. The estimate, given below, shows that in the temperature region 0 < T < T~ the effect of magnetic scattering on superconductivity is small and practically constant for all systems with 0ex '~ Tc~. If an external field is applied, or if the sample is in a magnetically ordered state, this type of scattering is reduced. So, in real systems like ErRh4B 4 and HoM06S 8 with 0ex < TM "~ Td, the effect of magnetic scattering can be incorporated into the renormalization of the transition temperature Tc~, and of the gap parameter A0 = A(T = 0) in the absence of LMs. The effect of the exchange scattering on the depression of Tc can be guessed from the dependence of Tc(x) in the series of compounds RExY~ xRh4B4 and RExY~ .~M06S8. Since y3+ is a non-magnetic ion, the addition of RE magnetic ions leads to a decrease in T~. From the Abrikosov and Gor'kov (AG) theory [8] it follows that: dT~(x) -dx

rd -N ( O ) j 2 ( q ) ( g - 1)2nj(J + 1) = 2

---

g2 0cx, 2

(3.1)

where j2(q) is an average value of j2(q) over the Fermi surface. It is understood, in (3.1), that the system is far from TM and is not affected by crystal-field effects. The corresponding data for To(x) in pseudoternary borides are published in [30, 40, 43], and for pseudoternary chalcogenides in [55, 56]. Using them and equation (3.1) we calculated 0cx, and our results are given in tables 1 and 2. Below, we shall discuss experimental data concerning the upper critical field (Hc2), from which we estimated h0 = 40 K, and 0ex = 0.9 K for ErRh4B4. Comparing 0ex and 0ex for this compound, we conclude that there is no great difference between them, which means that (3.1) is adequate for an order-of-magnitude estimate of 0ex- Using the AG theory, we obtained the superconducting transition temperature in the absence of LMs--Tc0 = 11.5 K and 2.5 K for ErRh4B4 and HoMo6S8 respectively. Similarly, we get small values for the characteristic parameter of the exchange scattering x~ = [r~(Tcl)A0] ~, 0.15 and 0.25 respectively. Here, the exchange scattering time is given by r~-I(T~) = 2g0ex.

194

L. B. Bulaevskii, A. L Buzdin, M. L. Kuli? and S. V. Panjukov

On approaching the magnetic transition (of second order), critical fluctuations grow, and in principle they might lead to an increase of T~-'. An expression for ~ ~, above and near the temperature 0, has been obtained by Rainer [102] by neglecting dynamical effects in the interaction of fluctuating spins. In the absence o f crystal field, it has the form z ; ~ = N(O)(g - 1)2 ~ g ( q ) j 2 ( q ) ( j q ~ _ q ) (3.2) q

g(q)

=

(47rk2q) ',

q >> ¢o l,

where the weighting function g(q) has a similar meaning (and q dependence) as the function (z~(q) - Xs(q))/z,(O) = n/(2q~0) for q ~ ~o I (see (1.1)). Z~ and Z, are the electronic paramagnetic susceptibilities in the superconducting and normal phases respectively. An important fact is that the static approximation is justified for the system under consideration where 0 ~ TM ,~ Tc~, A0. This is because the characteristic frequencies of fluctuating spins are small compared with the characteristic frequencies of the electronic subsystem. The correlator (Jq3_q), given in (3.2), satisfies the sum rule (,]qJ q) q

=

~ J ( J + 1),

(3.3)

q

where summation over q goes over the first Brillouin zone of the magnetic sublattice. As a result, z s~ increases as T decreases to 0, because of the increase of the correlator at small q. The small q gives the largest contribution to z~-~ because of the behaviour of g(q). In the Ornstein-Zernike approximation, but neglecting the longrange part of the dipolar interaction, one obtains (J~qJ~ q ) ~ ( T - 0 + 20exa2q2) -~. This leads to a logarithmic divergence of Ts I when T ~ 0 [85]. These arguments have at some time represented a crucial objection against the coexistence of superconductivity and magnetic ordering in re-entrant magnetic superconductors. On the basis of the above analysis superconductivity would be destroyed before magnetic ordering is established. However, a very important effect, the long-range part of the EM interaction, has been omitted in the above analysis. Its presence removes the logarithmic divergence of Zs ~in systems with an easy-magnetization axis. In that case, (JzqJz q )

~

( T - 0 + 20exa2q2 + 0em COS2tp) I,

(3.4)

where q~ is the angle between q and the easy-magnetization axis. The correlators (JiqJi-q), with i = x, y, do not diverge when T ~ 0, due to the magnetic anisotropy. Finally, we are in a position to conclude that the temperature dependence of Zs ~ is weak in the interval 0 < T < T~, and that zs(T) ,~ zs(Tc,) in this interval. The main conclusion of this section is that for compounds with small parameter x~, magnetic scattering can be taken into account with sufficient accuracy through the renormalization of A0. We shall accept that A0 = A0(1 - nx~/4) = 1.76 Td and ~0 = 1.76 To0. 3.2. Upper critical field He2 To calculate He2 for magnetic superconductors we must include both the orbital effects, of the induction la = H + B(0)S, and the paramagnetic effect of the exchange field h = h0S. Here, we are interested in those systems with TM '~ Tc~, where the magnetic scattering can be incorporated into the renormalization ofA 0. The


195

systems with close magnetic and superconducting critical temperatures TM ~ Tc~, have been studied in [103]. There are factors which make an essential difference between ferromagnetic superconductors and usual superconductors. One is the increase of magnetization (of LMs) in an applied magnetic field, which leads to a decrease of the thermodynamic critical field H c in ferromagnetic superconductors at temperatures near 0. Indeed, in the presence of an applied field Ha the normal magnetic energy of LMs is drastically decreased in comparison with the corresponding one in the superconducting state. As a result, He tends to zero when T ~ 0, because of the growth of paramagnetic susceptibility (of the LMs) )~m(T), as described by the Curie-Weiss law. In the compounds under consideration, far away from 0, ,~m is small, and the orbital effect of the magnetic field predominates. However, when T ~ 0 the exchange field h = hoS grows, since h ~ Zm H ,'~ H ( T - O)-j. Therefore, near 0, when h is of the order of A0, the exchange field suppresses the formation of a superconducting nucleus. As a result of this paramagnetic effect, He2 drops also, when T ~ 0, in compounds with h0 >> A0. It may happen that the decrease of Hc2 near 0 is faster than that of He, which means a change of superconducting behaviour from type II (He2 > He) near Tej to type I ( H c 2 < He) near 0. Here we shall confine discussion to the case when the external field is applied along the easy-magnetization axis, since only then does Zm diverge as T ~ 0. Moreover, in this case the peculiar behaviour of ferromagnetic superconductors, in magnetic field, is extremely pronounced. In the presence of a magnetic field four superconducting states are possible: the Meissner state (MS), vortex state (VS), non-uniform L O F F state, and intermediate state (IS). The Meissner state is characterized by the uniform superconducting order parameter, and the magnetic field does not penetrate the bulk sample. In the VS state the superconducting order parameter depends on (x, y) coordinates and the field is along the z axis. The magnetic field penetrates the bulk sample in the form of a vortex lattice. In the case of the L O F F state the superconducting order parameter depends on all coordinates, and it varies on the length scale of the order of G0for temperatures T ,~ Tc~. Such a state is energetically more favourable than the uniform one in the presence of a strong magnetizing field when h ~ A0. Actually, in that case there is 2h splitting of the energy o f the electrons in the Cooper pair. As a result, the superconducting condensation energy is decreased. At the same time, the energy splitting between electrons with opposite spins is small if they are paired with total momentum Iql ~ (2h/vv). In the case of a one-dimensional system, electronic energies of states PT and ( - p + q)~ are equal ifq = (2h/VF), while in the three dimensional case the splitting effect can be minimized by choosing an appropriate value of q. The pairing with total pair momentum q 4 : 0 causes the appearance of the non-uniform superconducting L O F F state. The magnitude of q can only be determined relatively easily near the transition pbint N - L O F F , where A(r) ~ 0 [15, 104]. However, the detailed structure of the L O F F state is not well known far below this transition point. Although the response of the L O F F state to the magnetic field has not been sufficiently studied, some results of [15, 104] indicate that the Meissner effect is probably absent in the L O F F state. The IS phase is realized in samples with the demagnetization factor Nz 4 : 0 in an applied magnetic field. It is characterized by laminar, or fibrillar, structure of alternately arranged superconducting and normal regions.

196

L. B. Bulaevskii, A. I. Buzdin, M. L. Kulid and S. V. Panjukov

Downloaded by [University North Carolina - Chapel Hill] at 15:02 05 December 2012

The possibility that the L O F F state could exist in clean magnetic superconductors was noted in [105]. Near the transition point T¢~ the magnetic superconductors studied so far are type II, so now we examine the conditions under which they become type I, near the point 0. To do this, we calculate the fields H~ and H~2 in this temperature region, and find the conditions for H~ > H~2. Let us find the thermodynamic critical field H~ for the N - M S transition, using the conditions Fs(H) = FM(H), where t42 + -a 87r 87r'

n~0

F~ -

_

8zc

1N(O)A~"

(3.5)

Here, H~0 is the thermodynamic critical field at T = 0 and h 0 = 0, and H a is the applied field. The magnetic energy FM of the sample in the normal state is given by

1MHa,

Zm

=

~m =

ngm,

FM -M

=

;~mH,

Ha =

n

T--O'

(3.6)

n ( 1 + 4=N:;c.),

where H is the internal field in the sample. Here it is understood that the applied field is along the z axis, and that the sample is of the ellipsoidal form with one of the principal axes directed along the z axis. Hence, the demagnetization factor N: appears. We consider also the case T ~ Tc~. Combining (3.5) and (3.6) one gets 4XMHa2 + Ha22 =

H~0,

(3.7)

where Ha2 is the external field. Using (3.6) and (3.7) one easily gets the internal critical field He, Hc =

H~o{[1 + 4X~m(mz + 1)][1 + 4xNz~m]} -½, (3.7 a) ~m

=

nfl2/( r --0),

Zm

=

nZm"

For temperatures close to 0 one has Hc ~ ( T - 0) when N: ¢-0, and Hc ~ (T - 0) ~when N: = 0. When Tis very close to 0 the magnetization is nonlinear function of H for H < H o and in that case the real value of the critical field H~ exceeds the value given by (3.7 a). The curve H~(T) is shown in figure 7. The line Hc(T), of the N - M S transition, continuously reaches the first order transition line N-VS. Details of the latter line are not known, although we know that it terminates at the point To (the tricritical point). For T > TOthe N-VS line describes the second order transition. In figures 7(a) and (b) the first order transition lines N - M S and N-VS are given by the/arc line, in the temperature region where H~ is higher than H~2 (for the N - L O F F transition). The continuous line from T = 0 up to T2 exhibits Hc(T), which characterizes the N - M S transition described by (3.7). The dotted line shows the boundary between N and VS phases. Now, let us find the second order transition line between the normal and superconducting phases. This line is the boundary between phases N and L O F F (or VS) if Hc2 > He, or is the 'super-cooling' line (with respect to the field) of the N phase when H~ > H~2. Here we discuss only clean superconducting systems. Any method


! 97

H (kOe

.f.,.

,0 .

J-' '\ H¢2

/

\

"°.// \

". H¢2

/~f

"\.

•

.1"i-Ii \\"°@i

.4'

\ 1

3

5

7

,S 9

3

(a)

5

7

3

(b)

5

(c)

7

r (t 1.8 the L O F F phase occurs. In real compounds like HoMo6Ss and ErRh4B4, where Q = [AoB(0)/

198


h0H*(0)] 2 >> 1, then ~ >> 1 a n d f ( ~ ) ,~ 1.07. Therefore,

H~z(T) = 0,

0.75 A°---~( T h0/~2

0),

T-

0 ,~

(3.10)

so that in the temperature region close to 0 the orbital effect can be neglected. Next, let us consider compounds with ~t ~> 1. In this case, He2, as well as He, drops linearly when T ---, 0, and He2 represents the N - L O F F transition line up to the tricritical point To = 0.55 Td, where it transforms into the transition line N-VS [106]. Close to 0, and if the condition 0ex > l'120cmNz(N: + 1) is fulfilled, then Hc(T) > Hc2(T). This means that the first order transition into the MS or VS state occurs, depending on the relation between Hc and lower critical field H¢~. Below, we show that, near 0, H~ is practically the same as H* (i.e. the case when LMs are absent). In this way, we can produce the phase diagram near 0, as shown in figure 7(a) and (b). Figure 7(a) shows that the curves H¢(T) and H~2(T) cross at the point Tl < To, while in figure 7(b) H¢(T) is higher than H¢2 up to To. The L O F F state occurs in the temperature interval T~ < T < To in (a), but is absent in (b). If the condition 0cx < 1.12 OomNz(Nz + 1) is fulfilled then the N - L O F F transition takes place (see figure 7(c)). The boundaries between L O F F and MS and L O F F and VS phases are not known. It is interesting that in this case the critical field H~ (the upper limit of the MS phase) near T¢~ is almost identical to ~*. At temperatures near 0 the Hcl (T) line separates the MS and LOFF phases. Here, Hcl is different from H¢*, and it seems to be the first order transition line. Regarding the line which separates the N and L O F F phases, the orbital effect is negligible (near 0) and the N - L O F F transition is of first order, according to Nakanishi and Maki [108]. If the orbital effect is included then the first order line N L O F F moves closer to the second order transition line. A very important fact emerges from the above analysis; that all three types of phase diagram (7 (a), (b), (c)) can be drawn for the same compound, by changing only the shapes of samples and the corresponding direction of the applied magnetic field. This is due to the fact that the ratio H¢/H~2 depends on the demagnetization factor. Below, we shall show that this possibility is realized on monocrystalline samples of ErRh4 B4, where spherical samples correspond to case (a), plates or cylinders with the field parallel to the axis correspond to case (b), and plates in perpendicular field correspond to case (c). Machida [109] has also recently calculated the critical field He, but neglected the demagnetization factor Nz. He obtained the first order transition line N - M S (near 0), and his expression coincides with (3.7 a) when N~ = 0. It is very difficult to realize the L O F F state in usual superconductors, since in real clean systems the orbital effect dominates (~ < 1.8). If non-magnetic impurities are introduced into the sample the mean free path decreases and the region where the LOFF-phase exists narrows [110]. In dirty systems, where 1 ,~ ~0 the L O F F phase is absent. For the calculation of H¢2 for dirty superconductors, see Sakai et al. [111]. When the system is prepared in the forfn of the polycrystalline sample, it is necessary to take into account the real anisotropy of the magnetic susceptibility. Then the transition has a percolation character, due to the random orientations of crystallites with respect to the applied magnetic field. The corresponding calculation of He2 for dirty polycrystalline superconductors, in the framework of the effective medium approximation for the percolation transition [112], has been performed in [113, 114], and is in good agreement with experimental results on polycrystals of ErRh 4B 4.

Coexistence of superconductivityand magnetism

199

Ha(kOe) 2

/// Ha2/

i/i

f ~

l///lll /

/

/

/

2

I

3

4

T (K)

[a)

Ha2

o z

o Z

/

Nc

raN

w IE

IE

\ (b)

(c)

Figure 8. (a) Phase diagram in the (H.,, T) plane for a spherical sample of ErRh4B4 above Tv. H, is the applied field, IS the intermediate state, MS the Meissner phase and the VS vortex state, (b) magnetization versus H. at T < 7"1, (c) magnetization versus H at

T< TI.

At temperatt,res T < ~ the condition (3.7) gives the thermodynamic field Hc of the transition N MS. In fact, the intermediate state (IS) is realized in samples with the demagnetization factor AT. # 0 in the external field H a < Ha2. By lowering Ha, from Ha= to Ha~ = Hc(1 - AT-)the relative volume occupied by normal regions decreases from unity to zero. The phase diagram (Ha, T), corresponding to the diagram in figure 7 (a), is shown in figure 8 (a). At T < T2 and by lowering Ha, the following sequence of transitions occurs: N --* IS - , MS. The dependence of the magnetization (M) on Ha is determined from the well known c o n d i t i o n / ~ = Hc [115], w h e r e / ~ is

200

L. B. Bulaevskii, A. L Buzdin, M. L. Kuli~ and S. V. Panjukov

an average field over the whole sample. Thus, at temperatures T > /'2 we have

4rtM -

Ha

1 - Nz'

4~zM -

Ha - Ha

4rrM -

1 + 4~znNzZm'

Nz

'

Ha < Hal' Hal < H < Ha2,

(3.11)

4/ZZmn a

Ha > Ha2'

assuming the linear dependence of M on H. The magnetization M is continuous function of Ha (see figure 8 (b)) and the function M(H) jumps (AM = (Xm + 1/4n)Ha) at h = Hc (see figure 8 (c)). At temperatures T > Tj the IS state does not exist, although in the region T1 > T > 7"2the line separating the IS and VS phases must exist. This line is shown dotted in figure 8 (a)). 3.3. Lower critical field Hal We now turn to the interesting question regarding the change in the screening of the magnetic field in the MS phase, which is due to the presence o f LMs, and so we study the MS-VS boundary line. Since we are interested in the change of superconducting and magnetic quantities on the scale of the order 2( ~>G0) in the field parallel to the z axis, it is then correct to write that part of the free-energy which depends on H in the following form:

)

#nB(r) S~(r) F = f d 3 r [ n ( ~ z m + 0era S2(r) + B2(r) 8-----~-

+

q~I 1 Qs(q)AqA_q -]- 0ex i(n(q)~(n'~) -- Xs(q) SzqSz' q 1 '

8u)"'1 4re ] (3.12)

where Qs(q) is the electromagnetic kernel of the superconductor. In the homogeneous superconducting state (A = const) with 2 >> 4, we can put zs(q) = 0 and take the local approximation for Q~(q) = (47r22L) l, 2L2(0) = 8rrnN(O)V2Fe2/3c2. In the case of a single vortex line at x, y = 0 and along the z axis, we obtain, by minimizing (3.12) with respect to S~, the following free energy functional (per unit length):

F = fd3r{pI~-~z + 8 ~ L ( A - - V t p ) 2 ]

- ~}

(3.13)

p = 1 -- Oem (Oem --J- Oex --]- ft2 ~ I

[12Xm// '

where ~o(r) is the phase of the superconducting order parameter. It changes its value by 27t in one turn of the vortex line.


201

Minimizing (3.13) gives the equation for B whose solution in the reciprocal space is given by

Bq =

tpo 1 +pq222'

tP° -

hc 2e"

(3.14)

Equation (3.14) shows that the magnetic flux is quantized in the usual way, i.e., the total flux of the vortex line is Bq=o = tP0. A very important fact emerges from (3.13), which is that the effective penetration depth 2 is decreased with respect to the London penetration depth 2L, i.e., 2 = 2Lp½,p < 1. AS a result of(3.13) and (3.14), one obtains the lower critical f i e l d / / ~ , ncl

=

~0° ~'Lp½ 4n2[ In ~ ,

(3.15)

which is similar to the usual expression for H d [116]. This means that the temperature dependence of Hd in the interval TM < T < Tel is practically the same as in superconductors without magnetic ions. We note that similar results for Hcl in ferromagnetic superconductors were obtained by Sakai et al. [111]. However in that paper p was calculated by neglecting the screening of the EX interaction by superconductivity, so that the last term in (3.12) was absent. Hence, the expression for p given in reference [111] does not include the term with 0~x. Without this term there is a possibility that 2 ~ 0 when T ~ 0, since then p ~ 0. However, the presence o f the 0ex term in the expression for p practically leads to the usual temperature behaviour of 2 in the systems with 0ex ~ 0m. If the case 2 ~ 0 ( p ~ 0) is realized, it would lead to a change in the character of the transition in the magnetic field near 0. This effect was predicted earlier by Tachiki et al. [18, 64], in the framework of the EM mechanism only (0¢x = 0). The main reason for the change is the change in the charater of the vortex interaction, caused by the sharp decrease of 2 when T ~ 0 (i.e., the transition type II-I)i As we established above, the inclusion of superconducting screening of the EX interaction leads to the usual behaviour of 2 in the system with 0~x ~ 0m. In fact, the strong decrease of H¢2 near 0 within the framework of the EM mechanism is only possible in such compounds where the condition B(0) > H*(0) is fulfilled. However, in ErRh4B4 the opposite case H*(0) > B(0) is realized, because there H*(0) ~ 10 KOe and B(0) ~ 6"5KOe, n ~ 1022cm -3, and # ~ 5"6#a). We emphasize here, as a result of the above analysis, that the change o f the type of transition near 0 in real compounds is not due to the change o f the effective G - L parameter K, as suggested in [18, 64], but is due to the sharp decrease o f H~2. This is caused by the strong paramagnetic effect of the EX field. 3.4. Experimental data for critical fields in ErRh4B 4 Behroozi et al. [68] have recently made magnetization measurements on an ErRh4B 4 monocrystal as a function of the internal field H, along the a axis and at different T. The corresponding results for spherical samples are shown in figure 9(a) and (b). In weak fields up to Hcl, the Meissner regime was observed, while in fields H > Hcl and for T > T~ ,~ 3.5 K, vortex lines penetrated the sample. As a consequence, the magnetization behaviour changes from diamagnetic to paramagnetic. At sufficiently high fields superconductivity is destroyed, and the susceptibility is well

202

L. B. Bulaevskii, A. I. Buzdin, M. L. Kuli? and S. V. Panjukov -I

L

-0.5

/

i

] - 0.5 I

'//~ /MEISSNER / /'

SLOPE

0

O

"2 t.o %

\\ \ '\\ \

~E

0.5

0 . 5 ~Z

z o (a)

o

\ 1-

\

\

/,4 i.l.l..l z < I

\

U W Z

\ \, 15

< 5"

\ \, \

5.51

\,

1.5

2.5 I

0.5

I

I \

I

I 1.5 2 INTERNAL FIELD (kOe)

x - l ] 0 2 ~J.~//K

15

10

(b) 5

0

0

1

23

4

5

6

78 T(K)

Figure 9. (a) Magnetization versus internal field along a axis in an ErRhnB4 single crystal at T > 3 K [68], and (b) inverse susceptibility along the a axis in the normal state according to (a).


203

-0.5

-0.5

0

0

O

O

"v

2 2

5-

Z C)

hi

3 3 zO n

I---

N t,

N

I---

b.I z t.3

Z

.~ 5

IE

5

0

I

2

3

IE

t.

APPLIED FiELD(kOe) Figure 10. Magnetization versus applied field for a single crystal of ErRh4Ba for T < 3 K. The applied field is along a axis.

described by the Curie-Weiss law (see figure 9 (b)). On cooling below 3.5 K the VS region narrows, while at the same time a jump in the magnetization appears when H = H c. Below 1.5 K the VS state does not exist and the magnetization j u m p reaches its full value, i.e., at H = Hc the transition N MS occurs. Experimental results for the dependence M(Ha), and for T < 7"1, are shown in figure 10. There it is seen that M(Ha) is continuous for all temperatures. Note that below 3 K the dependence M(H) is nonlinear already in fields smaller than Ha2. The temperature dependence of the internal fields directed along the a and e axes is shown in figure 11 [41]. One can see that the critical field along the e axis is determined by the orbital effect only, because gm is very small along the e axis (see figure 32, below). The experimental value (H~)(TM)/T~I)/(dH~)/dT) = 0-66 is in a good agreement with the G o r ' k o v theory. This fact confirms the theoretical prediction regarding the very weak temperature dependence of the" exchange scattering parameter Tsl in the temperature region 0 < T < T ¢ I . The behaviour of H ~ ) near T¢~, is also in agreement with the theoretical predictions for the orbital critical field H*(T). From these data we obtain the corresponding value for 40 and VF: ~(0a)= 210A, ~(0c ) = 160A, V(FS)= 1"4 × 108cms - I , a n d v ( v c ) = 1'1 × 107cms -l The data for H 1.8. This condition is enough for the existence of the L O F F state. Hc2(T), given by the expression (3.9), is represented by the dotted line in figure 7 a. Experimental results (squares) lie above it when T < 3 K, and coincide with it when T > 3 K. Now we consider the position of the line H¢(T) near the temperature 0. Above 1-5 K this line separates N and VS phases, and the expressions (3.7) and (3.7 a) give only a lower limit of H e. Below 1.5 K the line H¢ separates N and MS phases. However, the dependence M(Ha) is nonlinear here, as shown in figure I0. The expression (3.7 a) gives again only a lower limit of He, and the exact value of H¢ is not known. The magnetization measurements, reported in [68], show that below 3 K there is a magnetization jump. On that basis, we may assume that the situation with



205

Hc > He2 is realized here. In that way figure 7 (a) represents the phase diagram (in the ( H T ) plane) for the spherical sample of ErRhaB 4 investigated in [41, 68]. From the theoretical results given in §3.2 for the intermediate state it follows that below T2 = 1.5 K the transition from the N to the MS state, in a spherical ErRh4B4 sample, goes through the intermediate state in the region of fields Ha~ < Ha < Ha2. Experimental data shown in figure 10 are in good agreement with (3.11). When T < T2 = 1-5 K, the curve M(H,) is in three parts. The slopes for Ha < Ha~ and Ha~ < Ha < Ha2 are determined by the demagnetization factor only, i.e., the corresponding slopes are (1 - N.) ~and N~-I . In the temperature interval (1.5-3.5 K) and in the field Hc < Haz the curves M(H,) have parallel segments (figure 10) with the slope Nz-~ . At these temperatures the IS phase is realized. Above 1.5 K, the IS state does not survive down to the field Ha~, as it is transformed into the VS state, which is further transformed into the MS state. Experimental data for the spherical sample of ErRhaB 4 monocrystal are well described by the phase diagram shown in figure 8 (a). Magneto-optical or magnetic powder methods could be used to resolve definitely the question of the existence of the IS state in spherical samples of monocrystalline ErRh4B 4. In conclusion, we stress that the behaviour of the critical fields of ErRh4B4 uniquely

proves the dominant role played by the EX mechanism in the destruction of superconductivity near the point Tu. 4. Structure of coexistence phase in ferromagnetic superconductors 4.1. General methods for calculation of the free-energy functional In the coexistence phase both the superconducting and magnetic order parameters A(r) and Si are non-zero. Our task is to find their equilibrium values by minimizing the free-energy functional F{A(r), Si, A(r); T} with respect to A, Si and A. We shall divide the total functional into three parts: F{A(r), S,, A(r); T} = FM{S~, A(r); T} + Fs{A(r); T}

+ F~.~{S,, a(r),A(r); T},

(4.1)

where FM is the magnetic functional in the normal state and Fs is the functional of the superconducting subsystem in the absence of LMs. F~,~describes mutual influence of superconducting and magnetic ordering due to the EX and EM interactions. Moreover, we shall restrict our analysis to those systems with the property rsl ,~ A0, when the magnetic scattering is included through the renormalization of A0. The magnetic subsystem is described by the simplest approximation, i.e. by the MFA. The effect of the EX field h(r) and the magnetic induction B on superconductivity is given by F~.t. The influence of the rapidly varying components (of the order of G) of these fields can be neglected in comparison with those with q 0 the reversal of direction of Si is only possible because Si vanishes at the wall, because rotation of moments is energetically unfavourable. As a result, one forms a linear domain wall near TM [ 124],

S~(x) = Sth(xx/t/a),

Sy = Sy-~- O,

nOexS2a(T),

a(T) = 2,~/23 at'/2'

¢(S, T) =

(4.7a)

where a/x/t is the domain wall thickness. The generalization of the expression (4.7 a) for ((S, T) for arbitrary T is given in Appendix D (see [125]). In the case of the Ising ferromagnet the corresponding formula for h(T) has the form

?fiT)

=

h~_ ab ½I1 _ b d In S2(T)] -½ 3

"dln'T d

'

(4.7b)

where b is the fitting parameter. It is assumed to be small compared to unity, and the expression (4.7 b) is valid for t ,~ b. Using (4.6) and (4.7) for FM, together with Fintgiven by (4.5), we are in a position to find the equilibrium structure. For the striped structure, S~(r)

4_S ~ sin(2k + 1)Qr k=0 ~ 2k + 1

which finally gives Q(T) = [4.2/~(T)] ½,-, t -~. The necessary condition for the domain structure to be realized is Q-~ ~> a/x/t, i.e. the temperature interval of its formation is determined by the small quantity tc ,~ (a/~o)~, which also characterizes the region of strong fluctuations. In practice, this means that the existence region of the

210

L. B. Bulaevskii, A. L Buzdin, M. L. Kuli~ and S. V. Panjukov

sinusoidal structure is rather small. The direction of (1 in the (x,y) plane depends on the anisotropy of ( and vv about which there is currently insufficient information on. Let us find those values of the magnetic anisotropy parameter D for which the spiral structure becomes domain structure. In the case when D ,~ 0, S changes its direction by continuous rotation through the angle n. Such a rotating solution, with the wall thickness of the order of a(O/D)½and with surface energy (OD)½S2, is continuously transformed into the spiral structure when D ~ 0. As is well known, the domain structure corresponds to the case when its period Q- l is much larger than the wall thickness. This is fulfilled when (D/O) >> (a/Go) > 10 -2. So, a rather small magnetic anisotropy favours the domain structure. The results obtained for F, given by (4.5a), (4.5c), (4.6) and (4.7b), are an adequate basis for the study of ferromagnetic superconductors with weak coupling between superconductivity and magnetic ordering. This case is realized when F~,t N(0)A0:/2, i.e., when 0cx "~ 0~,~ = 0.135 TcIVFN(O)Q(O) It is easy to show that in this weak coupling limit the equilibrium value of Q(T) decreases with the decrease of temperature, which is mainly due to the growth of the wall energy. The value of Q is given by Q(T) = a(0) I1 - b d l n S 2 ( T ) ] -½ Ik_

[- 4"2 1½ Q(0) =

(4.8)

L¢0a(0)J

The corresponding results for the weak coupling limit (Fi,t '~ N(0)A2(0)/2) will be used in the interpretation of experimental data in HoMo6Ses. In the strong coupling case, i.e., 0,x >> 0~,¢),x if the temperature is lowered further, the exchange field increases, and the magnetic ordering strongly affects Cooper pairing in the temperature region t ~> (A0/h0)2. Now, the superconducting part of F must be calculated for the case of a strong exchange field. The important fact about the solution of (4.3) is that the exchange field is rapidly varying in space. For the characteristic wave-vector of the structure Q >> ~o 2, the superconducting order parameter is approximately constant in space. This allows analytic solution using perturbation theory with respect to the small parameter (Q~0) -j . 4.3. Coexistence phase with strong exchange field In the case of dirty superconductors, when Q¢0 > 1 a n d (hq'r)2 ,~ 1, one realizes the diffuse regime of Cooper pair motion, because the above conditions guarantee an effective averaging of the exchange field over the length ~0. This leads to the isotropisation of electronic motion in the presence of anisotropic non-uniform magnetic structure [22]. As a result, the coordinate dependence of g, f + , A is small. It can be accounted for by perturbation theory with respect to the small parameters (q~0)-t and (hz), with an arbitrary value of (h/A). The solution of (4.3) is achieved by expanding g, f + and h in Fourier series. The contribution of higher harmonies q = kO # 0 is small, because the parameters (q~0) -~ and hz are small. The detailed calculation is given in Appendix A. As a result we find the superconducting part of F, Fs(A) = Fint(m,

~m)

~__ -

cA02

- ½N(0)A2 In A--T N(0) (x2__~ m

1m) , 37r2

zmA /> 1

(4.9)


211

ZmI = ~q 7rhqhvvqq Ll(ql) L,(y)

=

2y(arctgy) 7r(y-arctgy) '

where A is the superconducting order parameter averaged over the sample. We have written the expression for F~,t for the case ZmA ~> 1 only, since just this case is realized in the coexistence phase. One can see from (4.9), that the effect of the EX field on superconductivity in sufficiently dirty systems is described by the parameter z~ ~. So the effect of the exchange field on superconductivity is equivalent to the effect of magnetic impurities with Zm as the exchange scattering time. Using this analogy, the condition zmA t> 1 corresponds to the gap regime of a superconductor with magnetic impurities. In §5, we shall see that this analogy is not complete. For instance, the behaviour of these two systems in a magnetic field is different. Finally, we know the complete free-energy functional with respect to A and S. The one-dimensional striped domain structure is favourable again, since (Azm)-~ is sufficiently small ( < 0.68) and the dependence of Fint on S(r) is only slightly different from the analogous expression (4.5 c). As a result we obtain F{A, S, Q}. Studying the clean systems [24] we accept that Az >> 1 and q~0 >> 1. We shall find F~nt for a one-dimensional striped structure in the case of the strong exchange field h >> A [24]. In calculating the properties of the DS phase it will be supposed that (h/v v Q) and ( A / v F Q ) a r e small. However, at the end of the calculation we shall show that these conditions are fulfilled in the entire region of the DS phase. In the case z = ~ , the functions A and g are coordinate independent in the first approximation. Then, the calculation o f f - (v, r) is straightforward (see Appendix B), and as a result one gets the self-consistency equation for the mean value of A, A In ~

=

-f-

=

ct + ifl =

;~ de) f

T- -

~

(4.10a)

rexp (to - ih)t, exp - (co + ih)t~] ~ + ifl L ~----i/~ to + ih ~ (1 + ct2 + if)½

(4.10b)

(exp 2tot1 -

(4.10c)

l) -l f0* dt exp f0 dt [to + ih(r)]

2x t

4 A ~ + to~

--

2d ,

tI

Vx

=

-Vx

w h e r e f is the mean value o f f - (r) over the domain, h(x) = + h inside domains and the x axis is along (1. The integration in (4.10a) is over the solid angle and frequency. The expression inside the square bracket in (4. l0 a) is non-zero in the narrow region of angles cos 0 ~< ~, y (7~h/'uFQ) ~ 1. Inside, this region electrons move in the strong exchange field h > A, while, for other angles, h rapidly oscillates along the electron trajectory. As a result° the mean exchange field, over the correlation length, vanishes for these angles, reducing f to f for the usual superconductor (h = 0). In the interval of angles cos 0 < y the integral over to diverges logarithmically in the region to ~ h and when A ~ 0. Therefore, we divide the integration region i__nto two intervals (0, c) and (c, + oo), where A ~ c ~ h. In the first interval f - is simplified, due to the condition to ~ h. This gives =

212

L. B. Bulaevskii, A. I. Buzdin, M. L. Kulid and S. V. Panjukov f-

=

AK2(#) (092 + A2K2(/0) ~,

K(#) = vvl~ sin -hd -, hd Vv/~

g =

~0 (in2 + A2K2(#)) ½ ,

(4.11)

/~ = cos 0.

In the interval (c, + 00), (1 + ~t2 + /12) can be replaced by unity. Integrating over ¢0 we get the self consistency equation for A, which leads to the following expression for gint

F~,,t =

che ½ N(O) ~~r2hAZ In - - ~

(4.12)

where c =

0"88,

e = 2'718,

h >> A.

The non-uniform part of A(x) can be taken into account by perturbation theory, where the small parameter appears to be y. The corresponding contribution to F~,t is proportional to ),2 (see Appendix B). The function A(x) is exhibited in figure 13. There one can see that A reaches its maximum on the domain walls, since in their vicinity electrons move in an alternating exchange field. The space variation of A(x) is rather small in clean superconductors (proportional to 7), while in dirty superconductors it is proportional to hr. The intermediate case has been studied in [126]. It is not difficult to conclude that the one-dimensional (striped) structure is favourable again, because the change in Fi, t is insignificant in comparison with the change in the wall energy for more complicated structure [24]. In the above analysis only magnetic structures with the wave-vectors Q >> ~o I are studied. It is not difficult to see that slowly varying (in space) magnetic structures with Q < ¢o 1 are of higher energy than the DS structures, if 0ex > Oem(a/~o). Indeed, in long-period structures (Q-~ > ~0) the mutual influence of superconductivity and magnetism leads to the increase of energy, which is of the order of 0exS 2. This is due

Z~(x)

m. X

S(x)

×

lira

Figure 13. Su ~erconducting and magnetic order parameters A(x) as a function of coordinate x along the Q direction shown schematically.


213

to the screening of the long-range part of the EX interaction by superconducting electrons. At the same time, the increase of energy in the DS phase is of the order OemS 2(aQ) ~ OemS2(a/GI) +. Thus, the MS and the VS phases are unrealizable in real compounds like HoMo6S s and ErRh+B4, since the DS phase has lower energy. Let us discuss now the important question concerning the applicability of Eilenberger's equations for superconductors in the presence of a non-uniform magnetic structure. They useful if the exchange and magnetic fields are slowly varying on the electronic wavelength, i.e., when (Q2/m(vQ)) ~ 1. This means that they describe the region of the angle cos 0 ~> Q/kF. Thus, the accuracy of description is of the order of(Q/kF) ~ (a/G) ½in the case of dirty systems (or in the case of weak fields), because all angles are important in reaching self-consistency equation (4.3e). For clean systems the accuracy is of the order of (Q/?kv) ~ (v~QZ/hEF) ~ (A/h) 2. Here the main contribution gives the angle region of the order of ?. Let us determine the equilibrium parameters S(T), A(T), and Q(T) in the case of strong exchange fields, using (4.6) for FM, and (4.9), (4.12) for F+ and Fi,t. The magnitude of S(T) is mainly determined by FM, since the influence of superconductivity is negligible because (F~m/0) ~ aQ is small. In the case of dirty superconductors with (a~o)½ ~ l ~ (VF/ho) the dependence of S(T) is analogous to that found in the absence of superconductivity, but the magnetic transition temperature is decreased by 60 = - I'400(GQ~) 1, i.e., 6S ~ S(a/G) ½. The change of S in clean

(o)

sg, +++

x\ > C0

(4.14b)

One can see from (4.14) that he2 >~ A0, i.e., the D S - F N transition takes place in a strong exchange field. The jump of S at T¢2 from S to S + 6S, leads to the appearance of latent heat W, which is of the order of superconducting condensation energy. The magnitude of W is given by the expression

F62FM(S, T)]

w = Tc26S L- ~-g-T

Arc2

sg4r)]

= c°N(0)A2

[ d In d ln r

, ATe2

where Co = 0.46 and 0-36 in dirty and clean superconductors respectively. SFN(T ) is the equilibrium value of the relative magnetization in the FN phase.

Coex&tence of superconductivity and magnetism

215

Using (4.14), the expression for Qc2 and the inequality S < 1, we find the condition for the DS phase to stay stable down to zero temperature, i.e., h < ho~. For the critical value ho~ we get ho¢ = 0"44Ao[¢o/~(0)] ~ when I ~ ~o, while it falls off with the decrease of 1 although not very rapidly. So, for 1 ,~ (~(0)~0) ½ one has h0¢ = 0.32 A0[~0/~(0)]~. Excluding the special case of very dirty superconductors one has always h0¢ > Ao. The supercooling temperature T~ ) of the DS phase is given by the condition of the absence of a free energy minimum with respect to A. In the case of dirty superconductors with (VF/h) ~ l >> (fiG) ~, we get S~ ) = 1.26 Sc2, and for clean ones S~) = 1.32 S~2. This means that the region of the supercooled DS phase is sufficiently large and may be spread over the entire temperature region below T~2. Therefore, for the D S - F N transition one needs an activation energy for the appearance of a critical nucleus in the temperature region T~ ) < T < T¢2. This energy is large compared with the temperature T~2, since the minimal size of the critical nucleus (of the normal ferromagnetic region without domain walls) must exceed the characteristic scale (VF/h). Below T~ ) superconductivity vanishes without activation energy but the complete transition to the equilibrium state may go slowly, since it needs the disappearance of a large number of domain walls. At the same time, pinning of domain walls on crystal imperfections prevents the transition process. The overheating temperature T~w) of the FN phase is determined by the condition for the appearance of a small superconducting nucleus in the FN phase. In the sample with magnetic domains this temperature corresponds to the appearance of a superconducting nucleus near domain walls (see §7). For the compounds with h0 >> A0, Tc~) practically coincides with TM. An estimate for the energy of formation of the critical DS nucleus in the FN phase, and below T ~ ), gives a very large energy of the order of (E2/A0) >> TM. This leads to the practical impossibility of the appearance of the DS phase by wanning the FN phase up to the point T ~ ). In that way, in the compounds with h0 >> A0 the DS phase appears below TM by cooling, and disappears at the first order transition point To2 when the FN phase appears. With increasing temperature the FN phase is practically conserved up to the point T¢~)(~ TM), and only here does it pass into the DS phase. The behaviour, of compounds with h0 >> A0 on wanning, gives us the information concerning the magnetic properties of the system in the absence of superconductivity. 4.5. Superconducting properties of DS phase In dirty superconductors with (h~)2 ,~ 1 there is a gap in the one-particle excitation spectrum in the entire region of the DS phase. In this case, the magnetic order decreases the superconducting order parameter and the gap. The decrease of the gap is similar to that in a superconductor with magnetic impurities (see [8, 127]). The gap is given by eg =

A(T){1 - (A(T)zm)-~} ~,

(4.15)

where Zm is given by (4.9). In clean superconductors, there is also a gap which is of the order of A0, but only in the region where the condition hoS(T) ,~ A(T) is fulfilled, i.e., near TM- However, in the temperature region where hoS(T) >>A(T) superconductivity has gapless character, as in the case of spiral magnetic order. This is due to the strong effect of the exchange field on electronic motion, which is perpendicular to the wave-vector Q of the structure.

216

L. B. Bulaevskii, A. L Buzdin, M. L. Kuli? and S. V. Panjukov

Let us find the electronic density of states e(E) in the gapless regime of the DS phase [24]. The function Q(E), for E ,~ A(T), is given by g(~o) from (4.11), by replacing ~o by iE, integrating over v directions, and taking the real part ofg(~o). One can see from (4.11) that the quantity A(T)K Icos 01 is the gap in the spectrum for those electrons moving at the angle 0 with respect to Q axis. This gap is absent for an infinite number of v directions determined by the conditions cos 0 = 0, and by the Bragg reflection of Cooper pairs on the periodic exchange field. So, in the presence of the exchange field h the net moment of the Cooper pair is k x = 2h/vv. Thus the Bragg scattering takes place when kx = 2mQ where m is an integer. This reflection causes the pairs to move inside the same domain, as the pairs with 0 = n/2. The functions f-+ vanish for these directions causing the gapless character of superconductivity. The density of states in the region E ,~ A(T) is determined by summing all gapless strips at the Fermi surface. It is given by Q(E) =

O,

hoS(T)

~

A(T)

~(E) =

? N ( 0 ) ~E l n 4A(T) z~E '

hoS(T)

~

A(T).

(4.16)

In the clean DS phase o(E) is finite at all E, and has a maximum at E ~ A(T) where o(A(T)) = x/-3N(0)/27. By cooling from TM, the gapless region appears at the temperature where S ( T ) ~ A0/h 0. The function Q(E) is shown schematically in figure 15. The dependence of Q(E) on the mean-free path l is non-monotonic for E ,~ A(T) when I decreases from I >> ~0 [127]. For v >~ h-~ the density of states increases proportionally to [1 + (2TA) ~], because electrons are removed from the gapless strips by scattering. Further decrease of v causes the disappearance of the gapless strips, which

P

Ao

E

Figure 15. Density of states e(E) for quasiparticles in clean FS at temperatures T~ and T2, where TM < T~ ,~ To), and T~2 < T2 < TM, the gapless DS phase.


217

leads to the isotropization of the system and restoration of superconductivity and a gap in the spectrum. As is well known, the gapless character of the clean DS phase can be detected by tunnelling experiments. The tunneling current Is of the FS normal metal contact as a function of voltage V in the gapless region of the DS phase is given by (d/rid V) (din/d V)

__ y2 V2 ( ln2 7[V

V + 2 - g7[2) , ~--~ - 2 In ~-~

V> (llso)-~, where lso = VErsO. In that way, when Q2 >> (llso) ~, the above proposed theory of the DS phase still holds. In the dirty limit the wave-vector of the magnetic structure is given by Q ~ (a~ol) -~, and the proposed theory is applicable when (lso/l) > (a~o/12)~. If this condition is fulfilled, then the energy of the DS phase is lower than that of the F N phase. Indeed, the energy increase in the DS phase, due to the interaction of superconductivity and magnetic order, is given by OcxS2aQ ~ OexS2a~(Gl)-~. On the basis of (4.17) we conclude that the energy increase in the FS phase is of the order of 0exS2[gn(0) -- Xs(0)]/gn(0) ~ OexS2zsoAo ~ OexS2(lso/~o). As is known, the spin-orbit scattering is a relativistic effect, and zs0 is an additional small parameter with respect to the usual scattering time z. Therefore, the relation (/so/l) is not smaller than 10, and the spin-orbit scattering starts to be important only when l is of the order of a. In the earlier analysis we explained that the effect of the exchange scattering is reduced to the renormalization of the parameter A0, when zsA0 > 1. Now, we shall examine the behaviour of systems for which Vs ~ -,~ TM ~ Tc~, in order to construct

218

L. B. Bulaevskii, A. L Buzdin, M. L. Kulik and S. V. Panjukov T/Tco

®

,k li I

//

@

i

O~

P

I

e~, /

Tco

Figure 16. Phase diagram of FS in the (TfTco, 0ex/Too)plane, shown schematically. Full lines separate N, S, DS and FN phases. Tel and T M a r e second-order transition lines, dotted line is 0 line, and dashed ones are overheating and supercooling lines of first-order transition. The second order transition line 0 separates N and F phases in the absence of superconductivity.

the complete phase diagram of ferromagnetic superconductors in the plane (T/T~o,

O./T~o). In the region (O~x/T¢o) ,~ 1, but 0ex >> 0(a~0)1122{ ~, the temperature TM lies above the S - F N transition point T~2. By lowering the temperature the following sequence of phases should be observed, r~l rM r~2 N -5i-~ S - - ~ DS --T+ FN. I, and II represent the first and second order phase transitions respectively. The last transition may be absent in systems for which Oex/T~o is sufficiently small, as explained by figure 16 and §6.2. The T¢2 line approaches the TM line with the increase of O¢x/T~o, since A is decreased at the magnetic transition point. This causes the T~2line to lie above the TM line. This leads to the following sequence of phases on cooling: Td

Tc2

N--iT-+ S--i--~ FN. The quantitative analysis of the situation near the tricritical point (the intersection of Tel and TM lines) is given in Appendix C. It shows that the crossing of lines Tu(0~x) and Tc2(Oex) occurs near the crossing point of lines Tel (0~x) and 0. The region around the tricritical point B is shown in figure 16.


219

We emphasize that a similar phase diagram, but without the DS phase, was obtained by Kaufman and Entin-Wohlman [27] who calculated the S-FN transition point To2. The overheating line of the FN phase was obtained by Reiner [102] who calculated the effect of magnetic scattering on Tc~, including magnetic correlations near the point 0 (see also [20]). This line lies slightly above the 0 line, if the long-range part of the EM interaction is not taken into account, because of the logarithmic divergence near the point 0. Taking into account the magnetic dipole interaction, the overheating line of the FN phase drops down below the 0 line. The supercooling line of the S phase is the TM line. In the complete phase diagram, figure 16, there are two tricritical points, A and B. At the point A two lines meet separating phases S, DS, and FN, while the point B is located at the boundaries of N, S, and F N phases. The Lifshitz point L is located on the overcooling line of the S phase. Besides the parameter Oex/T~o, another essential characteristic of ferromagnetic superconductors is the cleanness of the crystals, characterized by the parameter l/(a¢o) ½. By decreasing this the region of the DS phase is narrowed, because the points A and ~,) tend to the co-ordinate origin. At the same time the region of the first order S FN transition is widened. Now, we are in a position to give the general picture of the mutual influence of superconducting and magnetic orderings. Even in the case when TM ,~ Tel the magnetic ordering is the stronger effect so long as h0 > A0. This is because the magnetic ordering energy is of the order of 0 ~ h~N(O) (per LM), while at the same time the superconducting condensation energy does not exceed N(0)A02. On cooling, ferromagnetic order destroys superconductivity at sufficiently small temperatures if h0 > h0~. Nevertheless, in some temperature region superconductivity and non-uniform magnetic order coexist. In the system with h0 < h0~ the coexistence phase survives down to zero temperature (see §6.2). Almost in the whole co-existence region, excluding a narrow region in the vicinity of TM, the effect of Cooper pairing on magnetism is weaker than the converse effect of magnetism on superconductivity. The coexistence phase is not possible in the system TM ~ Tc~. In this case the two competing types of ordering are separated by the first order transition line T~2. 4.7. Effects of magnetic structure defects on coexistence phase We now discuss the very important question regarding the influence of different types, and concentrations of defects in magnetic structure on the coexistence phase [66]. In 'regular' crystals like ErRhaB 4 and HoMo6X 8 we could expect (a) chaotic orientations of easy-magnetization' axis, and/or (b)change of interactions between moments due to displacements of magnetic ions from the equilibrium positions. Both kinds of disorder may be due to the presence of random stresses inside the crystal. In pseudoternary compounds magnetic disorder is stronger. Here magnetic and nonmagnetic ions with different types of anisotropy are distributed randomly on the sites of a lattice (stannides and compounds RE~ xYxRh4 B4) or different magnetic ions are distributed randomly on the lattice (compounds Erl_xHoxRh4B4). We know from the general theoretical arguments that defects, acting as a random external magnetic field, destroy the long-range order of the non-uniform incommensurate superstructure in a system with less than four dimensions [128, 176]. The proposed sinusoidal or domain-like structures must be either completely suppressed when the disorder is sufficiently high, or deformed more or less strongly with respect to the ideal structure which is shown in figure 4. The deformations of the domain structure are, in this case, reduced to bends in domain walls and to a random distribution of domain thickness.

220

L. B. Bulaevskii, A. I. Buzdin, M. L. Kuli6 and S. V. Panjukov If xk is the coordinate of the kth domain wall d3q xk = kd + J(-~)3 Ax, exp iqr~,

(4.18)

r = (x, x±). Here, Axq is the Fourier component of domain wall displacements from their equilibrium conditions x~°) = kd. The magnetic moment is given by Sz(r) =

~ $2,+, exp iQ~0(r)(2n + 1)),

(4.19)

n

S2n + I

2S ni(2n + 1)'

-~

where, by using (4.18), ~o(r) has the form d3q ~o(r) = x - f(--~K)3 AXq exp (iqr)

(4.20)

The intensities of Bragg peaks for neutron scattering are determined by the correlation function, (S~(0)Sz(r))

=

~, 1S2,+, 12exp {iQ(2n + 1)x) n

x exp { - ( 2 n + 1)2Q: f ~d3q (Axq AX_q)(l - cos qr)}

(4.21)

where the brackets mean the average over random orientations of the easy axis, or over disorder of any other type, and over all domain wall displacements. Thus the correlator (AxqAxq) describes the domain structure in the presence of defects of the magnetic subsystem. According to general conclusions given by Sham and Patton [128] this correlator depends on the correlator of effective random fields as well as on the stiffness of the ideal domain structure with respect to the deformation of the domain walls. In dirty superconductors with Q i ,~ l ,~ ¢0, the part of the free-energy depending on Axq takes the following form: F{Axq} =

z{Q~q~[1-

eA~ ' ]_]+ ~Q q ~ + 20~mq~} AxqAx q, (21n~--~o// (4.22)

where q ~ (~01)~ is assumed. The x axis is along Q; the z axis is along the easy axis

and ~ is the difference in the domain wall energy for Q along the y axis and along the optimal x direction. Additionally there is a pinning of domain structure due to the discreteness of the magnetic lattice, and the corresponding energy should be added to equation (4.22). However, the pinning energy is very small, i.e. % ~ ~ exp { - 7t2aw/a} in the case when the domain wall thickness aw >> a, where a is the RE interatomic distance along the Q axis. For aw = 2a Bak and Pokrovskii have found % = 0.001 (see [173, 174]). Thus under the condition (a2/d2) > ep/~ the domain structure is incommensurate with the magnetic lattice and the pinning effect in (4.22) can be neglected. The equivalent condition (a/~o) "> %/~ is fulfilled in all compounds under consideration. We do not take into account pinning effects because of the discreteness of the magnetic lattice. In the absence of pinning the long-range order is destroyed, as is shown in [128].


221

Hence, the correlation function (4.21) falls of exponentially on the correlation length Re, and depends essentially on the relation between the correlation length R 0 of the effective random fields and on the domain thickness d. The case R~ ,~ d represents the point-like defects considered in [128]. The value R~ may be small, and in that case the long-range domain structure is actually absent. If R0 ~ d the value R~ is exponentially large, i.e. Rc ~ exp {Ro/d}. This limiting case represents the ideal domain structure. Considering a low concentration of point-like defects, when the condition R~ >> d is fulfilled, one easily obtains the peak intensity at the wavevector q = [(2n + 1)Q + r/, 0, 0] (using (4.21) and (4.22)), I2,+~(q) =

I,(0)[q2/~ + (2n + 1)4]-2.

(4.23)

The magnitude of R~ depends on concentration and type of defects, and can be found from the width of the experimental peak intensity. It is obvious from (4.23) that intensities of higher peaks (harmonics) fall off as (2n + 1) 8 (in monocrystals) independent of Re. The gapless character of the electronic spectrum in clean superconductors is preserved when R~ >> d/?. However, the contribution of Bragg reflections (of pairs) to the density of states Q(E), for E ,~ A(T), falls off rapidly with the decrease of R~. In that case there is the contribution to Q(E) from those Cooper pairs moving along domain planes. Then the large logarithmic factor in (4.16) should be changed by a numerical factor of order unity. For R0 >> d the long range order is preserved because P~ in (4.23) is exponentially large. This limiting case represents the "ideal' domain structure. 5.

The effect of magnetic field on coexistence phase

5.1. Change of parameters of DS phase in magnetic field H In the presence of a magnetic field the harmonic of the magnetization with q = 0 appears (Sz,q=0 ¢- 0), as well as the corresponding harmonic of the exchange field. On the basis of results in §3.2 one can see that the superconducting order parameter is uniform on the length scale G0, and the non-uniform superconducting L O F F state is realized. The lines separating the FN phase from these two coexistence phases are obtained by extrapolating the corresponding lines in figure 7. The F N - D S boundary terminates at the point To2, and F N - L O F F at Tff), which is obtained from the condition for the appearance of the L O F F nucleus in the FN phase. In clean superconductors this condition has the form hoS(T~ ~) = 0.75 A0, which means that when h0 ~> A0 the point T~') is very close to TM. The L O F F phase is favourable in the interval TM > T > T~'), and the region of its existence is negligibly small for systems with h0 >> A0. Because of this we shall only study further the properties of the DS phase in the presence of magnetic field. It is well known that in bulk monocrystalline samples the magnetic field only penetrates to a depth 2 into the sample. Therefore, the DS phase in the bulk is not affected by the field. The first order transition line FN-DS is determined by the condition FFN(T, H) = FDs(T, H = 0), where the free-energy los is known (see §4). However, the change of the DS phase in a magnetic field can be observed in thin plates, or in polycrystalline samples with an incomplete Meissner effect. There are indications that the latter case is realized in polycrystalline samples of HoMo6S 8 [51-53, 58].

222

L. B. Bulaevskii, A. I. Buzdin, M. L. Kuli4 and S. V. Panjukov Y Z


J i[j[r

b/.

reX

Figure 17. Geometry of the sample in which the DS phase in a magnetic field is considered. For simplicity, we prefer to consider a thin superconducting plate with thickness L < G, and with the field H parallel to the planes of the plate [23, 24, 126]. Then the orbital effect of the field is negligible, and we further suppose that the easymagnetization axis (z axis) and Q lie in the plane of the plate, as in figure 17. The greatest effect on the DS phase is caused when the field is along the z axis, and therefore we consider this case first. The domain thickness for domains with M parallel to H is increased, while it is decreased for those with M antiparallel to H. The modified thicknesses are denoted by d(1 __+ 6) respectively. The next step is to find the free-energy functional depending on A, S, 6, and T, as well as the equilibrium parameters of the modified DS phase. The relative magnetization Sz(r) in the modified DS phase has the following form: Sz(r) =

as

$6 +

~

{[1 - ( -

1)k cos (~k6)] sin (kQr)

k=l

+ (-- 1)~ sin (~k6) cos (kQr)}.

(5.1)

New harmonics have appeared as well as the zeroth harmonic of the exchange field, i.e.,/7 = hoS6. For dirty superconductors the theory given in §4.3 still holds for the modified DS phase, but with ~o replaced by the complex frequency u = co + i/7. Equation (4.9) gives the exchange scattering time Tm 1 for the modified DS phase.

rm'-- 8'4h°zS21 nQv-------~G(6)l)k

} (5.2)

G(6) =

~ k=l

-- ( -

cos (nk6) 2"1k3

In the self-consistency equation for A we have to integratef0(u) on a line parallel to but above the real axis/7, instead of integrating~(~o) over the real co axis. The final result does not depend on/7 (nor does A), as long as this path does not intersect the singular points off0(u). They are determined by f0(u) = (co - i/7)/A =

u(1 + u2) -½, u[1

-

(TmA)-l(l

t n t- U 2 ) - ½ 1 .

(5.3)


223

When A z m > 1, the nearest singular point o f f ( u ) to the real axis is the point u = iA[1 - ( A z m ) - ~ ] ~. As long as /T < h"c = A[I - ( A z m ) - ~ ] t the order parameter does not depend on/~. The free energy functional F has the form F{S, A, Q, 6, T, H}

=

- OS2 + Fo(S, T} + ((S, )t______QQ rtn H2 + 0¢xS262 - g#~SH6 8n eA02 (r~A - ½N(0)A2 l n - ~ + N(0) ~-~m

Azm > 1,

h- < A[1 - (Azm)-~]~,

7Z

Q = ~,

1) 37r2m '

(5.4)

]~= hoS6.

The superconducting solution is absent for/~ > /~c. In ~4.3 we noted the analogy between the effect of the domain magnetic structure on superconductivity, and the effect of magnetic impurities with magnetic scattering time Zm. NOW we see that this analogy is not complete. In the former case the parallel electronic paramagnetic susceptibility Z~(q = 0) is zero at T = 0, while in superconductors with magnetic impurities it is non-zero for all values of Ars. This difference is related to the fact that the superconductor with magnetic impurities is completely isotropic in spin space, while in the DS phase the direction of the magnetization inside the domains is preferred. Therefore, the responses of these two systems to a constant field are different. In the superconductor with magnetic impurities Z¢ is isotropic, while in the DS phase one has Z~ = 0, but X~ :~ 0. The component Z~ is analogous to the susceptibility in the presence of magnetic impurities [23]. Similar anisotropy of susceptibility was found by Ishino and Suzumura [129] for antiferromagnetic superconductors. Minimizing the free energy with respect to A, S, Q, and 6 gives 6 = (#H/2SOex), and shows that the magnetic susceptibility of LMs Z~m= (nl~:/2Oex) does not depend on temperature. The relative magnetization S(T) practically coincides with the corresponding magnetization in the FN phase. The wave-vector of the structure decreases with the increase of the field. The function Q(6)/Q(O) is shown in figure 18. The maximal change of Q is near the temperature where S(T) ~ (A0/h0), and 6 changes from zero to unity, changing H from zero to the 'overheating' field of the DS phase. The change of Q is about 20% in the thermodynamic field He. However, the restoring time of the equilibrium value of Q, achieved by varying H, may be large (see the end of §4.3). !.0

0

0

i

0.25

0150

I

0.75

tOO

6

Figure 18. Q versus 61tand 6± in a monocrystal, and versus 6p = laH/2S(T)Oexin polycrystal.

224

L. B. Bulaevskii, A. I. Buzdin, M. L. Kulid and S. V. Panjukov [J.H ho 20ex ~o

s(C) c2

Sc 2

-(w) 0 :~c2

~

T¢2

T(w) ~c2 rq,4 ~ T

0

~

S

Figure 19. Region of the DS phase in the (H, T) plane, where 1 is the 'overheating' line of the DS phase, 2 is Hc line of the FN-DS transition, 3 is He: line, i.e., the 'supercooling' line of the FN phase, and 4 separates DS and S phases with homogeneous magnetization. This figure is a part of figure 7 b near the point TM. In the case of the clean superconductor, Z~ is again zero when h ,~ A, because of the gap. In that case the free energy functional is given by equation (5.4). In the gapless regime the response of electrons to the weak (constant) exchange field is non-zero, and the dependence of the free-energy on/7 may be obtained using the following expression: 6F = --aft

_ iN(0) 2roT ~,, ~ I ~dF~ 4,

(5.5)

where ~ is the domain-averaged value of g. The main contribution to the sum over oJ in (5.5) comes from the region Ico[ ~ A, /~,~ h. Making use of (4.11), in which co + ih-, we write (5.5) in the following form: 6___F - i N ( 0 ) f ~ dco f~ d# co + i/~ 6/7 = ((co + i/~)2 + A2K2(p)) ~" Integrating over/7 and co gives, r(/7) - F(0) =

(5.6)

N(0) Re f~ dp [AZK2(/~) In /~ + (/~2AK(#)_ A2K2(#))½ +

fi2

- /~(/72 -- A2K2(/~))t].

(5.7)

It follows from (5.7) that the superconducting solution is absent for /~ > A. For /~ < A the main contribution to the integral over/~ comes from the gapless strips on the Fermi surface. Summing up these contributions we finally get,

F(fi) -

F(0)

=

N(0)h 2 I -

h~>A,

0.61

h2 Ih'l "] vvQAJ

(5.8)

/~> A0. The theoretically expected temperature dependence of the ferromagnetic peak F and its satellites (2k + l ) O for the sequence of transitions S - D S - F N in regular ferromagnetic superconductors is shown in figure 20. ( f ) The influence of magnetic field on the DS phase. The magnetic field H ( < He), directed along the easy-magnetization axis, lowers the temperature TM and moves the point Tc2 toward higher temperatures. If such a field penetrates the sample in the DS phase, Q increases and even peaks 2kQ appear. The phase diagram (H, T), in the case when the field penetrates into the thin slab, is shown in figure 19. The DS phase is completely suppressed by the field H > H~w) which is parallel to the easy-axis. The effect of the field perpendicular to the easy-axis on the DS phase is much weaker than that of a parallel field [23, 24, 126]. (g) The FNphase. After the D S - F N transition, and on cooling down to T~ ), the metastable DS phase may survive, thus increasing the conductivity of the system with respect to the normal state. 6.2. Properties of HoM06S 8 and HoMo6Se 8 HoM06S8. The comparison of experimental data for H o M o 6 S 8 and the theoretical predictions (a)-(g) is rather difficult, because all measurements have been performed on polycrystalline samples [51-53, 58]. According to the prediction (b), Lynn et al. observed the growth of ferromagnetic fluctuations in small angle neutron scattering on cooling down to 0.69 K. At temperatures below 0.69K the peak at Q = 0.03/~ -~ was observed. The intensity of neutron scattering as a function of angle and at different temperatures is shown in figure 21 [53]. These data unambiguously show that non-uniform magnetic structure appears in the superconducting state. Below 0.65K superconductivity and nonuniform structure disappear, although Lynn et al. reported an anomalously strong

228

L. B. Bulaevskii, A. L Buzdin, M. L. Kuli~ and S. V. Panjukov I

I

I

(~ I000

-

N ?

o T = 0.72K

NO

• T=0.70K A T=oBBK + T=0.66K

o'~0 -

I

HoM%S8 Ei =3.72MeV

o

N,~

o T=o~K

_


'E

~,

.+/+'+\ 500 -

_

_

I

o

aot

t

a02

I

003

]

oo~

oos

Figure 21. Measurements of small-angle scattering in the visinity of the re-entrant transition in HoMo6S 8. The peak at Ikl = 0-027A develops upon cooling and disappears below 0.64 K [53]. small angle scattering. They suggested that this is due to the scattering on domain walls. In figure 22 we show the neutron scattering intensity, which corresponds to the wave-vector Q = 0.027 A ~, as a function of magnetic field and temperatures, on cooling and on warming. The non-uniform magnetic structure does not appear upon warming of the F N phase, as upon cooling it is suppressed completely by a magnetic field higher than 400 Oe. Figure 22 also shows the overheating and supercooling temperatures T~ ) and T~ ) . They were obtained from a.c. susceptibility measurements. In [53] it was observed that by lowering the temperature quickly down to 0.62 K the intensity of the Qth peak (0.027 A -1) is doubled, in comparison with that shown in figure 22. After approximately one hour it is decreased to the level of 360 counts per minute. Lynn et al. [58] have measured small angle neutron scattering corresponding to the wave-vectors q = 0.009 A I and 0.03 A - i. The result is that upon cooling from TM (0.74-0.75K) down to To2 (0.67-0.7K) only the non-uniform structure with Q = 0"03 A ~is present, and ferromagnetic scattering (q = 0.009 A ~) only appears below T = 0.7 K (see figure 23). One can also see from this figure, that upon warming from the F N phase above T¢2, the intensity of the Q = 0.03 A - ~peak grows insignificantly, but the ferromagnetic peak is conserved up to T = 0.72 K. Also, upon cooling below 0.69 K, the volume of the sample occupied by the non-uniform structure and its superconductivity are preserved, down to T~ ) ,~ 0.62 K as a metastable state, although it is much smaller than the total volume. Therefore, upon cooling and in the temperature region between 0-74)-62 K the sample consists of the F N and superconducting regions with non-uniform magnetic ordering, where several hours are needed to relax to the equilibrium state [58]. The experimental results shown in figures 22 and

Coexistence of superconductivity and magnetism I

I

1

r

r

229

r

HoM%Se

600

E i = 3.72 MeV

H=o H=~eol (/)

400

Z

3OO

Worming

~

H=4°°Oe

200

I00

(c) " ~ iT~) ,~.. Tc2 Q30

,

0.40

,

050

,TT,

060

(170

,

0.80

090

T(K)

Figure 22. Temperature dependence of the scattering intensity at Ikl = 0.0275A J in HoMorS8. An applied field suppresses the inhomogeneous state (which is not observed on warming). Re-entrant superconducting transition temperatures, on cooling and on warming, are indicated by the arrows on the T axis [53]. 23, are in full agreement with the theoretical predictions shown in figure 20. Figures 21-23 give information concerning the basic peak of the non-uniform magnetic structure. Higher peaks (harmonics) were not noted in [51-53]. The results for the Bragg peak (100) in HoMo6S8 are presented in figure 24, with the additional magnetic scattering (below TM = 0.67 K) which is proportional to S2(T). The difference in this parameter upon cooling and upon warming is obvious from figure 21. We note here, that the temperature behaviour of intensity upon warming is that typical of a standard ferromagnet with the effective spin S~fr 1 with a second-order transition at the Curie point. Nevertheless, these experimental data do not allow unique determination of the magnetic structure in the coexistence phase. They show only that the non-uniform magnetic structure has transverse component, since only that is fixed by neutron scattering measurements. The data also show that the Fourier component with Q = 0.03 A -] is present. The coherence length Rc for the magnetic modulation is 1500A according to data [53], and Pynn et al. [130] reported the estimate Rc > 3000A. The absence of higher odd peaks in experiments on polycrystalline samples can be explained by the fact that for such samples the intensity of higher peaks in small angle scattering is very small. The width of higher peaks is expected to be large due to the irregularity of the domain structure. Let us discuss the question concerning the magnitude and temperature depen-

230

L. B. Bulaevskii, A. I. Buzdin, M. L. Kulik and S. V. Panjukov I

I

T~ Tt

0 •

1000

I

o

O=O.030~A -1

H=O

A A O=0.00gA -1

A

200

8000

C~ 0

0

6ooo l-Z

Z 0

0 0

IOG

4000 •

2000

'c2

T!C)

0.65

"x, "~ "Y~ 070 T(K)

075

Figure 23. Neutron scattering intensity at k = 0.030A ~ (modulated component) and at Ik] = 0.009 A (ferromagnetic component) versus temperature in HoMo6S8, showing the presence of the modulated component only, on cooling from 0-74 K down to 0.70 K. On warming, the modulated component is very weak up to 0-73 K [58]. dence o f the w a v e - v e c t o r O. One can see in figure 21 t h a t iOI is a l m o s t i n d e p e n d e n t o f t e m p e r a t u r e . I f we a s s u m e t h a t Q,xp = 0-03 A -~ ,,~ Qcz, we are in a p o s i t i o n to estimate the q u a n t i t y fi by using the expression (4.13a). T a k i n g for VF the value 1.8 x 1 0 7 c m s -1 f r o m the n u m e r i c a l c a l c u l a t i o n o f F r e e m a n a n d J a r l b o r g for the b a n d structure o f t e r n a r y chalcogenides [57], we o b t a i n ~0 = 1-5 × 10 -~ cm. I

9000 >-

[

--

[

I

I

I

I

-- = ~ . . ~ / H e a t i n g

[

I

[

•

8000

~z 7000 L~ 6000 z__ 5000 Y

4000 n

I t

3000 2OO0 1000

HoMo6SB -

0

[

0

f

i

I

I

J

[

,i If

o.l 02 a3 oz a5 0.6 07 08 09

I

l.o

T(K) Figure 24. {100} peak intensity versus temperature. This is due to the nuclear Bragg scattering above TM ~ 0-67 K, while below TM additional intensity is magnetic in origin, and is proportional to S2(T).

Coexistence of superconductivity and magnetism ~0

,

,

. ,o,tk D_m~-_

"- 2(]

o

o

~

231

O=0.030A -1

~,

is0

T=0.730K

"%k

s;o

700

,ooo

H(Oe)

Figure 25. Peak intensity at q = 0 . 0 3 / ~ -1 in H o M o 6 S 8 versus magnetic field for OII H and 0 3_ H [58]. Experimental results of Ishikawa and Fischer for H c 2 ( T ) n e a r Tel give ¢ = (G0/)½ 300 A and l ,~ 60 A. The estimated values for ho, ~o, and l given in table 3, show that the investigated samples of HoMo6S 8 can be described by the theory of dirty superconductors with Q-~ ~ l ~ vv/h. Using (4.13 a) we get the reasonable value ~ ,~ 2-5A. At the point TM we get QM ~ 0.04 A-m from equation (4.5). In fact the decrease of Q has to be small upon cooling. It is difficult to observe this behaviour, because the main change of Q occurs near TM, where the intensity of the Q peak is fairly small. According to the statement (e), hysteresis occurs at the D S - F N transition. So, the F N phase is conserved up to TM, and the DS phase survives down to T~ ) ~ 0.62 K. Using the data from figure 23, we obtain S~ ) = 1.26 Sc2 for T~ ) = 0.62 K and Tcz = 0-7 K, which coincides with the theoretical prediction given by (4.14 a). For the right-hand side of(4.14a) one obtains a value of 6 A at 0ex ~ 0ex ,~ 0.15 K, while the left-hand side is about 12 A for S~ ) ~ 0"35 (taken from figure 24). This difference of a factor of two may be considered as acceptable, as a rough estimate was used for 0~x. The sharp peak in the specific heat near Tc2, observed by Woolf et al. [131], gives evidence for a first-order transition at this point. According to point (f), a magnetic field decreases the transition temperature of the S-DS transition. The corresponding decrease of the peak intensity with the increase of the magnetic field at T = 0.735 K, was observed by Lynn et al. [58]. The depression of the non-uniform magnetic structure, in the presence of a magnetic field in HoMo6S s below TM was observed in [51-53, 58] (see figure 22). The peak intensity at Q quickly drops as the magnetic field increases above 100-200 Oe. According to the theoretical predictions in (f), the parallel critical field is about 100 Oe. Likewise, the anisotropy of the Q peak intensity was observed experimentally [58] to depend on the direction of the scattering wave-vector q with respect to the field H (see figure 25). This result is also in agreement with the prediction of ( f ) concerning the anisotropy

232

L. B. Bulaevskii, A. I. Buzdin, M. L. Kulik and S. V. Panjukov I

I

I

I

I

I000 HoM%S8

900

_

T= ae7K

Ei:3.72meV

800

700 o') Z

g

6oo

0

H=0

500 H=100Oe

t.00

H=2000 e

300 200

x\

'°°Io 0

oLo=

0.02

0.03 IRI (,L-')

oo~

o.os

Figure 26. Scattering intensity versus [k[ for several values of field in HoMo6Ss, showing that the intensity is shifted to smaller k by increasing the field [53]. behaviour of the system in a magnetic field. So the contribution to the neutron scattering at q ]1 H gives those crystallites in which one has Q [] H, and the magnetization is perpendicular to Q. In the case q _L H, there is a field component along the magnetization axis, which means that its effect on the coexistence phase is stronger than in the case q [I H. This leads to a faster destruction of the DS-phase in those crystallites which give contributions to the scattering with q _1_ H. In figure 25 one can see the increase of the peak intensity with the increase of weak magnetic field. Only in stronger fields does the peak intensity fall off according to the theoretical predictions. The cause for such a behaviour of the Q peak in a weak magnetic field is unclear. The concurrent decrease of ]Q] in a magnetic field was observed in [53], as shown in figure 26. In that way, experimental results for HoMo6S8 are in agreement---on basic points --with theoretical predictions for the DS-phase. H o M o 6Se8. Let us analyse the very recently reported properties of the second ferromagnetic superconductor HoMo6Se8, from the family of ternary chalcogenides. Lynn et al. [54] observed recently that a polycrystalline sample of HoMo6Se8 becomes superconducting at Tc = 5-5K, and the superconductivity survives down to the

Coexistence of superconductivity and magnetism O(T) O(o)

S2 4-

0.8

233

-

+

o

1.3

0.6 +

1.2

0.4

1.1 0.2

1.0 0.5

I

L

l

I

0.6

0.7

0.8

0.9

T/O

Figure 27. Temperature dependence of the integrated intensity (which is proportional to the square of the ordered magnetic moment at Q) and the characteristic wave-vector Q, for HoMo6Se8. The solid lines represent theoretical curves. lowest temperature achieved in their experiment, T = 0-04 K. Neutron scattering measurements give evidence that at TM = 0.53 K the non-uniform magnetic structure appears. Likewise, they show that the wave-vector of the structure decreases monotonically from Q(TM) ~ 0"09A - j down to 0 . 0 6 2 A - l at T = 0.4K, while below T = 0.4 K it remains constant (see figure 27). Ferromagnetic peaks, as well as higher harmonics (2n + l)Q, were not observed. The intensity of the Q peak tells us that the square (relative) magnetization S 2 grows upon cooling down to T = 0"3 K, and thereafter enters the saturation regime. The remarkable fact is that in HoMo6Se8 the re-entrant behaviour of superconductivity was not observed. This means, that in this compound the parameter h0 does not exceed the critical value h0¢ (see ~4.4). Then, from (4.14) it follows that 0~x is smaller than the critical value 0~) given by 0~) =

0"077A0vFQ(0)N(0),

(6.1)

in the case when Q-l ~ l ~ G0. By using experimental data for Q(0) and Tc, we get 0~Cx ) = 0.34K. The exchange parameter 0ex for HoMo6Se8 can be estimated from the data on the depression of T¢ under the influence of exchange scattering. So, for non-magnetic LuMo6Se8 one has To0 = 6.2 K, which leads to 0cx = 0.14K (for HoMo6Ses) using (3.1). Assuming that 0ex ~< 0~, one easily concludes that the condition 0e~ < ~,) is indeed fulfilled. From (4.9) it follows, that (8h2/vFQAo) = 0.6(0~x/0~,~) ,~ 1, which means that the effect of magnetic ordering on superconductivity is negligible at all temperatures. The estimates given above show that the condition 0"60~x < 0~]) is probably realized in HoMo6Ses. Experimental results for Q(T) are in accordance with the theoretical predictions for the case when h0 < A0. The theoretical value of Q(T) is given by (4.8) with b = 0.2 (see figure 27). We note that on cooling Q stops changing in the region where the magnetization is growing. This fact confirms our conclusion regarding the weak influence of magnetic ordering on superconductivity. It also shows that the temperature dependence of Q is connected with the temperature dependence of the surface energy of the domain walls (see equation (4.7)). From knowledge of Q(0), and at G0 = 470A we get ~(0) = 2.7A. The long time needed for the restoration of the


234

L. B. Bulaevskii, A. I. Buzdin, M. L. Kulik and 8; V. Panjukov

equilibrium state--about 50 hours [54]--may be explained by the large time needed for the formation of the equilibrium number of domain walls at such low temperatures. This is due to the fact that the wall formation process needs a large activation energy. At temperatures T > TM, and in an applied magnetic field, HoMo6Se8 may become a normal ferromagnetic metal. The life-time of such a metastable state has to be very large at T > T~ ~ ~ 0.4-0.5 K. According to the theoretical results presented in §3.1, the D S - F N transition in an applied magnetic field should be first order. In that way, experimental data for HoMo6Se 8 are in agreement with the theoretical predictions. 6.3. Experimental data for ErRh4B 4 The measurements of resistivity in a static magnetic field, as well as of magnetic susceptibility in an alternating field carried out by Maple et al. [132] on polycrystalline samples of ErRh4B4 are shown in figure 28, which shows the re-entrant behaviour of conductivity in this compound. The N-S transition at Tc~ is of the second-order, while that at To2 is of the first-order with pronounced hysteresis. The character of the magnetic transition has been investigated by Woolf et al. [43], who measured the specific heat of ErRh4 B4. The results are shown in figure 29, which shows that the transition temperature Tc~ is 2-8 K less than that for the non-magnetic computed LuRh4B 4. This is due to the exchange scattering of electrons on LMs. At the point To2 there is a pronounced anomaly in the specific heat, the detailed form is given in the inset of figure 29. The first neutron measurements carried out by Moncton et al. [34], showed the monotonic growth of ferromagnetic Bragg peaks, starting from temperatures slightly above 1 K. Subsequent neutron measurements also performed on polycrystalline samples [36], gave evidence for a peak at the wave-vector Q = 0.06 A - ~ in the small angle scattering, and in the temperature region 0.6 K < T < 1 K. The intensity of this peak was smaller on warming than on cooling.

¢-

I

~

o

a

F-

v

~g

J

-2

,.~15o

F

c~

tO

ioo

~ 50

/

n,-

o

o

I

2

f 3

I 4

1 5

i 6

l ?

I 8

J

i 3 9 IO T(K)

Figure 28. A.c. susceptibility and resistance versus temperature in ErRh4B4 [132].


235

36 32

32

28

t

28 ''" %

ErRh4B4

24

~ 2o

2l.

IL.

iJ 12

E 20

8

Tc2 ~'% 0.93 K) ~" ............

4

16

0

I

I

0.6

t

I

0.8

I

)0

I

I

l

12 T(K)

I

I

14

I

1.6

12 ErRh4B4, °, . . . .

4-

~'

--i'" ~"

I

Tel

rc ,-...

0 0

,,

LuRh4B 4

.

,o.°O,°'°'"

0~3Ki L..~ . ..... r . . . . i'" i'"l l i 2 /. 6 8 10 12 I/. 16 18 20 T(K) T(K) T(K)

Figure 29. Molar heat capacity C versus temperature in polycrystalline samples of ErRh4B4 and LuRh, B4. The inset show the anomaly near To2in the ErRh4B4. Very important results o f neutron and conductivity measurements on the monocrystalline sample of ErRhaB4 were recently reported by Sinha et al. [37] and are shown in figure 30. They reported a very surprising result; the simultaneous appearance of ferromagnetic Bragg peaks with four satellites around some of them. The positions of satellites around the ferromagnetic peak (101) in the plane b*-c* are determined by the vectors Q = ( _ 0 . 0 4 2 b * + 0.055e*), while the magnetic moments are oriented along the a axis. This means that the magnetic structure is transverse. The intensities o f two satellites are higher than those of the other two, giving evidence for the one-dimensional character of this non-uniform ordering. At the same time, in some regions of the sample the structure with Q = + (0.042 b*, 0.055 e*) is realized, while in others the structure is Q = + (0.042 b*, - 0 . 0 5 5 e*). The dominance of one of them is connected with the existence o f strains inside the sample. Higher harmonics (satellites) were not observed. Theoretically their intensities should be smaller than 0.02 IQ. Experimentally, however, the magnitude of Q does not depend on temperature. The coherence length Rc for magnetic modulation exceeds 1000 A [67]. Figure 30 shows that the ferromagnetic peaks and the satellites appear simultaneously although the total intensity of the satellites is small compared to the basic Bragg peak, and does not exceed 0.1 IB,,u. It is clear, from the measurements of Sinha et al., that the non-uniform magnetic structure disappears simultaneously with superconductivity. According to the experimental results of Mook et al. [133], long-range ferromagnetic order is absent at T = 1-1 K, and the ordering is widespread in the region with the size of 200 A. The genuine long-range order is established at lower temperatures, and at T = 0.8K the size of correlated magnetic regions exceeds iooooA.

236

L. B. Bulaevskii, A. I. Buzdin, M. L. Kulik and S. V. Panjukov

T(K) 0.5 0.6 0.7 0.8 0.9 ].0 I.I 1.2 I

6000 o

A

I

Z

I

~,~ ~ "~

5O0O

r

i

]

I

Er RhaB4 single crystal Ferromagnetic Intensity IFH

-

c

4000 0

~3000 2000 Z W

I000

"G 80 [I ~

~z~o

'1 Y

Satellite Intensity Is

ocooting

W

z ~ 20
0.3, and below Tcz, Ho moments order along the e axis. For x < 0.3, Er moments order in the basal plane, and a multicritical point is found in the phase diagram near the Er concentration of the order (1 - x) ~ 0.7. In the region 0.3 < x < 0.9, the sharp transition S - F N at To2 is first order. By careful analysis of neutron scattering data, Woolf et al. [62] found that the magnetization disappears discontinuously at Tcz. Extrapolating to zero magnetization gives the Curie temperature 0, as shown in figure 33. However, the cause of the direct S - F N transition, with no coexistence phase, in these compounds is not only the vicinity of transition points Tel and 0. As in the case of ErRh4B4, pseudoternary compounds have a disordered component of magnetiza-

Y

v

SUPERCON~

t. 3 2

MAGNETC I ALLYORDERED

I 0

0r

J

0~

2

I

0.I4

J

X

0'.6

~

0.i8 l

I.'0

Figure 33. Phase diagram for Erj_xHoxRh4B4 pseudoternary alloys [144]. The dashed curve shows a magnetic transition at 0 which would occur in the absence of superconductivity.

240

L. B. Bulaevskii, A. I. Buzdin, M. L. Kuli? and S. V. Panjukov

tion suppressing the coherent non-uniform magnetic structure in the coexistence phase. The measurements of hyperfine fields of a I~B nucleus by N M R , carried out by Kohara et al. [147], show a broad distribution of hyperfine fields in Erl_xHoxRh4B 4 with x = 0-6 and 0.8, and in the presence of external magnetic field. This is due to the chaotic orientations of Er moments and/or irregular distribution of Ho moments. In that way, the behaviour of compounds with irregular magnetic subsystems is in accordance with the assumption that magnetic disorder has a detrimental effect on the coexistence phase.

7.

Does superconductivity exist in the domain walls of the FN phase? 7.1. Formulation o f the problem

Matthias and Suhl [148] were the first authors who noted that the appearance of superconductivity is more favourable around domain walls than inside domains. The magnetization is non-uniform around the domain wall, and the electrons move in the field with changing direction. Thus, there is a possibility o f the appearance of localized type superconductivity around walls, in the case when Cooper pairing is suppressed inside domains. The first attempt to describe this effect quantitatively was made by Kopaev [69]. He considered a ferromagnet with a linear type of domain wall, as described by (4.7). In the centre of such a wall the magnetization is very small, and Kopaev proposed that superconductivity may exist in the region of small magnetization. He assumed that the superconductivity was localized on a scale much smaller than ~(T), and in the framework of this assumption, found the criterion for the appearance of superconductivity. However, proximity effects were not taken into account. The idea for the existence of superconductivity inside domain walls (superconducting domain walls (S-DW)) was renewed in connection with experimental studies of conductivity of the F N phase in ErRh4B 4. Fertig et al. [33] observed that below To2, and down to zero temperature, the resistivity of the sample is about 40% of the resistivity above Tc~. This normal value is achieved only at high magnetic fields H > 5 kOe. An analogous effect was also observed in HoMo6Ss, but there the resistivity below To2 is higher: ~(T < To2) ~ 0 ' 9 ~ ( T > Tcj). It reaches its normal value in fields H > 0.7kOe [19, 171]. Tachiki et al. [70] proposed that the decrease of resistivity below T¢2 is due to superconductivity localized inside the walls. There it is supposed that the thickness of the wall is much larger than ~(T), and only the EM mechanism is taken into account. In [70] the rotating Boch wall were considered, and in [71] the formation of S - D W was studied qualitatively within the framework of the EX mechanism, and with the domain thickness much smaller than ~(T). This assumption is indeed correct, since in real compounds the wall thickness is much smaller than ~(T), and at the same time superconductivity is localized on a scale which is equal or larger than ~(T). Therefore, the type of domain wall (linear or rotating) does not affect the final results. In this section we find the conditions for the appearance of the S-DW, approximating the magnetization near the domain wall by the step function S~(x) = S sgn x. The wall lies in the plane x = 0. We shall also neglect the exchange scattering of electrons on localized moments assuming the condition z~ t ,~ Tcj. Magnetic scattering narrows the region of the existence of localized superconductivity, and therefore, the conditions for the appearance of S - D W are most favourable in compounds with


241

Our analysis is based on Eilenberger's equations (4.3), and in what follows we shall find the condition for the appearance of superconducting nucleus localized around the domain wall. For this, it is sufficient to solve equations (4.3) in linear approximation. From (4.3), putting g = sgn co, h(x) = hoSE(T) sgn x, Ay = - B ( 0 ) S E So (sgn t) dt, and Ax = Az = 0, one obtains the linear integral equation for A(r). The condition for the existence of the solution, with the boundary conditions A ~ 0 when x ~ _+ ~ , gives the critical value S ~ ( T ) for the relative magnetization SE(T) in the F N phase. Superconductivity localized on domain walls does exist if SF(T) < S~(T), and if this interval does not coincide with the interval for a superconducting solution in the bulk. Since we are interested in the solution in the temperature interval T < T M ,~ Td, it is correct to replace the summation over co by integration. As a result, the value S~c~ does not depend on the temperature. The conditions SE(T) = S ~ gives the temperature for the appearance of a localized solution around the domain wall in the F N phase. In fact, this transition may be of the first order at the point given by the conditions Sv(T) =ScD and SCD > S ~ . We shall study the problem o f the type of the transition to the localized state more completely below. Let us consider the relative role of the EX and EM interactions in the problem of localized superconductivity. Comparing the orbital EM term evA~c, IAI ~ eB(O)SE~o/C, with the EX term h = hoSE in (4.3 a), we conclude that their relative role is determined by the parameter Q = (2eB(O)VE~o/hoc)2. In the case of dirty superconductors 40 should be replaced by (~0l) ½. Since in real compounds such as ErRh4B 4 and HoM06S8, Q < (B(O)Ao/hoH*2(O))2 ,~ l it is correct to retain only the EX term, as is pointed out in [71]. This case is the subject of the next section. 7.2. Localized superconducting solution First, we shall study the case of clean superconductors. To find the f u n c t i o n f f r o m (4.3), we use the continuity condition o f f at x = 0, and the boundary conditions f ~ 0 f o r x ~ _+ ~ . The linear integral equation for A in the momentum representation has the form 2 A(k) = ~ f d G A(q)K(k, q)

K(k, q)

1

f~d~

dp f~/~ (7.1)

x exp(i(kxx-qx2)f2 =

f-~~ l x -

~o + ik,(1 - /t2) ½costp,

2l) c ° s ( )w1. ,- ,T Z - h(t) , [ ~ dt , k,

=

(0, ky, ks),

where all moments are expressed in units 2h/VE. Further, we extract the part K0 from the kernel K, which corresponds to the uniform exchange field h(x) = h = constant. Ko(k,q)

L oo

]

= 2 n f ( k x - qx) l n ~ - + L(k) , (7.2)

L(k) -

1

2 f ~ l n [ 1 - #2(k~ + k~)]d/~

I

242

L. B. Bulaevskii, A. I. Buzdin, M . L. Kulik and S. V. Panjukov

In the part (K - K0) it is correct to integrate over co in the interval ( - oo, oo). This gives the following equation for A(k) [L(k) - E]A(k)

=

E =

K,(k,q)

-

1 9 2-~ J dqx K1(k, q)A(q)

(7.3 a)

- In 2h A0 2(kx + qx)

1 - ~ kx - +2 - q~)

d~sgn#/

x [P(/~, k) + P(/t, q)] P(#, k) = o~+_ =

(7.3 b)

arcsin [2(1 - / t k x ) / ( l a t + l + 1 - pkx +_ kl(1 - p2)~,

I=-I)] K, =

K-

K0.

In the absence of the wall/(i = 0, and the function L(k) has the maximum value at Ikl = 1.2. This gives the bulk solution A(k) ~ 3(k - k0), Ikl --- 1, 2. The critical value of the exchange field for this solution to be realized in the bulk is h~a) = 0.754 A0 (see [15, 104]). In (7.3) k± is the parameter and the 'ground state' eigenvalue E(k) should be minimized with respect to k±. This minimization procedure gives the critical exchange field h~D~ for the localized solution. The localized solution with h~D) > h(c~ still exists, due to the one-dimensional character of this eigenvalue equation. The numerical solution of this problem gives kx = 1.1, h(cWD ) = 0'794 A0, and the solution A(x) is shown in figure 34, curve (a). Note that the localization length is large in comparison with ~0, and the difference between htcW~and h~a) is small. So the localized superconducting solution in the clean

/

J

6

lI

b~ lla

a\

\b

(a) 0.265 x/r.o. (b) 06 x/

III 2

///

h(x) +h

X

-h Figure 34. Coordinate dependence of the superconducting order parameter A(x) in the presence of magnetic domain wall.

Coex&tence of superconductivity and magnetism

243

case, corresponds to the bound state of a particle in a weak one-dimensional attractive potential. In the case of dirty superconductors (l ,~ C0) the upper critical exchange field for the formation of a superconducting nucleus does not depend essentially on the L O F F wave-vector k± [110]. Therefore, in studying the S - D W we may assume that A(r) and jr(r) depend on x only. From (4.3) we get Usadel's equation f o r f ( x )

(

.2)

Iogl + ih(x) sgn co - -}v~z ~

f(x) = A(x),

(7.4)

with the continuity condition for f(x) and its derivative at x = 0. As a result we obtain equation (7.3 a) where L(kx) and K, (kx, q~) are given by

L(kx) = ½1n(l + k 4) b(t)(l + fl)½ dt

K,(kx, q~) = 4Io~ [1 + (t + kx)2][1 + (t + qx)2]

b~(t)

=

½[(1 + t~) ~ -

(7.5)

t].

Numerical solution of hcD (w) gives • h(w) c D = 0'717 A0. The function A(x) is presented in figure 34, curve (b). The value of h~D) is markedly different from the corresponding critical exchange field for the bulk solution ,cB~'(w)= 0.5 A0. Then, the localization length in dirty compounds is of the order of C ~ (Col)~. 7.3. Type of transition to S-DW state and region of its existence What we found above is the 'overheating' field h~'~ of the normal state near the domain wall. However, the transformation into the S - D W state may be of the first order. To solve this, let us examine the sign of the fourth order term of the superconducting functional at the temperature T = Tc~. The second order term is given by the expression F{A} = K(k, q) =

A

N(O)2 /dkx f dqxK(k, A (2n)

q)A(k)A(q)

2n[L(k) - E]a(kx - qx) + Kl(k, q),

= 2 ~ 'VF

l>> ~o,

A =

~'

k± = q~

(7.6)

l ~ Co,

where L(k) and Kl(k, q) are given by (7.3) and (7.5) for clean and dirty systems respectively. To calculate the fourth order term in A(k) we may take h(x) as fixed equal to h0 sgn x. Moreover, it is necessary to take into account the change of the magnetization S(x) under the influence of superconductivity. The first term, at fixed h(x), is of the order N(0)CA4/A02. The additional term is always negative and we shall see that it dominates if h0 >> A0. Actually, it is of the order of (6F/6S)2/(f2FM/fSZ), where F(S) is determined by (7.6) and FM(S) is the magnetic functional. The derivatives should be taken as S = S ~ . At small (A0/h0) we have S ~ ,~ (A0/h0) "~ 1, and the additional negative term is of the order of (0exN(0)A 4C/A02S 20), which dominates since S 2 ,~ 1. Then, at h0 >> A0 the transition must be of the first order and hcD may be

244


higher than h~c~. Besides that, the stable S - D W exists at h = h0 S(T) < hcD if the F N state (inside the domain) is stable. As known from the previous analysis, the F N phase is stable at h > h¢2. Then, the S - D W state exists in such compounds where hcD > h~2. In regular ferromagnetic superconductors the value of h~2 is determined by the expression (4.14) for the first-order phase transition. Certainly h~2 > h ~ ) there, but the relation between h¢2 and hCD is unknown. It seems, however, that h~2 > h~D too, because h0D may only differ from h ~ ~ by a numerical factor which is o f the order of unity. In irregular ternary magnetic superconductors, like Ho,_xY~Rh4B4 and H o l _ x Y x M O 6 S S , the coexistence phase should be absent for values x > x¢ which are sufficient to destroy the coherent DS phase. This is due to the random magnetic interactions. For x > x¢ the first-order transition S - F N takes place. At the temperature T~2 of this transition, the superconducting condensation energy should be equal to the magnetic energy --. OS 2. So, at the point T¢2 we obtain

{ 0ex

(7.7)

hcz "~ Ao \ON-~)A~J "

We see that he2 < A 0 ~ h ~ ) if 0ex < OA~/h2. The latter condition may be fulfilled in real compounds with h0 >> A0 if (0ex/0) is small, such as in compounds where the EM mechanism gives the main contribution to the Curie temperature 0. In HoMo6Ss, 0ex ~ 0.08K, 0 ~ 0-7K and (O~x/ON(O)A~) ~ ,.~ 1.4. Then in Ho~_xYxM06S8 we may hope to obtain the S - D W at concentrations x > xc. The results concerning the S - D W are shown in figure 35 for regular (a) and irregular (b) compounds with dominant EX mechanisms. In any case the temperature interval where the S - D W exists is very narrow, i.e., 0 - T ~ O(Ao/ho) 2. This means that in real compounds with h0 >> A0, experimental observation of the S - D W appears very difficult. The results obtained above indicate that the additional conductivity observed in HoM06S 8 below To2, is probably related to the metastable DS phase and/or to the inclusion of a nonferromagnetic superconducting phase. Such a phase in ErRh4B4 might be tetragonal body-centred ErRh4B4.

FN (a)

~

FN (b)

[~

I

II

Tc2

"- ''4T,M 'cD

~ I T(Wl "C D Tc 2

S

I N

=

Tc,

S

IN -]'Cl

=-

Figure 35. Regions of stability of phases N, S, DS and FN in (a) regular, and (b) irregular magnetic systems. In case (a) the temperature Tc~ of S-DW formation lies in the region of the DS phase. In case (b) the DS phase is absent, T(c~could lie in the region of the FN phase and S-DW is possible there.

Coexistence of superconductivity and magnet&m 8.

245

A n t i f e r r o m a g n e t i c superconductors

8.1. Interaction of superconductivity and antiferromagnetic ordering As already pointed out in §2.1, most of the known magnetic superconductors belong to the antiferromagnetic (AF) class (see tables 1 and 2). In the introduction we noted that superconducting and antiferromagnetic orderings affect each other weakly. So Cooper pairing hardly changes the electronic spin susceptibility at wave-vectors of the order of the antiferromagnetic ordering wave-vector G. In that way, the EX interaction is practically the same in superconducting and in normal-antiferromagnetic phases The short-range part of the EM interaction is also practically unchanged in the presence of superconductivity to an accuracy of the order (a/)],L) 2. Considering the effect of antiferromagnetism on superconductivity quantitatively, we have to study two principal mechanisms. (a) The splitting of electronic levels under the influence of the exchange field generated by LMs. As a result the gap opens at a small part of the Fermi surface only, thus lowering the total density of electronic states [10, 129, 149, 150, 152, 153]. (b) The magnetic scattering of electrons on spin fluctuations above the N~el temperature TN, and on spin waves below TN [102, 152-154]. Let us start by considering the effect of the exchange field on the superconducting order parameter. Generally speaking, it is necessary to use Gor'kov's equations for superconductivity in the presence of an exchange field rapidly oscillating on an atomic scale. Eilenberger's equations are applicable only in the quasiclassical case when external fields vary slowly on an atomic scale. However, with an accuracy up to numerical coefficients, Eilenberger's equations give a correct description of the interaction between superconductivity and antiferromagnetism. So, in the functional F~,t we have to replace q by the antiferromagnetic wave-vector G/2. In the case of dirty superconductors one concludes, using (4.9) for F~=,, that the superconducting order parameter is decreased by 6A under the influence of antiferromagnetism, i.e., 6A --

A0

h2 ~ - -

TN

AovFG

~--.

Tc

(8.1)

This uses the fact that for the antiferromagnet vvG ~ N J(0). The expression (4.12) for F~,t must be used in the case of clean superconductors. As a result, the decrease of A0 when h > A0 is given by 6A Ao --

h

h

(8.2)

In A o

As follows from (8.2), the effect of the AF exchange field on superconductivity is very weak, because (h/vvG) ,~ 10 2. Comparing (8.1) and (8.2), one concludes that nonmagnetic impurities in the AF state increase the depairing effect of the exchange field by (h/Ao) times, at low temperatures T ~ TN. The breakdown of the Anderson theorem in the AF superconductors, as well as the suppression of superconductivity by non-magnetic impurities, has been investigated by Morozov [155], by Zwicknagel and Fulde [149] and by Suzumura and Nagi [156]. Generally speaking, in the AF superconductors a non-uniform component of A appears with the wave-vectors 2kG, besides the uniform one. From Appendices A and B it follows that their amplitudes are small, i.e., Aq ~ (h/vvG)A ,~ A. This means that pairing effects with non-zero momentum may be neglected in the study of

246

L. B. Bulaevskii, A. I. Buzdin, M. L. Kuli? and S. V. Panjukov

superconducting properties. This is the reason that results [150] based on uniform pairing, and those [10, 149] where non-uniform pairing is also included, are the same. The characteristic inverse time of magnetic scattering z~-~ at T > TN, is of the order of TN. In the static approximation, with a contact exchange interaction and isotropic Fermi surface, it is given by the expression (3.2). Approaching the A F transition point the maximum of ( J q d q) moves toward larger q, i.e., to q ~ G where the g(q) factor is small. According to the sum rules (3.3) the contribution of long-wavelength fluctuations is decreased, thus decreasing z-l. In that way the AF order leads to an increase of the superconducting order parameter, in the case of isotropic Fermi surface, at temperatures below TN [102]. However, as shown by Ramakrishnan and Varma [154], when the form of the Fermi surface is more complex, for instance if there is nesting effect, Ts I may be increased near TN. Then, the AF ordering may either increase or decrease the superconducting order parameter, depending on the form of Fermi surface. We emphasize that the orbital effects of the AF ordering vanish. The change in A is given by 6A O(a/,~,L) 2 ~. A0, so that the orbital effects are negligible. We also conclude that, depending on the cleanness of the crystal, the destructive effect of magnetic scattering in the AF state exceeds, or is equivalent to, the effect of the exchange field. Therefore, in practically all existing AF superconductors, Tc > TN, except in pseudoternary systems RE(Rhl xIrx)4B4 with RE = Ho at x > 0.6, and RE = Tb at x > 0.2 [31, 157, 158, 173], where the situation with TN > T¢ is realized. Thus we see that the essential difference between A F and ferromagnetic superconductors is that in ferromagnetic superconductors a new magnetic structure appears in the coexistence phase. In antiferromagnets the superconductivity does not alter the magnetic structure, except in the case of A F with weak ferromagnetism (see below). ~

8.2. Upper critical magnetic field Hc2 in A F superconductors The effect of the occurrence of antiferromagnetic ordering below TN is remarkably pronounced in the temperature dependence of the upper critical field He2. In the I1| I0 ]-

i

I

I

I

I

I

I

I

I

.~'~.

\

\

\.

\ \

\ \ 0

1

2

I

I

I

I

3

z.

5

6

I

7

I

8

I'N

9

I0

TEMPERATURE (K)

Figure 36. Upper critical field versus temperature in polycrystalline TmRh4B4 according to measurements of resistance [44].


247

2O0O H

[oe]

1500

o

~00000 0 0

I000

0

%o0 0

500

015

1

1.5

2

2.5

TtK] Figure 37. He2 versus temperature in polycrystalline SmRh4B4 according to measurements of conductivity [37]. presence of a magnetic field the exchange field h = h0ZmH appears. Hc2 depends on the exchange scattering, the exchange field h (paramagnetic effect), and the magnetic induction B (orbital effect). The susceptibility ~m is finite at TN, and He2 never goes to zero in AF superconductors, except at Tc. In compounds where zs J "~ TN "~ To, the exchange scattering is unimportant, and H¢2 falls off as the temperature decreases to TN. This is due to the increase of the susceptibility Zm, which reaches its maximum value at TN. The curve H~z(T) for a polycrystalline sample of TmRhaB 4 is shown in figure 36, and corresponds to the above qualitative analysis. The quantitative analysis, presented in [113, 114], gives h0 = 14K and 0ex ~ 0.1 K, while 0,x ~ 0.3K in this compound. In the case of compounds with z~-~ ,~ To0, the exchange scattering dominates in the determination of H¢2. This scattering weakens as antiferromagnetic ordering increases. As a result, the behaviour of H¢2(T) is more complicated. The curve He2 for SmRh4B4 [31'] is shown in figure 37. It is obvious that the appearance of the antiferromagnetic state is manifested through the slope discontinuity, rather than through the appearance of a bell-like curve. Grey et al. [159] measured the maximum of the Josephson current versus temperature in a tunnel junction with SmRh4Ba electrodes. The discontinuity of the slope at TN, which mimics that for H¢2(T), was observed. These data revealed that the pair density follows the same behaviour of H¢2, and is enhanced below TN due to the suppression of magnetic scattering. 8.3. Coexistence phase of superconducting systems with weak ferromagnetism In contrast to the usual antiferromagnets where the magnetic moments of sublattices are fully compensated, there is no full compensation in the case of weak ferromagnetism (WF) [167, 168]. This is connected with the relativistic interaction which weakly destroys the antiferromagnetic structure established by the exchange interaction. In the language of magnetic functionals for antiferromagnets with two sublattices, the appearance of WF is connected with the presence of the invariant Fwr = D[M~ × M2], where D is the vector, and M~ and M2 are sublattice magnetizations. The form of FWF shows that Mt and M2 tend to be mutually perpendicular which breaks down the collinear antiferromagnetic structure. As a result of the compromise between the

248

L. B. Bulaevskii, A. I. Buzdin, M. L.. KuliO and S. V. Panjukov

exchange and relativistic interactions, a cant emerges between M~ and M2, which leads to the WF structure. The existence of the invariant FwF is due to the crystal symmetry; the corresponding theory of the WF structure based on symmetry arguments has been given by Dzjaloshinskii [160]. It results in the appearance of the net, although weak, magnetic moment M(~/~L), where L is the antiferromagnetic order parameter, and fl ,~ 1, because the interactions responsible for this phenomenon are relatively small. In the case of ferromagnetic superconductors, the only coexistence phase is the DS phase. Contrary to this situation, the phase diagrams of superconducting systems with weak ferromagnetism (WFS) contain much more information. Weak ferromagnetism is due to the presence of the mixed invariant ( ,-~ ML) in the free-energy functional, which is determined by symmetry. A suitable candidate for WFS in ternary superconductors seems to be the tetragonal body centred (t.b.c.) phase of ErRh4B4. This structure is presented in figure 6b, where the labels 1-4 indicate the magnetic sublattices of the structure. Hence, the magnetic ordering of this system is described by one ferromagnetic vector M = M~ + M2 + M3 + M4 and three antiferromagnetic vectors; L~ = Mt + M2 - M3 - M4, L2 -- Mt - M2 + M 3 - - M 4 and, L3 = Ml -- M2 - M3 -t- M4. The symmetry point group allows the determination of the form of magnetic functional and of the possible types of ordering. As follows from the symmetry analysis [161], the weak ferromagnetism appears when LI lies in the basal (x, y) plane, and the corresponding mixed invariant is fl(MixLly + M~,L~x). The Ll direction is determined by the anisotropy of the system and its explicit form is not essential, although here we will assume that L~ is directed along the y axis. Then M~ is along the x axis, and the subscript 1 will be omitted in further text. The corresponding magnetic functional is

Fm

=

f d3r { nOex[ ½AI2 + ~c 02) 2 + ~b m 2 + flmxly + a2(Vl)21

B2

+ ~

}

BM + 2rcM 2 .

(8.3)

where I and m are normalized to the maximal value M(0) = ~i IMg(0)l, i.e., I = L/M(0) and m = M/M(0). The parameters b and c are of the order of unity, while fl ,~ 1. In the absence of superconductivity (8.3) describes the WF with rn = ill~b, and with the transition point TN. The characteristic value of/~ is 10 -2 10 -3 and consequently the ferromagnetic moment is very weak. Thus it cannot destroy superconductivity, because the exchange field is smaller than Tc and the maximal induction Bmax = 47tM does not exceed the upper critical field He2. Therefore, one can describe the interplay between superconductivity and magnetism by means of perturbation theory using F,,t in the following form:

Fint =

q~I½Q~(q)AqA. + 0¢,,Zn(q)Z~(q)mqm_q ] Zn(0)

(8.4)

which uses the notation introduced after equation (3.12). The total functional of the system is F = Fm + Fs + Fret, where F~ has the usual form which is inessential for


249

further consideration. Minimizing F with respect to m one gets the relation between m and I. Then, eliminating m from F gives, r{Iq}

R(q)

= "~ O~x ½A - ~f12 + a2q 2 +

=

1 -

/~2R(q) 2b(b

¥

~lql q + F s

(8.5)

zs(q) 4nQ~(q)0=m -+ X,(0) [q2 + 4nQ~(q)]0e x,

where 0era = 2rcM2(O)/n and quartic terms are omitted. Comparing this expression with the corresponding one for ferromagnetic superconductors, we see that the only difference between these two cases is the much smaller interaction energy between superconductivity and weak ferromagnetism (,,~f12/b2). The other terms are approximately the same. From (8.5) one concludes that in the description of the WFS we may use ferromagnetic superconductor theory with effective magnetic stiffness a = ab/[L An interesting point is that the inequalities ~ a. If by further lowering the temperature the induction B = 4riM exceeds Hc~, then the MS phase transforms into the vortex phase. The full phase diagram for the WFS systems in the plane (T, fl) was obtained by Buzdin et al. [161}, and is presented schematically in figure 38. 9.

Conclusions

In concluding this review we shall reiterate the most interesting physical aspects of magnetic superconductors with localized moments. (a) In magnetic superconductors in magnetic fields, and near the magnetic transition point, several different phases can be observed. All known magnetic superconductors are type II superconductors, but at temperatures slightly above TM in ferromagnets, and near TN in antiferromagnets, they can become type I super-

250

L. B. Bulaevskii, A. I. Buzdin, M. L. Kulik and S. II. Panjukov

T TN

A DS

FS

D /3"-Figure 38. Phasediagram of weak FS in the (T, fl) plane, where fl is the parameter of weak ferromagnetism. DS is the superconducting phase with magnetic domain structure for the AF vector I and the F vector m; directions of I (and m) are opposite in neighbouring domains. VS is the spontaneous vortex phase, and FS is the Meissner phase with weak ferromagnetism. conductors. Such behaviour depends on demagnetization factors, and on the relation between parameters of the EX and EM interactions. This effect was observed in ErRh4B4. (b) In clean superconductors in a magnetic field directed along an easy-magnetization axis, the phase with a non-uniform superconducting order parameter, the LOFF state, can be realized in the same temperature interval. In usual superconductors such a phase is practically unattainable, which means that magnetic superconductors offer a unique possibility for its experimental investigation. In particular, the LOFF state should be observable in samples of ErRh4B4 prepared in the form of plates and placed in a field which is perpendicular to the plate. As we have pointed out, detailed theoretical analysis of the LOFF state does not exist as yet. (c) The coexistence phase with domain-like transverse magnetic structure should be realized below T u ( < Tel) in anisotropic ferromagnetic superconductors with regular lattices of magnetic ions. This is the DS phase, and experimental data give evidence that it is realized in HoM06Ss, HoMo6Se8 and ErRh4B 4. In clean re-entrant compounds superconductivity has gapless character in the coexistence phase, although in ErRh4 B4 it is present only in a small part of the sample. This is probably due to the presence of magnetic disorder in real samples. Unfortunately, HoM06S8 and HoMo6Se 8 have been synthesized as polycrystals. (d) In superconductors with weak ferromagnetism the DS phase, the spontaneous vortex phase, and the Meissner ferromagnetic phase are possible. To date, weak ferromaggnetism in ternary superconductors has not been observed, although the body centred tetragonal phase of ErRh4B4 with antiferromagnetic ordering was discovered recently, and it is possible that it will be found to be weakly ferromagnetic. (e) Laser irradiation can be used to destroy superconductivity, partially or completely, without seriously affecting the magnetic ordering (see [162]). Thus, there is a real possibility that the magnetic structure can be investigated in the absence of superconductivity, and in the temperature interval where the coexistence phase is realized. Such experiments would allow the investigation of the influence of superconductivity on magnetic ordering. We note that the DS-FN transition may be observed


251

optically, because in the FN phase the reflection of light depends on its polarization, while in the DS phase it does not. We stress here that a number of experimental and theoretical problems remain unresolved, such as the puzzling behaviour of ErRh4B4 below TM. Most researchers in this field believe that this compound plays a key role in the investigation of the problem of coexistence of ferromagnetism and superconductivity. The hypothesis concerning the realization of asperomagnetism in existing monocrystals of ErRh4B4 can explain the experimental facts, although the assumption of strong magnetic disorder in sufficiently regular monocrystals is at first sight only slightly possible. In near future we shall, no doubt, find the answers concerning magnetic ordering in the FN phase of this compound by using diffuse neutron scattering, M6ssbauer spectroscopy in a magnetic field, or N M R spectroscopy of hyperfine fields on JtB. Analogous measurements have been made by Kohara et al. [147]. Experiments using applied uniaxial stresses to monocrystals may help to clarify this situation.

Acknowledgments We are grateful to V. L. Ginzburg for his continuing interest in this problem, for reading the manuscript and for his critical remarks. We would like to acknowledge A. A. Abrikosov for reading the manuscript and for his valuable comments. We are also grateful to S. V. Brazovskii, S. S. Crotov, L. R. Golubovi6, D. E. Khmelnitskii, D. I. Khomskii, D . A . Kirzhnits, Yu. V. Kopaev, I.O. Kulik, A.I. Larkin, V. L. Pokrorskii, A. I. Rusinov, and E. F. Shender for useful discussions and critical remarks, and we would like to thank Dr H. R. Ott, from ETH-Zfirich, for a useful discussion about the experimental data of ErRh4B4. The help given by Dr P. Senjanovi6 and Dr N. ~vraki6 in reading the manuscript is highly appreciated. One of us (M.L.K.) would like to thank Theoretical Department, P. N. Lebedev Physical Institute, Moscow, for the hospitality he enjoyed during his visits. This work is supported in part by RZN Serbia and by Serbian Academy of Arts and Sciences. Appendix A Superconducting functional for dirty superconductors The field h(r) is a periodic function with period 2~Q-t which allows the following Fourier expansions: 1 ih(r) = ~ ~ h2k+,[exp (iOr(2k + 1)) - exp ( - i Q r ( 2 k + 1))], k=0

f (u, r) = f0(v) + ~ fk(q) exp (ikOr),

(A 1)

k#0

A(r) = A + ~ Akexp(ikQr), k~0

and analogously f o r f ÷ and g. Neglecting in (4.3) the terms quadratic in gk andfk with k # 0), which are small because (hz) 2 is small, one obtains for k # 0,

gk(V)

-

fo(v) [ A ( v ) - A ( 2g0(v)

v)]

252

L. B. Bulaevskii, A. I. Buzdin, M. L. Kulik and S. V. Panjukov i

L(v) = ½&(v)I1 Sk(v)

k(vO)

2

]

(A2)

1 ~o + ~ g 0

_ 2Zfo(V)go(V)/~k

=

1 + kEz2(V(~) 2

arctan Qlk~- J

/~k = hk (1

/-

Using (A 2), one obtains the equations for go and f0, gg0(v) + f~0(v)

=

(A 3)

1

(~ + ~-~g,o)fo(V) -- (A + l fo) go(v)

= ½ ~o ~7 ~&(v)

After averaging over the angle of u one obtains (L) 2 + (&)~ =

l,

~of0- Ago =

- 2f0g0Tm' }

+ k2z2(v(~) 2

Using (A 4), and the condition COD >> A0, we obtain from (4.3 e) the following selfconsistency equation for A (at T = 0). A°

ln-~--R(x)

= O, x = (zmA)-l

R(x) = --4-' x 1.

Now we can determine the free-energy functional. It follows from (4.4) that it has the form F{A} = f d 3 r I ~ +

FI(A)I,

(A6)

where F~(A) does not depend on Z explicitly. This representation, together with the condition that minimization of F{A} with respect to A should give the self-consistency condition (A 5), allows us to determine F{A} unambiguously. It can be obtained by multiplying (A 5) by 2N(0)A, and integrating over A from zero to A. F~(A) =

eAo~

(A 7)

-- ½N(0)A2 In A2

Fint(A, Zm) = N(0) (:~Am

312m),

"t'mA > 1.

Coexistence of superconductivityand magnetism

253

Integrating over v directions, in the case of an isotropic Fermi surface, gives the expression (4.9). From (A 5) and (A 7) we see that the effect of magnetic structure of the DS phase on superconductivity is similar to the effect of magnetic impurities with magnetic scattering time rs [8, 127]. By this reasoning, the effect of magnetic scattering on the DS phase can be taken into account replacing by ,~ l in (A 7) by (ZmI + T~-I). rs is the exchange scattering time, and is due to the spin excitations in the DS phase.

Appendix B Superconducting functional for clean superconductors The first two equations (4.3) with ~- ~ = 0 can be rewritten in the following form:

f~ = {a+ib+ f~dxA(x)g(x)exp(+_ fod~[~o+ih(O])}

where t = 2x/vx, a + ib = f+(O). Using f ± (0) = f-+ (2h), where ti = 2d/% gives:

the

1 a + ib =

-+ exp(+ 2~Otl) - 1

condition

,2j0,, dx a(x)

of

periodicity

g(x)

x exp (__+ fo d~ [09+ ih(z)]). From (4.3 c) there follows the equation for

(B2)

g(t)

g2(t)+{a+ib+f~dxA(x)g(x)exPfodZ[og+ih(O] } x {a-

ib-

[:dx A(x)g(x)exp-

;o dr [¢o + ih(0]}

1.

(83)

The integral terms are of the order (A/h) ,~ 1, if the inhomogeneous parts of A and g are small. This will be proved to be true, because 7 = (rch/VFQ) is small, so assuming this, we use perturbation theory. To first order in (A/h) g and A are constant, where g = (1 - a2 b2) ½. The integral terms with this correction may be omitted in (B 1). Thus, (4.3 e) and (B 2) give -

A ( x )2-

A

Aln ~A =

N(0)

f2,Dde) f_d

[f (x)

a ]

(A2 + (.o2)½ ,

(B 4) f

(t)

=

a + ifl

(1 + ~2 + f12)~

exp ( -

f~ dr [~0+ ih(T)])

where ~ and fl are given by the expression (4.10 c) and A is the mean value of A(x) over the domain. Replacing A(x) by A in (B 4) gives the self-consistency equations (4.10), and the functional (4.12) to first order in 7Making the Fourier transform of (A(x) - A) in (B4) gives the self-consistency equation for Fourier components Ak(k # 0)

254

L. B. Bulaevskii, A. L Buzdin, M. L. Kuli? and S. V. Panjukov A2, In 2kvrQ ~-eA°

-

(-- 1)* ~

7

h In ~ ,

A2k+,

=

(B 5)

0.

The final expression for F~°, is Fint

=

nZ hA 2 che ½ N(0) 3-~vQ In - ~ 2kvv Q + N(0) ~ A2, A2k In - e--A0 + ( - 1) k 4vr:k2------~ In k#0

.

(B6)

In the equilibrium state the sum over k in (B 6) is proportional to ~;2, which can be neglected because 72 ,~ 1. Fin, is calculated also for the spiral state S(r) = S(cos Or, sin Or, 0) when h >> A [14]. It has the form ~p*r,~ = N(0) nhA-----~2In 8h ~t int "UF Q e½--~ •

(B 7)

Comparing (B 6) and (B 7) one concludes that the difference between them is small.

Appendix C Phase transition near tricritical point Near the transition point T~] the G - L functional may be used for Fs, taking into account the magnetic scattering [127], Fs = N(O)b,(o) [ T "Te] - TC' A 2 b2(~ )

=

_

b,(o)

=

1 -

½~/(2)(½ _~ O) - -

7!-

b2(0) 20T¢ l b] (0)

A4

Q qj(3)(½ + Q)

e q / ' ) ( ½ + a),

e

=

]

,

(C 1)

(2=rsL,)

',

where ¢(") is a polygamma function. The magnetic functional FM can be taken in the form (4.5 b), while the functional Fiat, in the case of a clean superconductor, has the form [27], Fiat

=

0ex X

Xn(q) -- Zs(q) SzqS z

.

~.T0i

'

"

(C 2)

x.(q) xs(q) ~.(0) -

-

n2AZ - cosh-I (nO). 4vr q T¢l

Minimizing over q gives the wave-vector Q of the structure, and the magnetic transition temperature, Q3 =

n2 N 8vv Tel a 2 cosh(n0)

TM =

0

1 -- 3 1_8~"FL-S~h(nQ)_I J"

(C3)


255

The expressions (C3) are applicable when Q ,> Co l, i.e. for (Tet - O)/Te~ >> (a/¢o) 2. If this condition is not fulfilled, then we have a second-order transition to a superconducting phase with ferromagnetic order. However, in that region of parameters the first-order transition S - F N takes place at higher temperature (as will be shown in what follows), and the Lifshitz point (L) lies on the supercooling line of the S phase. The first-order transition line S-FN, where Fs = FFN, is given by

Comparing (C 3) and (C 4) we find the condition for the first-order phase transition; -

-

Tel

1

(D6)

Jl d In S2(T)l ((T)

= aoSZ(T)

1 + 20

-d~nT

J

The transformation of Q ~ [((T)] -½ on cooling is shown in figure 27, and is well described by the approximate formula for ((T), which interpolates (D 5) and (D 6), ((T)

= aoS2(T)Ob

d In $21 1 - b din T] (D 7)

b

J* 0

There is a relation between a and ao, i.e. a = 3aob½(O/O~xxf2).

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

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