Interplay between magnetism and superconductivity

This is a review of the recent developments in the study o f magnetic superconducting materials. Particular emphasis is given to our recent theoretical work which is a result of collaboration of H. Umezawa, H. Matsumoto and M. Tachiki.

Interplay between magnetism and superconductivity H, Matsumoto and H. Umezawa Key words: superconductor, magnetic, materials

The problem of the interplay between magnetism and superconductivity has attracted much attention from physicists since as early as 1957. ~ This is because these two phases tend to destroy each other; the superconducting diamagnetic effect tends to screen the spin effect while the spin effect tends to break the Cooper pairs through scattering of the conduction electrons by the spin magnetic moments. Interactions

Briefly speaking the two systems (the spin system and the conduction electron system) interact through one, the socalled s-f interaction (the interaction between conduction electrons and localized spins), glaBIt~ta~.S, and two, the electromagnetic interaction which consists of the usual minimal interaction, ~ / ( i f , fft).a, and the dipole • . ~ ---9-, . interaction, - glaab'S. Here a ~s the eleetromagneUc vector potential,#is the induction field (~ = V × -~), ~ is the electron, S is the localized spin, I is the s-f coupling constant, g is the Land6 factor of the localized spin and #~ is the Bohr magneton. Until recently, most of the previous experiments were performed using alloys with magnetic impurities. These experiments showed that the presence of the s-f interaction easily quenched the superconductivity? Revival of the subject (since 1978)

This situation has been changed by the recent discovery of rare earth ternary or pseudoternary compounds, a'4 A systematic study has been made on two groups of compounds s ; one is the R h 4 B 4 - g r O u p (ie RERh 4 B4 ; RE - rare earth) and another is the Chevrel compounds of the type REMo6X 8 (X - S or Se). Many of these compounds become superconducting at T < Tc, despite the presence of the rare earth magnetic moments. This seems to suggest that the s-f interaction effect may be small in most of these compounds. Fig. 1 shows the temperature behaviour of Rh 4 Ba-group and that of the Chevrel compounds with X = S. The wavyline means that no stable compounds have been made. The figure shows the following behaviour: in the case of ErRh4 B4 and HoMo 6 $8, when the temperature is cooled below Tc (or Tel, the superconducting transition temperature), a ferromagnetic normal phase appears at a temperature Tc2 (Tc 2 < T c l ) . This is called the re-entrant pherromenon.3'4's The authors are at the Theoretical PhysicsInstitute, The University of Alberta, Edmonton, Alberta T6G 2JI, Canada•Paperreceived 19 February 1982.

Fig. 2 presents the temperature behaviour of the ac susceptibility Xac and resistivity for ErRh4 B4.3 This clearly exhibits the re-entrant phenomenon. The hysteresis at the magnetic transition shows that this phase transition is of first order. In some cases an anti-ferromagnetic phase appears at T = TN (< To) and coexists with the superconductivity. In the Chevrel case, there are many compounds LaRh4BI CeRh4B4 PrRh4B~ NdRh 4B4 Smkh4B4

Eu[th~B4 C;dkh ~B4 TbRh ,B4 Py}th ;Bt }t.}th ,,B~

a

[',,XI ,,hI'r\1 . . . . I.a,%,, >;- t"

NJ:%, ,-,.

*;J\l -. l%Xl,,, m, II,,Xl,,, m l.:rkl,.S.

b

c Tem peratu re, K

Super

Ferro-normal ~

Antiferro-super

Spin-sinusoidal Fig. 1 a - Phasediagram for RERh4B4, b -- Phasediagram for REMo6S8

0011-2275/83/001037-15 $03.00 © 1983 Butterworth & Co (Publishers) Ltd. CRYOGENICS . JANUARY 1983

37

IO0 s

75 50

%©

o

L

-25 -5@

......................

•

3007

V~

I

150 I

,ooI t @

1

I

I

I

I

2

3

4

5

_t

6

7

8

9

10

Temperature. K

Fig. 2

ac susceptibility and resistivity for ErRh4B43

which do not become either ferromagnetic or antiferromagnetic. The reason for this may be that the s-f interaction in the Chevrel compounds is particularly weak. The transition temperature from the superconducting to the magnetic state is lower or equal to 1 K. Considering the large magnetic moment of the rare earth atoms which enhance the dipole interaction, this magnetic transition temperature (~ 1 K) is of a magnitude consistent with the dipole effect. There appears a spin-sinusoidal phase in a small range of temperature above Tc2 in the case of ErRh4B4 and HoMo6Sa. A periodic spin structure with a long period (• 100 A for ErRh4 B4 and * 200 A for HoMo 6 Sa) was observed in neutron diffraction experiments just above Tc2.6'7 Recent experiments s using a single crystal of ErRh4B4 determined Table 1.

Experimental values of ErRh4B 4 and HoM06S 8 ErRh4B4

HoM06S 8

Tetragonal 9

Rhombohedral 1°

a = 5.30 A

a = 6.45 A

c = 7.39 A

~"~89.5 °

N = 9.64 x 1021

N = 3.726 x 1021

J(free)

15/2

8

gJ(free)

9

10

~(free)

9.59

10.60

41rC

2.31 K

1.09 K

5.6 (neutron) 11

9.06 (neutron) 12

Crystal Structure

exp

iU.(eff)

the direction of the wave vector and the magnetic moments, confirming that the spin periodic structure is a sinusoidal one. The transition temperature to the spin periodic phase will be denoted by Tp. These experiments showed also that a ferromagnetic component coexists with the spin periodic part. This is currently one of the most controversial aspects of this subject. Some of the experimental data for the parameters of ErRh4 B4 and HoMo 6 $8 are presented in Table 1. The Curie constant C was calculated by the use of the formula C = [(g/.tB)2J(J + 1)N]/(3kB) for free rare earth ions. The crystal structure of these compounds is shown in Fig. 3 (for Rh4 B4-group, a, and for the Chevrel group, b). More information is obtained from the study of pseudoternary compounds such as RExLul_ x Rh4B 4.5'17 Since lm is non-magnetic, an increase in x would be expected to reduce Te. For all the RE's studied so far the depression rate, (dTc/dX)x,,.o, is qualitatively proportional to the De Genne factor (g - 1) 2 J(J + 1). The deviation from the De Genne's curve is usually assumed to be caused by a crystal field which lifts the 2J + 1 fold degeneracy. A particularly large deviation of the depression rate from the De Genne's curve has been observed in Ce, Eu and Yb. Several mechanisms have been suggested, notably, the Kondo effect, valence mixing and the partial gapping of the Fermi surface.

The s-f interaction Let us now briefly recall some characteristic effects of the s-f interaction, ignoring the electromagnetic interaction. The s-f interaction modifies the single electron state of the conduction electrons through the self-energy diagram of the form t~'~, where the wavy line is the spin-spin correlation function X for the localized spins. In the case of magnetic impurities, X is mostly dispersive (ie, imaginary). However, when the magnetic moments form a lattice, as in the case under consideration, it acquires a significant real component. In both cases the result depends on (12/10×, where Vis the coupling constant of the BCS interaction. As is well known, this self-energy contribution leads to a decrease of both the energy gap and Tc:. Despite the divergence of Xla at T = Tm (the Curie temperature), a careful calculation shows that this divergence does not always quench the superconducting state near Tin. The order parameter A does not vanish even at T =Tm when (I2/V) is smallJ 9'2° RE Rh4B4

REblo6S 8

8.4 (M'ossbauer)13 9.62 (X-1, high T) 3

10.85 (;(-1, high 714

7.65 (X-1, high field) 14

6.0

Tcl

8.7

1.82 K 4' 16

%

0.93 K (warming) 15

0.668 K (warming) 16

0.87 K (cooling) 15

0.612 K (cooling) 16

K 3' 14, 15

(;(-1

high field) 4

~

(single) 0.775 K (warming) a

O RF

0.710 K (cooling) a

T,

38

* 1.0 K

a ,x, 0.7 K

ORh

0 s

O Me

•

RE

• B b

Fig. 3 a - Crystal structure of RERh4B4, b -- Crystal structure of R E M o 6 S 8

CRYOGENICS . J A N U A R Y 1983

The s-f interaction also mediates the spin-spin interaction. The spin-spin interaction (among the localized spins) mediated by the s-f interaction is called the RKKY interaction. The coupling is proportional to 12 Xa where Xa is the e~ectron-spin susceptibility. In a superconductor X,,.(~.) at k "-" 0 is substantially suppressed in contrast to ×oiO ~) m the normal state. This suppression is caused by the energy gap of the electron. In addition since Xo(~ damps at high Ik~l,it will have a maximum at a certain value of Ikl (say kp) in the case of the superconducting electrons. In this way the RKKY interaction, modified by the superconducting effect, creates a kind of spin periodic phase with wave number kp at T < TAS = CI ZXcr(kp). This is called the Anderson-Suhl mechanism. 2z A detailed analysis of the spin-periodic phase caused by the Anderson-Suhl has recently been revived. = The modification of the RKKYeffect by the superconductivity is weakened when there is a strong spin-orbit coupling of the conduction electrons, 23 with the result that the RKKY-interaction is almost the same in both phases. When the coupling among the localized spins arises solely from the s-f interaction, the spin-spin correlation X has the form C[T- C~RKKY ] - 1 , with ')'RKKY = I 2 x o • Since 12 Xo = [2X (Pauli) = I2N(0) in the normal state, where N(0) is the density of states at the Fermi surface, we have Tm/C = VN(O) (12/V). Let us apply this consideration to the ferromagnetic normal state at T < Tc2. Calculating C for Er and using Tm ~- 1 K, we have 47rC/Tm ~- 2. Then, using the typical value VN(O) = 0.3, we obtain (/2/V) ~- 20, which is more than sufficient to destroy the superconducting state at T > Tc2 through the pair breaking mechanism. This rough argument suggests that the electromagnetic interaction is significant. As was pointed out previously, T m ~- 1 K is about the magnitude given by the dipole interaction associated with the large magnetic moment of the rare earth atoms. Therefore, the origin of the spin-sinusoidal phase in ErRh4 B4 and HoMo6 Ss may not be the Anderson-SuN mechanism.

The model We may therefore proceed with the assumption that the electromagnetic interaction plays the dominant role and the weak s-f interaction effect can be considered as a correction. In the following, we concentrate mainly on the electromagnetic effect, although some of the s-f interaction effect may be renormalized into the physical parameters. Notation

Some of the symbols for the various physical quantities are summarized in Table 2. Dimensionless parameters used in the theoretical analysis are listed in Table 3. In the following, a derivative operator (say F(defined through the Fourier representation:

F(-iV) exp(ikx-*-*)=

iV)) is

F(k) exp(ikX~

(1)

When the Fourier representation of a function g(x+) is given by

g(x~) = f d3k Table 2,

exp(i~x-~ g(k*)

(2)

Symbols of physical parameters

T c (or Tel)

Superconducting transition temperature

Tc2

The re-entrant temperature

Tp

Periodic phase transition temperature

TM

Fictitious re-entrant temperature from Meissner to normal

Tm

Fictitious Curie temperature

C

Curie constant

D

Stiffness constant

J

Spin of rare-earth ions

N

Density of the rare earth ions

'Weak s-f interaction

•L(0)

London penetration depth at T = 0 K

We have seen some phenomena which suggest that the s-f interaction is weak in most of the compounds under consideration. A primary reason for the weakness of the s-f interaction is the cluster structure of these compounds. In the Rh4 B4-group and the Chevrel group, the Rh 4 B4 or Mo 6 X 6 units form 'clusters' which surround the rare earth atoms (Fig. 3). The conduction electrons are mainly the 4d-electrons of Rh or Mo. The strength of the s-f interaction depends on the mixing of these conduction electrons and the magnetic moment carried by the 4f-electrons of the rare earth atoms. This mixing is small primarily because the distances between the rare earth atoms and Mo or Rh atoms. The detailed nature of the s-f interaction, however, requires a tedious band-theoretical calculation. 24 The results of such a calculation show that the s-f interaction is indeed quite weak in both the R h 4 B4-group and the Chevrel group. In the Chevrel compounds the s-f interaction is particularly weak. Furthermore, Knight shift experiments 2s show that the spin-orbit coupling interaction of the conduction electrons is quite strong in the ternary compounds under consideration. As it was pointed out previously, this strong spin-orbit coupling may suppress the superconducting effect on the RKKY interaction.

~(0)

Coherence length at T = 0 K

/o

[~(O)XL (0)] ~

CRYOGENICS . J A N U A R Y 1983

Table 3.

Non-dimensional parameters used in the paper

t

T/T~

tp

tp/Tc

tM

TM / T c

tm

Tm/Tc

KB

Landau parameter (~(0)/XL (0))

C

d

47rC/Trn D/Tm X2 (0)

U

gI.LBJN/(~P/X2 (0))

Nmo

NkB Tm/(d#/12) 2 (c = (uo,/J)2 [4~J(J + 1)/3Nmo] )

do

DITto 12

(d = do/~ B )

Uo

gl~BJN/(~p/12o)

(u =/¢BUo)

39

we have

t7(- iV) g(-x) = f dak exp(ik~ F(/~)g(k-+)

(3)

= f day F(Z-y"*)gO'~ where

1

F(~) - (2n) a

fdak F(-~) exp(ik-~

ignored. The terms denoted by the dots in (7) can then be given by a sum of smeared-out non-linear products o f / ° carrying derivatives: G(~) f w(x - Y l , x - Y 2 , • • .) ]°(Ya)/° (.v2) . . . . The derivative G(~) is such that G(0) = 0. The linear current] ° has been calculated in the case of single vortex and it is a smooth function with range XL. Thus the terms represented by the dots in (7) and (8) will be ignored. However, the calculation of free energy takes into account some of the non-linear effect (core energy). Denoting the magnetic field by h, the Maxwell equation reads as

Equation (3) shows how F ( - iV) describes a non-local effect.

~X

h-* = ~4 r r ~1

(9)

C

The Maxwell equation

The magnetic induction is To study the electromagnetic properties it is convenient to use the Maxwell equation: - ~)uGu = J~'

(4)

Here the four dimensional notation was used.

Ju(x) = ~uv(-iO)a~, + . . .

(5)

~uu (P) is the photon self-energy term and its calculation requires particular care in order not to violate certain exact relations. On the one hand the single loop approximation for Zuv is inadequate because it does not give rise to the low energy pole singularity (in ~uv), which is required by the Goldstone theorem. On the other hand, while the approximation involving a chain of such loops does indeed create such a singularity, it does not satisfy the relation pUEm,(p) = 0, which is required by the conservation law of current. We thus modify the result of the chain diagrams by means of the condition pU~uv = 0. When we consider low frequency phenomenon (ie, time dependence is slow), we approxima_te ~uv(P) by the pole term. The result has the form cu~,( - iV)av. In particular, the spatial component is proportional to c ( - i ~ i ~ [ie, ci/(- i~) ~ c ( - i~)8~i] . Briefly, the result for c(k9 is c(~) = e x p ( - ~2 ~'~[2)

(6)

with ~ being the coherence length. Further consideration leads to the persistent current I having the form: 4rrX[

41rX~

~

(11)

We use the mean field approximation to calculate the magnetization: g/sBJ Jn~(~)l = g#B JNBj [ ~B T 'h~m(~) 11 ~ r n

(12)

where h-'m is the molecular field and N is the density of the localized spins. The molecular field, ffrn, acting on the rare earth, is a sum of the exchange field and the magnetic field generated by the persistent current: ~m = V(-- i ~ m

+ ~

(13)

where 3'(- i~) is the exchange interaction coupling,. 3'(~) is usually expressed as 3'(~) = (Tin - Dkz)/C

(14)

where Tm is a fictitious Curie temperature. Contribution

o f t h e phase

f(x)

Equations (9) and (7) show that f(x) contributes t o "-* h and --~b (and therefore, to/-]) only through ~ x ~ f, which vanishes unless f carries certain topological singularities (ie, pathdependent.t). We first study those cases in which no topological objects exist: f = 0. Using (9), (10) and (7), we obtain

"'"

- ~

"'"

where the c ( - i~)a term is the spatial component of Y~av term mentioned above. XL is the London penetration depth and fis the phase of the electron wave function. Let/° denote the current obtained from the Maxwell equation with the terms denoted by the dots in (7) being 40

(E-T.)

n

Here au is the vector potential, and a represents derivative operators.

I -

where r~ is the rare earth magnetic moment:

;,(5) = z g.,, jg'(.,)

Let us first consider the linear au-term in the current Ju:

(10)

b = h + 47rr~ = ~ x a

~" = - 47rF(-i~)t~

(15)

F ( - i~) = e ( - i~)](- X ~ 2 + e ( - i~))

(16)

Then (13) gives ~m = [3'(- i~) - 41rY(-

iff)] 7-n

(17)

This implies that the Fourier component of the effective exchange interaction constant in the superconducting state is changed.from 7(q) to ~ ( ~ :

CRYOGENICS. JANUARY 1983

~'(q~) = 3'(~ -

41rF(q'+)

(18)

~2

The second term represents the screening due to the persistent current. 26

r0

Screening effect and spin-periodic phase Let us consider a ferromagnetic superconductor. T(q) has a maximum at ~ = 0 as shown by the dashed curve in Fig. 4. In the superconducting state the low ~ component of the exchange current is strongly screened by the persistent current, as is shown by the solid curve in Fig. 4.3' (q) has its maximum at a value of b~l (say Q) indicating that a spinperiodic order with wave number Q will appear in the superconducting state. 26'27

splr@l

The transition temperature Tp is given by / /

Tp

=

C'~(Q)

Fig. 5

When (14) is used, we find Q _ lX/XL

8Tp - T m

I~-~Cll/4-- Tp =

(20)

2(4rrCD/XL) Y~

(21)

for large • and (4rrC/D)X~ >> 1. (For other choices of parameters see the original paper) e ) This spin-periodic phase tends to be a spin-spiral phase unless the effect of the magnetocrystalline anistropy is strong. When the magnetocrystalline anisotropy is strong, the component of the spins perpendicular to the easy axis of magnetization is quenched, and a sinusoidal phase becomes more favourable than the spin-spiral order. 2s This sinusoidal order is considered to be the periodic structure of spins obserbed in ErRh 4 B4 and HoMo6 Ss in the neutron diffraction experiments mentioned above. However, this does not explain the existence of the ferromagnetic background which is observed in the neutron diffraction experiments. Fig. 5 presents the free energies of spin-spiral, ferromagnetic normal and paramagnetic super (ie, the Meissner) phases. The energy is normalized in such a way.that the energy of the paramagnetic normal phase is zero. Note that the phase transition for (Meissner -+ spinspiral) is second order, while the transition (spin-spiral -+ ferromagnetic normal) is first order, in agreement with the neutron diffraction experiments.

~'/P/t ~Jc I I I

I I

0 Fig. 4

p

Schematic plot of the screened exchange interaction ~(p)

CRYOGENICS

• JANUARY

1983

Schematic plot of free energies vs temperature

The screening effect and ultrasonic attenuation We have seen that the persistent current screens the exchange. type coupling• Since the persistent current is transverse, only the transverse component of the exchange interaction is screened: therefore, while the transverse spin-fluctuation is diminished by this screening effect, the longitudinal spinfluctuation is not affected at all: x,(k-5 - T - c

x,(k-) =

(22)

c T - C

(23) - 4n]

Calculation shows that in the normal state we have x

c

d) -

T - CT(k) XII (k*) = - C _ _ T - C [ 7 ( k ) - 4~r]

(24) (25)

Although XII is the same in both the normal and superconducting phase, X±(k) in the superconducting phase differs from the one in the normal phase; the difference arises from the screening effect in ~'(k), Thus while x±(k -+)in the region of small k acquires a sudden increase through the transition (super -+ normal), Xll(k) does not change at all. In magnetic superconductors, most of the sound wave energy is dissipated into the spin system through a coupling between the spin and sound wave. The attenuation coefficient is given by a(k)

rm/C-4~

Ferromagnehc-normai

(19)

= A.M

2 Ix(k) l =

(26)

with a constant A. Here x(k) is the susceptibility in the direction of the magnetic field and k is the wave vector of the sound wave. Thus a(k) with the condition Hlk measures X±, while the one with/~l hk measures X,. (To be more precise, we must consider the saturation effect of the magnetization when H is large.) When H is increased beyond Hc2, the transition (super -~ normal) occurs, and the ultrasonic attenuation measurement should exhibit the above behaviour of 7,x and XI1.29This has been confirmed by

41

singularity. This effect tends to induce the ferromagnetic state in the region (with the range ~ ?~L)around a singularity; this is a local ferromagnetic state coexisting with the superconductivity. The range of the ferromagn~ic region is determined by the non-local factor F ( - iV). When the singularities are lines, we obtain vortices, while surface singularities are associated with surface ferromagnetism.

Ero.eHOo.iRh4B 4 15 MHz, L Mode 0.7K 4- O.O05K 0.2

The vortices (the line singularities)

Let us begin with a single vortex with unit flux along the z-axis: o

x i y ~ ' ) = 27re-+z82(~

(28)

The last term in (27) has a finite value at x = 0 because of the presence of the non-local factor F ( - i~). Hereafter the vector notation for field variables is omitted, since the fields are directed along the z-axis.

Hcl

-0'2

L 0

h 2

L

L 4

H, k0e Fig. 6 Magneticfield dependenceof ultrasonicattentuation (from reference 30, Fukase& Woods)

ultrasonic attenuation experiments, a° Fig. 6 presents data from the Fukase-Woods experiment. These experiments represent a direct confirmation of the screening effect in ~'due to the persistent current, thus proving that the electromagnetic interaction is significant in the compounds under study. The ultrasonic attenuation measurement provides us with an excellent method for measuring He2 and also H c x. Screening of the magnetic scattering of the neutron

The magnetic scattering of the neutron is caused by the interaction mediated by the magnetic moments of the neutron and the rare earth ion. This magnetic field is screened by the persistent current. This screening effect introduces the screening factor [s(q)] =(s(q) = [~-~/(~'2 + X~2 c(~))]) in the scattering intensity, and, as a result, the scattering intensity almost vanishes for the scattering wave number much smaller than 2/XL31 [see, Fig. 7].

Since the mutual interaction of two vortices is proportional to the magnetic field, we first study the magnetic field, a2 Fig. 8 is t h e typical form of the magnetic field h(r) vs r with r = Ix l. Note that, when Tis near Tc2, h(r) shows a damped oscillation, causing an oscillation between attractive and repulsive regions. Although a similar behaviour of h(r) was obtained for a non-magnetic material with small K such as niobium, 33 the oscillatory behaviour is too small to be observed. However, in the case of magnetic superconductors, the presence of the magnetic moments substantially enhances the oscillatory behaviour; even when KB is as large as five, the ratio Ih(r) l/Ih(0)l at the first minimum point is more than one hundred times larger than the ratio for niobium. The presence of a strong attractive interaction among vortices at T near To2 means that these magnetic superconductors tend to become type II/lor type I, although it is type 1I/2 at higher temperatures. The oscillatory behaviour ofh(r) can be understood in the following way. The flux quantization for the magnetic induction leads to

bd2x =

md2x = - -

d2x + 4rr

(29)

2e

Since m increases near the magnetic phase transition temperature Tc2, h should diminish and create a region of negative h over a short range, giving rise to the oscillatory behaviour mentioned above.

The mixed states

K=I

The vortex is a topological object with a line topological singularity. Consider a single vortex at x = 0. Then the phase-function f is proportional to the cylindrical angle 0, which is multi-valued: "~ x ~ 0 ~x6(~). This shows how f(x) with a topological singularity_.contributes to the magnetic field through the term V × V f=/= 0. Assuming f ~ 0, we have the molecular field

1.0

~=

~

K= 2 0"5

x =:5

h~m = 3'(- i~)m - 4rrF(- i~)~n + (q~/2rr)F ( -

i~)~x~f

(27) 0

The last term is usually positive and compensates part of the second term which screens the rare earth spins. Thus the screening effect is weakened in the vicinity of a topological

42

I

I

I

I

2

5

4

5

kLq

Fig. 7

Wave n u m b e r dependence

of the screening factor

[S(q)] 2

for severalvaluesof KB31

CRYOGENICS.JANUARY 1983

Fig. 9 shows the theoretical magnetization curves for several temperatures near the magnetic phase transition temperature.34 The parameters are KB(O) = 3, Tna/Tc = 0.15, Tp/Tc = 0.05 and 4rrC/Tm = 1.5. Note that the magnetization curves abruptly jump around H = Hcl. This jump is caused by the attractive interaction (ie negative h(r)) and decreases with increasing temperature. This jump means that the system becomes the type II/1 at T near the magnetic phase transition temperature, implying the tendency of type II/2 ~ type II/1 for decreasing temperatures. For another choice of parameters, the attractive interaction becomes so strong that the transition series is given by type II/2 ~ type II/1 -* type I with decreasing temperature, a2 Such behaviour has been observed experimentally in the magnetization measurements (Fig. 10) on ErRh 4 B4 .3s

x 105 30

16 ~rB = 50

25 \ / ~// i

: 02

t tm

: Oi

d

= OI

c=l

12

08

/

2-0

0.4 ///I

2

I0 ~_

0

0

~I~

/~

0

-0.4

2.0

I

4i0

]

50

4.0

3'0 r~

Fig. 8

Note also the presence of the repulsive domain outside of the first attractive domain in the h(r)-curve. When this outside repulsive tail is sufficiently strong, it prevents the vortices from entering suddenly at H = Hc~. In this case, the transition at Hc~ becomes second order, while the first order transition occurs at a value of H higher than Hc~ (cf Fig. 1 1). We tentatively call this type II/3.36

/

0830

10

\

6.0

5-0

60

XL~Ot

He2 a n d Hcl vs T Calculations of He2 and Hct vs T have been made 32'36 and a variety of curves were obtained, depending on the choice of parameters 36 (Fig. 12). Very briefly, these curves are similar to the bell shaped curve observed by Ott et aim

Magnetic field distribution of a single vortex 32 1,4

(Fig. 13) and Adrian et al 3s (Fig. 14). (in Fig. 14, He°2 means really Bc2 .)

1.2

Usually, the curve B (magnetic induction) vs T is a smooth -020 ~

IO

T=686K

", -~-"~-.._'~.4 ~ 020 -

0'8

~

~

1.2 ~.,~ ~.6 I.e 2.o 2.2 c2

H,lO5Am-I

o.9. N~ ~o

78 L22,

06

020

i=

\\

,
04

E

02

080

% - 0 2 0 P /'T'~ M (P) ~ - ~ - ~

yil

s

o ~_ ,;,Qt, o'~ , ,,,,P I ~c~ -02

0

02

0.4

06

0.8

FO

H/(#o/X2L{O))

'

'"s

T:I.45K

---_ ,'.o '--L~--,'8 --

040 ;,

U__ 222

o. o

r,,

,~'

212

",,c~°1

1

Fig. 9 Magnetization curve. (Parameters are KB = 3, t m = 0.1 5, c = 1.5 tp = 0.05) 140

Now consider the vortices forming a triangular lattice:

160

~Y x Vf(~) = 27r~z .E 6 2 (x*-~-~) !

where ~/are the centre points of the vortices. We assume unit flux for each vortex.

CRYOGENICS

. JANUARY

1983

__

(30) Fig. 10 Magnetization vs magnetic field for Er0.73Hoo.27Rh4B 4 at 6.86 K (upper section), 4.64 K (middle section), and 1.43 K (lower section) 35

43

1'2

d

= 7.5

c

= 1.6

This behaviour has been observed both experimentally and in theoretical calculations. Note that our calculations take into account the saturation effect, which causes a variety of fine structures in the He2 and He1 vs T curves at T near Tp or To2.

tm= 0"16 = 0.14 l M = 0-11 ! = 0'15 KB = 2.5 fp

• 1.0

0.6

0.8 0.5

l l

t--1

/ /

9

0.6

l

/ /

..e..

0.4

i \ \

t¢ ,,¢

\

0.4

0-3

0.2

0.2

~J

0.1

Hc, 0

-0'2 I

0

I

0.02

I

]

0.04

I

I

0106

I

I

0 [0 8

I

I0

0.10

N

H/ [ ~/×~(o)] Fig. 11

~

0.8

44

= 75 = 1"6

lm = 0 ' 0 0 8 8 6 7

Magnetization curve for type 11/3

curve rather similar in shape to the non-magnetic superconductors when the s-f interaction effect is small. (Note that B = H in the non-magnetic case.) The Bc2 curve of ErRh4 B4 is considerably lower than the Hc2 curve of LuRh4B 4 . This decrease may be caused by the s-f interaction. However, the shape of the Be2 vs T curve for ErRh4B4 is a smooth one similar to the one obtained in the case of the usual nonmagnetic superconductors. Therefore, the unusual shape of Hc2 vs T curve can be attributed mainly to the magnetization since Hc2 = Bc2 -47rMc2. When T is close to Tc2, M tends to increase with decreasing temperature, causing Hc2 to decrease. This is the reason for the bell shaped Hc: vs T curve. The decrease of Hc2 is enhanced as u = g#BJN/[¢/X~(O)] is increased because u is the amount of magnetic moment per vortex. The qualitative situation is schematically described in Fig. 15. (Here He°2 is Hc2 for the corresponding nonmagnetic superconductor.) The reduction of Be2 at TI, tI (the transition temperature for type II ~ type I) may become recognizable, depending on the parameters. It should be noted that the pair breaking mechanism arising from a strong s-f interaction, can of itself create a bell shaped He: vs T curve./s'37 In other words, both the electromagnetic and s-f interactions tend to create the bell shaped Hc2-Curve. However, the two mechanism can be distinguished by the fact that the reduction ofHc2, due to s-f interaction, also induces a reduction of the same order in Be:. This is because s-f interaction modifies the electron energy which controls Bc:. Since the s-f interaction is weak in the case under consideration, we have confined our study to consider mainly the electromagnetic effects. (For a detailed comparison with experiments, however, the effect of the s-f interaction may be required). Briefly, magnetic superconductors behave like non-magnetic ones with geff ~ K/(1 + 4700 re.a8 When T is closer to Tc:, xeir becomes smaller, causing a decrease ofHc2/Hcl (ie, Hc2 comes closer to He1).

J c

9 %~-~

\

= 0.008867

u

= 0'09370

~ /

o.6

d

X

"B = l'75

0.4

0"2

2"0

J c

= 75 = 1.6

~rn

=

d u

0"16

= 0006009 = 0' 1339

"(B = 2 5 I-6

0.8

0.4 H(0) c

0 0

TTO'2 /M.J L i p

0.4

0.6

0'8

1.0

!

Fig. 12 Temperature beheviour o f critical fields, a - first order reentrant, b - second order reentrant, e - no reentrant

CRYOGENICS . J A N U A R Y 1983

is a necessary condition for the appearance of self-induced vortices. The creation of self-induced vortices is a selfconsistent mechanism because the vortices are stabilized by the magnetization, while the magnetization is stabilized by the magnetic field induced by the vortex current. Therefore, we need two conditions; (32) and

ErRh4B 4 resistive He2

7 _r2-

6

- -

f~

5 I

\

\N

-- . . . . . . . . . . . .

/

•

.

.

.

.

.

"1.k

\

.

it

&

\\

//

3 2 I

s-f interaction effect 0

I~ /

I

I

I

I

I

I

t

I

2

5

4

5

6

7

8

0

~",

I

9

l0

Temperature, K Fig. 13

2-4 I-6

?

v E 0.8 < %

-

0

H(O) c2

Hc2 vs T f o r ErRh4B414

47rMc2

x =0-0 .

.

%2

~

, x =0.40

.x ° 0 . 6 0

x °0.50

x =0"70

2-0 12 •

04 0

Hcl

f

0 2

4

6

2 4

6

0

2 4

6

0

2

4 6

Fig. 15 Schematic illustration of various effects to the temperature behaviour of Hc2

Temperature, K Fig. 14 Hcl (open squares), Hc2 (open circles) and H22 (full circles) as function of temperature for Erl_xHOxRh4B4 35

006

The critical field in the type I-region, H*, is illustrated in Fig. 16. At Tp, the change of sign in the curvature is observed in the spin-spiral phase. The results for the spin sinusoidal phase are not so dramatic; instead the H* curve decreases almost linearly with decreasing temperature.

,~/

J c

005

I

As seen in Fig. 9, at t = 0.1 the Meissner state disappears and a positive spontaneous magnetization appears at H = 0, indicating the appearance of self-induced vortices, by which we mean vortices spontaneously stabilized by the induced rare earth magnetization in the absence of an external magnetic field. 34'39'28 Briefly speaking, when n vortices per unit area enter the sample, the average molecular field acting on the rare earth ions is expressed as

=7.5 =16

t m = 0.16 d = 0.01168 u = 0"07496

004

Possibility of self-induced vortices

hm = n~ - 47rM + 7 ( 0 ) M

T

'I,II

8

/ I .'/~ /

A--B = 1.4

/

o.9 %
)-

003

,~o 0 02

(31) 001 F-

When h m is positive, the sample is uniformly magnetized and has the saturation value of Ng#BJ. We can see from the calculation of the free energy that 4rrMacts on electron motion as an external magnetic field. We write the fictitious upper and lower critical fields for a similar sample with no magnetic spins as Hc% and Hc°l respectively. Then,

SpiraiZp 0

// 0

I O.lO

]~" t

SinusoldQI

I[ 0.12

] 0.15

tM

I t

I I 0.15 0.16 0i7 tp

t

tt° 1 < 4rrM < He°2

CRYOGENICS

. JANUARY

(32)

1983

Fig. 16

Critical field H~ in the type I region

45

decreasing temperature (at H e a = 0 in this case) and in a second order transition to the normal ferromagnetic state (at He2 = 0 in this case). o~

( - 4 , r M O) = 7 H 0 / 4 r

Phase diagrams [ i e Hm :o ]

Here we discuss some of our results for the phase diagram. 4° We introduce the following parameters: Io = (~(0)XL(0)) U2 , KB = XL(0)/~(0), Uo = NgtaBJ/(¢/l 2o), do = D/Po Tm, Nmo = NkB Tm/(¢/12o) 2, tm = TIn/To, t = T/Tc and J = 15/2. The results are consistent with those of Varma et al. 28 Since the phase diagrams depend on the choice of parameters, we present only a few of them. In Fig. 18 the phase diagram in the t-K B plane is given. The parameters are tm = 0.15, Uo = 0.15,Nmo = 0.2 and do = 10 -a . The solid and dashed curves indicate the second and first order transition boundary, respectively. For KB very small, the Meissner phase changes to the spin-periodic phase with decreasing temperature. For an intermediate rB, the system makes a series of phase transitions from the Meissner phase to the spinperiodic phase and then to the normal ferromagnetic phase with decreasing t. For KB around KB = 4, the system makes more complex phase changes: Meissner -+ spin-periodic -+ self-induced vortex -* normal ferromagnetic phase. For KB large, the phase transition follows as the Meissner ~ spinperiodic -+ self-induced vortex.

r

L

% ~t

r

H ° (=4~rM) Fig. 17

Criterion for the self-induced vortices

Meissner 015 ~ , -~. • Sniral

....

~ ....

We also examined the t vs x phase diagram, where x is the concentration of magnetic ions. Note that tm is roughly linear in x. In Fig. 19 we have chosen ~ B = 2.5, do = 8.95 x 10 -3, tc = 1 - 0.5x and tm = 0.701 (x - 0.15). The temperatures are normalized by Tc ~ at x = 0. The tc and tm lines cross at x = 0.92.17 The tm line for x < 0.92 is indicated by the dotted line. For x < 0.92, a decrease in temperature changes the Meissner phase to the normal

I / I

0.10

/ I Vortex

0.05 J I

0 0

Ferro

I I

J 2

I'0

3

4

I 5

I 6

K

Fig. 18 Transition temperatures vs/~B. Parameters used are t m = 0.15, #o = 0.15, Nmo = 0.2 and d o = 10 -3 40

hm > 0

Meissner

(33)

0"5

Since (1/4rr) (n¢ - 4rrM) is the magnetization (M °) induced by the persistent current and since T(0) = Tm/C = 4~c, (33) reads as - 47rM° < 4~rM/c

•. "~,/ ...'i I ...' ,,

(34) Spirol

Thus, when c > 1, (32) and (38) becomes one condition

~

Ferro

// H°m < 4rrM < He°2

(35)

rtex I

where H ° is determined by - 4rrM ° = 4nM/c. This is shown in Fig. 17. This condition is fulfilled when u and KB are both large. The phase transition from the Meissner to the vortex state is first order while the transition from the vortex state to the normal state is second order. This results in a first order transition to the magnetic ordered state with

46

02

I

1 %

I

I

I

I

I

1.0

0'5 X

Fig. 19 Transition temperatures vs x . Parameters were assumed to be t c = 1 - 0.5x, t m = 0.701 (x - 0.15),/~o = 0.236 (x = 0.15) (tc, Nmo4~ 6.92 X 10 - 2 [ (x - 0.15)/t c] 2, do = 8.95 X 10 -3 and K s = 2.5

CRYOGENICS. J A N U A R Y 1983

ferromagnetic phase at to2, which is shown by the dashed line. The phase transition from the paramagnetic to the ferromagnetic phase is second order, while the phase transition at tc~ is first order. The tc2-1ine is slightly lower than the tm-line. This decrease is due to the condensation energy of the superconductor, as can be seen from Fig. 20. This behaviour has been observed 17 in Hox Lu l-x Rh4 B4 in which the Ho ions have magnetic moments while the Lu ions do not. Another interesting case is Eq_xHoxRh4 B4, because the easy axis of Er magnetic moment and that of Ho magnetic moments are mutually orthogonal; the easy axis of the Er spin is in the basal plane orthogonal to the c-axis, while the easy axis of the Ho spin is along the c-axis. According to the experimental results 41 (Fig. 21), when x = 0.1, we may expect around T = 0.7 that the magnetic phase transition, due to the Er-spin polarization, will occur along the easy axis, of the Er-spins, while around T = 0.3 another phase transition, due to the Ho-spin polarization, will occur along the easy axis of the Ho spins. This behaviour seems to prevail in the recent Woods-Fukase ultrasonic experiment (Fig. 22). 3°

2 Superconducting

//o//°

E [

./

/ L 0.0

i 02

I

Magnetically ordered

I 04

I

I 06

I

I 08

I

I 1.0

X Fig. 21

Transition

temperature vs x for E r l _ x H o x R h 4 B 4 4 1 Ero 9Ha 0 IRh4B4 o

Cooling

•

Warming

Surface ferromagnetism Let us now turn our attention to topological surface singularities. As the preceding general considerations have demonstrated, these singularities make it easier for a ferromagnetic state to appear in the neighbourhood of a surface at a temperature higher than Tc2, because the singularities reduce the screening effect due to the persistent current. Let us begin with the case in which an external magnetic field is applied in parallel to the surface of a sample. The presence of a surface can be described by a particular choice of the phase function f(x) which creates a surface singularity in i~ x ~tf. We take the x-axis perpendicular to the surface and its zero point at the surface. H o is the magnetic field. The calculation 42 leads to the following result (Fig. 23). At high temperatures above Tm, the penetrating magnetic field

T E o

o

0 o

16

:2; 0

J k 14 p~

0

0.5

1.0

Temperature, K

rm

B

r

Fig. 22 Temperature behaviour of ultrasonic a t t e n t u a t i o n for Ero.9Ho0.1Rh4B430

(H = 0}

decays monotonically with increasing x. When the temperature is decreased, an oscillatory decay of the magnetic field appears at a temperature To a little above Tm. When the temperature is further decreased down to T s which is higher than Tc2, the surface makes a phase transition to an ordered state, in which the magnetization is bound in a region of range )t L in the superconducting state, in the absence of a magnetic field. Thus, when Tc2 < T < Ts, a surface ferromagnetic phase appears. Normal-ferro

Thin film A ferromagnetic phase can coexist with superconductivity in a thin film (the depth ~ XL), for the same reason as that outlined above, when T < Ts, where Ts is higher than Tc2. The value of Ts depends on the thickness of the film.43

Anomalous phase just above Tc2 in ErRh4 B4 and HoMo6 S8 Fig. 2 0

Schematic temperature behaviour of free e n e r g y

CRYOGENICS . JANUARY

1983

Let us now return to the unusual phase just above the magnetic phase transition temperature Te2. Since the phase

47

Self-induced surface magnetisation

Oscillatory d,ecay

i

Simple decay

i

7;, r. I

r°

r

°

h(x)

No external magnetic field

o

~

x

o Fig. 23

Schematic temperature behaviour of surface magnetization

stabilized when the negative line (or surface) energy and positive normal core energy are in balance. The self-induced vortices mentioned previously may not be suitable for the case under consideration, since the flux lines go through the material and therefore it may not show perfect diamagnetism. Furthermore, according to this model, the phase transition from the Meissner phase to the vortex phase should be first order, and the phase transition from the self-induced vortex phase to the normal ferromagnetic phase should be second order; both features contradict experiment. Therefore, if vortices are created spontaneously, they must form closed loops. Though this kind of calculation has not yet been performed, the nucleation of the closed loop is expected to give rise to a second Order transition and at Tc: a first order jump may be expected as the flux suddenly expands over the entire system. Even these kind of closed vortices is in contradiction with experiment. Closed loops of vortices may form a lattice and such a lattice structure is not observed in neutron scattering; the periodic wave vector seems to point 7000

for T < Tc2 is ferromagnetic, we expect Tm > Tc2. We have seen that both the electric resistivity and the acsusceptibility (Xac) show a hysteresis around Tc2 and that the neutron diffraction experiments detected a periodic signal with period ~ 100 A for ErRh4B4 and * 200 A for HoMo6 Sa. Recent experiments, using a single crystal, 8 show that this is a spin sinusoidal phase, which suggests a strong anisotropy effect. This latter experiment determined the direction of the wave number vector and the direction of magnetization and also confirmed the coexistence of ferromagnetic order with sinusoidal order. (This coexistence has also been suggested by previous experiments which used powder samples. 6'7'n'12) Fig. 24 shows both the intensity curve of the ferromagnetic part and the satellite part given by Sinha et al; the latter is the effect of the spin-sinusoidal phase. There is a considerable amount of the hysteresis in both the ferromagnetic and the satellite intensities. The hysteresis appears in the region 0.7 K ~< T ~< 0.9 K. The hysteresis of these two intensities behave in an opposite manner. It has been found that the system is completely diamagnetic in this hysteresis region. Any model for this anomalous phase should consider: one, the coexistence of the ferromagnetic and spin-sinusoidal magnetization; two, the above-mentioned behaviour of hysteresis and; three, the completely diamagnetic behaviour of the bulk sample. The appearance of the ferromagnetic component in the superconducting phase suggests many possibilities. For example, coexistence of spacially different domains of ferromagnetic and spin-sinusoidal phase; the latter one corresponds to the theoretical prediction of the spin-sinusoidal phase caused by the screening effect mentioned previously. This model has been advocated by Varma and other physicists. In this model, the ferromagnetic domain should be stabilized in such a way that no flux comes out of the domain. Another possibility is to assume that the ferromagnetic component is created by the presence of certain topological singularities, because we have seen that topological singularity permits coexistence of ferromagnetic and superconducting phases in a region around the singularities. As possible singularities, we can consider either lines (corresponding to vortices) or planes. (We do not consider a volume because it corresponds to the domain mentioned in the first possibility). These objects either extend to the surface boundary or form closed objects, in the sense that the flux lines form closed loops. Such an object may be

48

ErRh 4 B4 Single crystal

6000

Ferromagnetic intensity

5000

ZrFM

o Cooling

o

•

Lx Warming

I

[

I

I

I

I

I

I

I

I

4000

2=

3000

2000

I000

o 80 >, {3 25

60 40 20 c~ I

dc resistance

40 =L qz"

o coo,og

20

I

~

~

War~ingI

x IOz 3.0

Intensity ratio

/m.~_ ,.~

2.0

~

~

[I O.8

I 0.9

o Cooling Warming

i.O

0.5

I 0.6

0.7

I 1.0

I H

I I-2

Temperature~ K

Fig. 24

Temperature behaviour of neutron scattering intensity 8

CRYOGENICS. JANUARY 1983

only in one direction. This leaves us with the possibility of surface singularities which create surface magnetization. Since no flux should come out of the sample, these surface singularities should form closed surfaces. (We may imagine a pile of rectangular-box surfaces.) Since the surface magnetization is induced below Ts, the surface energy becomes negative below Ts. As a preliminary study, a laminar structure was examined in which a series of surface singularities with an equal interval, say a, are placed along the xl -axis. This determines ~ x ~ f which carries an arbitrary constant factor, p, because the strength of the singularity is not restricted by the law of flux quantization:

15

±

~, IO

~

05

x ~ f(x) = 2nu Y~ 61 (x - n a ) e z

•

-

(36)

n

It is obvious that this model gives a ferromagnetic component coexisting with a spin-sinusoidal phase with Q =- 2rr/a (Q: wave number). In both cases, the hysteresis is expected because of the non-linear interaction among spins. Another way to make the presence of a ferromagneticlike domain consistent with the completely diamagnetic nature of the system is to assume that this seemingly ferromagnetic domain is not the usual ferromagnetic state, but a periodic spin alignment with a long wave length coexisting with the superconducting state. Considering the resolution of the neutron scattering, we find that this large wave length cannot be smaller than several thousands of an angstrom. Taking this view, one assumes two kinds of spin-periodic alignment, one having the above mentioned long wave length and the other having a short wave length (--~ 100 A). We are studying the theoretical question: can the nonlinearity of the spin polarization and the inclusion of the s-f interaction can produce these two quasi-stable spinperiodic phase. An uncomfortable aspect of this view is that the magnitude of long wave length seems to be too large. More experimental information may be required in order to decide if this intermediate state is a mixture of two phases or a new single phase.

Final comment

0

-0

I

t

I

05

10

15

Temperature, K Fig. 2 5 Temperature GdMo6S845

behaviour o f H c 2 f o r D y M o 6 S 8 and

can be theoretically reproduced by various choices of parameters. 45 An example of the He2 vs T curve is shown in Fig. 26. We have chosen the following parameters; Kn = 2, Ngl.tBJ/((~/l~(O)) = 0.1,4zrC/T N = 2.4, T N/T c = 0.23 and Tm/Tc = 0.07, where C is the Curie constant, TN the N~el temperature and Tm the paramagnetic Curie temperature. The solid curve shows the calculated values of He2 and the solid circles the experimental values of Hc2 for DyMo6 $8 obtained by Ishikawa and Fischer. 4 All the values are normalized to the maximum value ofHc2. The maximum value is calculated to be 0.416 x (~/12(0)) which corresponds to 1.5 K 0e in the experiment. As seen in this figure, the Hc 2 -curve has a dip at the temperature at which the antiferromagnetic order occurs. This ordering temperature is a little lower than the N~el temperature in zero field owing to the effect of the internal magnetic field of the vortices.

When the vortex loops are stabilized and the anisotropic energy is small, it is possible that the formation of amorphous-type singularities (random distributions of line singularities) becomes favourable. In this case one might see a spin-glass type phase in magnetic superconductors.

T

Mixed state in antiferromagnetic superconductors We have discussed the phenomena caused by the electromagnetic interaction between the persistent current and the rare earth magnetic moments mainly in ferromagnetic superconductors. This interaction is important also in antiferromagnetic superconductors, since the large magnetization of rare earth ions is induced by an external magnetic field in the mixed state. In the mixed state we consider the usual triangular vortex lattice. For the rare earth spin system, we take a two sublattice model. Following the same procedure as that of ferromagnetic superconductors, we calculated the free energies for the Meissner state, the mixed state and the normal state in antiferromagnetic superconductors. Using these free energies, we determine the upper and lower critical fields.

"~

05

0

i 0

i

i

i

i

J

05

~

t

1, 10

!

Depending on the samples, the experimental He2 vs T curves take a variety of forms. Two sets of experimental data 4 are shown in Fig. 25. Most of the experimental curves

CRYOGENICS . JANUARY

1983

Fig, 2 6 Hc2 vs T curve for a n t i - f e r r o m a g n e t i c s u p e r c o n d u c t o r . P a r a m e t e r s are ~ B = 2 , / z o = 0.1 41felT N = 2 . 4 , T N / T c = 0 , 2 3 a n d TM/T c = 0.0745

49

Recently, it was pointed out that the antiferromagnetic molecular field acting on superconducting electrons may affect the temperature dependence o f He2.46

References 1 2

3

Other unusual phenomena There are currently many kinds of samples and a variety of unusual properties which are not mentioned in this report. The Ce and Eu compounds seem to behave quite differently from the others. This may be related to their valence; Eu (also Sm and Yb) can be 2+, while most of rare earths have a valence of 3+. Some of the unusual properties may be related to the Kondo effect, suggesting a rather strong s-f interaction. In the case of EuMo 6 $8, the superconductivity is absent at zero pressure but is induced with increasing pressure. This has been attributed to the Kondo effect 47 or to the partial gapping of the Fermi surface.4a In the case of Dy(IrxRhl_x)4B4 with x = 0.6 ~ 0.8, the magnetic transition temperature is higher than the superconducting temperature. A similar phenomenon has been observed in Y3 Co4.

4 5

6

7

Ginzburg, V.L. Zh Eksperim i Teor Fiz 31 (1956) 202 [Soviet Phys JETP 4 (1957) 153] Maki,K. Superconductivity R.D. Parks (ed), Marcel Dekker, New York (1969) 1038

Fertig,W.A., Johnston, D.C., Delong, L.E., MeCallum, R.E., Maple, M.B., Matthias, B.T. Phys Rev Letters 38 (1977) 987 Ishikawa, M., Fisher, ¢,., Solid State Commun 23 (1977) 37; ibid 24 (1977) 747 Maple,M.B. Jde Phys C6 (1978) 1374; Ishikawa, M., Fisher, ~, Miiller, J. J de Phys C6 ( 1978) 1379; Maple, M.B. Ternary superconductors, G.K. Shenoy, B.D. Dunlap, Fradin F.Y. (ed) North-Holland Pub (1981) 131; Ishikawa, M. ibid 43 Moncton, D.E. JAppl Phys 50 (1979) 1880; Moneton, D.E., McWhan,D.E., Schmidt, P.H., Shirane,G., Thomlinson, W.,

Maple, M.B., Mackay, H.B., Woolf, L.D., Fisk, Z., Johnston, D.C. Phys Rev Letters 45 (1980) 2060 Lynn, J.W., Shirane, G., Thomlinson, W. Shelton, R.N. Phys Rev Letters, 46 (1981) 368; Lynn, J.W., Shirane, G., Thomlinson, W., Shelton, R.N., Moncton, D.E. Phys Rev

8 9

10

11

B24 (1981) 3817 Sinha, S.K. XVII International Conference in Low Temperature Physics (Los Angeles 1981) Vandenberg,J.M.,Matthias, B.T.NatlAcadSci USA 74 (1977) 1336 Yvon, K. Current topics of material science, North Holland Pub, (1978) 4 53

Moncton, D.E., McWhan,D.B., Eekert, J., Shirane,G., Thomlinson, W. Phys Rev Letters 39 (1977) 1164

12 13

Summary

The theoretical predictions made so far include: spin-spiral or sinusoidal phase caused by the screening effect of persistent current z6'27 ; anomalous behaviour of x±(k) in ultrasonic measurement (confirmation of the screening effect 29 ; with decreasing T, transition of type 11/2 type II/1 or type 1I/2 ~ type II/1 ~ type 132,36 ; scaling rule K~ = KB/(1 + 47rX)w for 47r(Ms - M n ) Hvs 4n,$/1 +41r×

curve in the regi°n T • Tc38;

appearance of type I1/336 ; self-induced vortices phase a4'39 ; appearance of superconducting Bloch wall s° at T < To2. (This may explain why resistivity at T < Tc2 is smaller than that of normal (See Fig. 2)); screening of the magnetic scattering of neutron fbr q ~ 2/?tL 31 ; and appearance of surface magnetization at T < T s.42'43

14

Kimball, C.W., Potzei, W., Pr/ist, F., Kalvius, G.M. Phys Rev B21 (1980) 3886 Ott, H.R., Fertig, W.A., Johnson, D.C., Maple, M.B., Matthias, B.T.J. Low Temp Phys 33 (1978) 159

15

Woolf, L.D., Johnston, D.C., MacKay, H.B., McCallum, R.W.,

16

Maple, M.B.JLow Temp Phys 35 (1979) 651 Woolf, L.D., Tovar, M., Hamakar, H.C., Maple, M.B. Phys Letters 74A (1979) 363

17 18 19 20 21 22 23 24

Among these predictions, the first three have been already confirmed by experiments. It is now a thrilling moment to see how many more of these predictions will be proven.

25

We are grateful to many of our freinds, in particular, Drs M. Tachiki, O. Sakai, T. Koyama and S. Takahashi at the Research Institute for Iron, Steel and Other Metals at Tohoku University and Dr A. Kotani at Osaka University, Japan, for their active collaboration with us in research and for their valuable discussions. We would like to thank Drs S.B. Woods and T. Fukase for providing us with their recent experimental data prior to its publication. Our thanks go also to Dr C.Y. Huang for informing us about his work on EuMo 6 $8. This paper is based on our lecture note prepared for the CAP Fall symposium on Condensed Matter at the NRC Institute, Ottawa, 1981. We are grateful to Dr E. Fenton who organized this symposium. This work was supported by the Natural Sciences and Engineering Research Council, Canada, and by the Dean of the Faculty of Science, The University of Alberta.

27

50

Lynn, J.W., Moncton, D.E., Thomlinson, W., Shirane, G., Sheiton, R.N. Solid State Commun 26 (1978) 493 Shenoy, G.K., Dunlap, B.D., Fradin, F.Y., Sinha, S.K.,

26

28 29 30

31 32 33

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