An analysis of the metamagnetic character of the mixed state in bct ErRh4B4

Physica C 158 (1989) 32-64 North-Holland, Amsterdam

A N A N A L Y S I S O F T H E M E T A M A G N E T I C C H A R A C T E R O F T H E M I X E D STATE I N bct

ErRh4B4 O. W O N G and H. U M E Z A W A Department of Physics, Universityof Alberta, Edmonton, Alberta, Canada T6G 2Jl J.P. W H I T E H E A D Department of Physics, Memorial Universityof Newfoundland, St. John's, Newfoundland. Canada A 1B 3X7 Received 25 August 1988 Revised manuscript received 20 February 1989

A theoretical analysis of a magnetic field induced transition in the mixed state of an antiferromagnetic superconductor is presented. The electromagnetic properties of the superconducting mixed state are calculated within the framework of the boson method while the magnetic properties of the rare earth ions are represented by means of a Landau expansion. The form of the magnetic free energy and the Landau coefficients are chosen to model the metamagnetic behaviour observed in bct ErRh4B4. Phase diagrams and magnetisation curves are presented for various choices of parameters and a comparison with the recent observations on bct ErRh4B4 is made. The analysis of the antiferromagnetic phase is extended to include the possible formation of a ferrimagnetic domain around the vortex core and its effect is discussed.

I. Introduction The interplay o f superconductivity and m a g n e t i s m has p r o v i d e d a fruitful area o f theoretical a n d experimental research for nearly three decades. Early theoretical [ 1 ] a n d experimental [ 2 ] studies revealed that the presence o f small a m o u n t s o f magnetic impurities dissolved into a superconducting metal had a d r a m a t i c effect on m a n y o f the superconducting properties. Such effects include a r a p i d suppression o f the superconducting transition t e m p e r a t u r e with increasing i m p u r i t y concentration, the p h e n o m e n a o f gapless superconductivity and, in the case o f i m p u r i t i e s with a negative exchange interaction, the p h e n o m e n a o f reentrant superconductivity due to the K o n d o effect [ 3 ]. The discovery in the mid-seventies o f a class o f rare earth ternary or pseudo ternary superconducting alloys [ 4 ] led to a n u m b e r o f i m p o r t a n t a n d unexpected d e v e l o p m e n t s in this area. These included the coexistence o f antiferromagnetism and superconductivity as well as c o m p o u n d s which exhibited both ferromagnetic o r d e r and superconductivity. The latter c o m p o u n d s are frequently referred to as ferromagnetic superconductors although in all instances the appearance o f the ferromagnetic o r d e r quenches the superconductivity, a n d hence the two phases do not coexist as the n a m e might imply. Instead what is observed, in a narrow t e m p e r a t u r e region just above the re-enterant temperature, is a m o d u l a t e d spin phase with a wavelength o f a p p r o x i m a t e l y 100 .~ which coexists with the superconductivity [ 5 ]. This c o m p r o m i s e between the ferromagnetic o r d e r and the superconductivity has its origins in the electromagnetic interaction between the rare earth ions a n d the screening effect o f the supercurrents [ 6 ]. Yet another example which reveals a subtle interplay between the magnetic a n d superconducting degrees o f freedom arises in the class o f c o m p o u n d s referred to as the heavy fermion c o m p o u n d s [ 7 ]. Some o f these compounds become superconducting at low temperatures, others become antiferromagnetic while some r e m a i n in the normal state down to the lowest observable temperatures. Recent experimental studies also [ 8 ] suggest that the interplay o f magnetism and superconductivity constitutes a key element in the recently discovered class 0 9 2 1 - 4 5 3 4 / 8 9 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )

o. Wong et al. / Mixed state in bct ErRh4B4

33

of copper oxide superconductors, although the nature of this interplay and its role in the mechanism generating the high transition temperatures in these compounds is far from clear. In the case of the magnetic superconductors experimental studies of the ternary borides and the Chevrel compounds have been limited largely to studies involving polycrystalline samples. This has tended to obscure much of the detailed behaviour concerning these intriguing compounds. Indeed in the few cases where measurements have been performed on single crystal samples [ 9,10 ] the results reveal and extremely complex behaviour arising in large part from the substantial anisotropy with the magnetic interactions. An example is provided by the magnetisation measurements on the pt phase of ErRhaB4, in which the tricritical point, observed on the phase boundary between the superconducting mixed state and the normal phase in the single crystal measurements [ 10 ] is not observable in any direct sense in the polycrystalline samples. One would also expect the magnetic anisotropy of the rare earth ions to play an important role in the case of antiferromagnetic superconductors. In particular it is well known that magnetic anisotropy in antiferromagnetic systems can give rise to a magnetic field induced transition from one magnetic configuration to another. Such effects are generally referred to as metamagnetic transitions and are observed in a large number of magnetic systems [ 11 ] and may be of a continuous or a discontinuous nature, The phenomena of metamagnetism arise in systems in which the energy associated with the magnetic anisotropy exceeds that associated with exchange interaction and for which the exchange interaction comprises both a ferromagnetic and an antiferromagnetic component [ 12 ]. Since the magnetic properties of the ternary borides and the Chevrel compounds are characterised as weak exchange interaction together with a relatively large crystalline electric field, one might expect these compounds to exhibit behaviour characteristic of a metamagnet. This is indeed borne out by a number of measurements on both polycrystalline samples and on single crystals. Perhaps the most direct demonstration of metamagnetism in this class of compounds appears in the recent magnetisation measurements on the bct phase of single crystal ErRh4B4 [ 13 ]. Like the corresponding pt phase of ErRhaB4 the bct phase undergoes a superconducting transition at 7.8 K, however, whereas the pt phase manifests a re-entrant ferromagnetic phase at about 1 K, the bct phase instead undergoes a transition to an antiferromagnetic state, at about 1 K, in which the superconductivity persists. While a number of materials exhibit the coexistence of antiferromagnetism and superconductivity, this particular example is of particular importance in that it is one of the few examples of a single crystal antiferromagnetic superconductor. Furthermore the magnetisation curves measured for the sample below the N6el temperature show a discontinuous behaviour characteristic of a metamagnetic transition. In addition to these results other measurements on polycrystalline samples also exhibit characteristics which are suggestive of a metamagnetic behaviour. Recent resistivity measurements on the antiferromagnetic superconductor [ 14 ] TmRh4B4, for example, show a reentrant behaviour very similar to that observed in the pt phase of ErRh4B4 on the application of an extremely small field ( ~ 3 kOe) suggesting the appearance of field induced ferromagnetism. Also magnetocaloric measurements on GdMo6S8 and TbMo6S8 [ 15,16 ], likewise indicate the presence of a field induced transition from an antiferromagnetic state to a ferromagnetic or spin flop phase. Neutron scattering measurements on polycrystalline samples of the Chevrel ternary alloy DyMo6S8 [ 17 ] indicate the presence of ferromagnetic domains at fields of around 200 kOe, substantially below the observed upper critical field curve of 1200 kOe. While the experimental work cited above indicates the ubiquitous nature of metamagnetic behaviour and the importance of field induced transitions in antiferromagnetic superconductors, relatively little theoretical work has been done to determine the manner in which the superconductivity and metamagnetism can interact. One aspect of such systems which has attracted some theoretical interest is the effect of a field induced transition on the nature of the magnetic structure in the presence of a single vortex or vortex lattice, and the effect of this on the thermodynamic and magnetic properties of the mixed state. In a series of papers Krzyston [ 18 ] has considered the effect of field induced ferromagnetism on the spatial distribution of the magnetisation in the presence of a single vortex, within the framework of the Ginzburg-Landau theory. He concludes that, under certain circumstances, it is possible to induce ferromagnetism within the neighbourhood of the vortex core and

O. Wonget al. /Mixed state in bct ErRh4B~

34

argues that this binds the vortex to the surface of the superconductor resulting in an additional energy barrier to the flux penetration which results in a two stage flux penetration. Iwasaki et al. [ 19 ] on the other hand have considered the effect of induced ferromagnetism on the spatial distribution of the magnetisation in the case of a vortex lattice and by means of a simple phenomenological model of the free energy have argued that the appearance of a ferromagnetic domain around the vortex core can significantly modify the nature of the vortexvortex interaction. Based on this model they are able to account for the anomalous decrease observed in the magnetisation curves for the bct phase of ErRh4B4 at low temperature. In this paper we present an analysis of the mixed state in a metamagnetic system by means of the boson theory of superconductivity. Specifically we focus our attention on the magnetic measurements made on the bct phase of ErRh4B4, although some of the results we obtain may be applicable to other antiferromagnetic systems. The purpose of the present work is to extend the formalism as developed in refs. [20,21] to incorporate the effect of a metamagnetic transition with regard to the rare earth ions. Based on this we compute the magnetisation curves and the phase boundaries for various parameters sets and examine a number of features arising from the metamagnetic character of the system. The paper is laid out as follows. In the next section we briefly outline the description of the superconducting mixed state in the context of the boson method. In section 3 we introduce the free energy associated with the magnetic degrees of freedom in the form of a Landau expansion and discuss its properties in relation to those of bct ErRh4B4. In section 4 we evaluate the free energy of the mixed state on the basis of the results of the preceding sections under the assumption that the relationship between the spatial variation in the magnetisation field and that of the induction field is linear. The calculation of the free energy presented in section 4 follows closely that presented in refs. [20,21 ]. Section 5 contains the calculated upper critical field curve, phase boundaries and magnetisation curves for various choices of parameters. In section 6 we show how the preceding arguments may be extended to include the possible formation of a ferrimagnetic domain around the vortex cores and the effect on the observed properties. In the last section we present a discussion of the results obtained, what conclusions we can draw and what extensions of the present work are appropriate.

2. The mixed state

The application of the boson method of superconductivity to the mixed state has been the subject of a number of review articles, in relation to both nonmagnetic [22 ] and magnetic [24] superconductors. The procedure allows us to compute, within the framework of a well defined approximation scheme, the free energy of a vortex lattice as a function of the lattice spacing and hence to compute the equilibrium vortex density as a function of the applied external field. From this we obtain the magnetisation curves and the phase boundaries between the Meissner state, the mixed state and the normal state as a function of applied magnetic field and temperature. In this section we present an outline of the essential aspects of the boson method together with the assumptions and approximations introduced in the course of the present analysis. We consider a system whose dynamics is described by the following Hamiltonian

H=~ d3x(~,*(x)~(-i(V- ieA hc (x)))~,(x)- V~,*~(x)~,*~(x)~,~(x)~,,(x) ) 1

1

--~ ~ f d3xl f d3x2s,(x,)J(xl-x2)xj(x2)

O. Wonget al. / Mixedstate in bct ErRh4B4

35

-- ~i f d3x(Isi (x)'9/*(x)trV(x) +si (x)"B(x) ) +Hcf.

(2.1)

The first two terms appearing in eq. (2.1) constitute the standard BCS Hamiltonian with ~/and ~/* denoting the second quantised electron operator fields, which satisfy the usual anti-commutation relations, namely {~(X,

t), 9It(y, t) )}=~3(x--y) .

(2.2)

This includes both the minimal coupling of the electrons to the electromagnetic field as well as the phonon induced BCS interaction. The third term represents the electromagnetic contribution to the Hamiltonian, while the fourth term represents the paramagnetic coupling of the electrons to the magnetic field. The fifth term represents the coupling between the rare earth ions arising from the exchange interactions not involving the BCS electrons. We have defined si(x) as

si(x) = Y. gjJl~BS~Si 3(x-Rn),i

(2.3)

where S~ and R~ denote the spin operator and the position of the ith rare earth ion in the nth unit cell, respectively. The sixth term represents the exchange coupling between the rare earth ions and BCS electrons, the s - f interaction. While the seventh term represents the straightforward paramagnetic coupling of the rare earth ions to the magnetic induction field. The final term Hcf denotes the effect of the crystalline electric field on the rare earth spins which is assumed to constitute the main source of the anisotropy of the rare earth spins. Let us consider first the electronic degrees of freedom. We begin by introducing the four component operator field 7'(x), which is defined as

[~u,(x) ~

~,(x)_/~'~(x) | - [ ~ ' I (x) l "

(2.4)

\ - ~d(x)/ The equation of motion for the operator field ~ ( x ) is given in the BCS approximation as

(i~ -e(-iV)% +Az, -tt.tr)~(x)=(l ~ s~(x)-#aB(x)-(I ~ si(x)-I~aB(x) ) ).tr~P(x) ,

(2.5)

where the matrices cr~and z~ are given in appendix A and the parameter p is defined as

I~=(l ~ s,(x)-ltBB(x) ) ,

(2.6)

which represents the spin splitting of the electrons due to the polarisation of the rare earth ions as well as the magnetic induction field B. The BCS order parameter A is defined as

A= V(9,,(x)~u~(x) > .

(2.7)

In general/t will have a complicated spatial structure. In addition to the spatial variation of the magnetic induction field the magnetisation of the rare earth ions will, in general, contain both an antiferromagnetic and a ferromagnetic component, the magnitude and direction of which may vary due to the spatial variation of the induction field B(x). While it is certainly the case that the antiferromagnetic component o f / t can suppress the superconductivity either through changes in the density of states [ 25 ] or through the modification to the electron-phonon coupling [26 ], the consequences are far less pronounced than those arising from the pair breaking effect associated with the ferromagnetic component of p. The fact that the superconductivity in ErRh4B4 persists in the ferrimagnetic state down to the lowest temperatures suggests that the strength of the s - f coupling,

36

O. Wong et al. / M i x e d state in bct ErRh4B 4

between the rare earth ions and the conduction electrons, is sufficiently weak that, in this analysis, we can safely neglect the antiferromagnetic component of/1. Two other simplifying assumptions are made, with regard to /l, in the course of the present analysis. Firstly we assume that the contribution of the magnetic induction field B(x) t o / , is negligible in comparison with contribution arising from the rare earth ions and secondly that we can replace/* by its spatial average in the evaluation of the electron spectra. This assumption will be discussed in somewhat more detail in section 6 when we consider the appearance of the ferrimagnetic domain structure. The term appearing on the right hand side of eq. (2.5) represents the effect of the spin fluctuations. The effect of the spin fluctuations is to renormalise the BCS coupling constant V in the manner outlined in ref. [20] resulting in a reduction of the superconducting temperature. However, based on the calculations presented in ref. [21 ] we regard the effect of such fluctuations as small and we assume that they may be incorporated into the analysis through a temperature and field independent renormalisation of the BCS coupling constant. On the basis of the above assumptions the electron propagator S(p), defined by the outer product

(T{g~(x)~(y)})=

(2~) 4

d4pS(p)e~p
(2.8)

can be calculated by means of eq. (2.5) as S - ~(P) =Po - e ( p ) z 3 +z/z~ +/w3 •

(2.9)

Where we have chosen # to lie parallel to the z axis. The superconducting order parameter A can then be calculated from eqs. (2.7) and 2.9) as A= (2~)3 V f d3k~(k)

2l(tanh~(E(k)+~)+tanh~(E(k)_~)),

(2.10)

where we have approximated zl by its value on the Fermi surface, and defined

E(k) = x / ¢ ( k ) 2 + A 2 . Eq. (2.10) can be rewritten in the form COD

I=VN(0)

f l d( e ~

t a n h ~fl ( E + / ~ ) + tanh

~ (E-#) ) ,

(2.11)

0

where o)D denotes the Debye frequency, N ( 0 ) denotes the density of states at the Fermi surface and E = (~2+z/2)~/2. The above equation may be solved numerically for a given value of/z and fl and the superconducting gap d obtained [20 ]. The expectation value of the magnetic induction field in the mixed state may be calculated from the Maxwell equation as VA < B ( x ) ) = 4_~x +4~VA ~ , e

(2.12)

i

where j(x) denotes the current operator whose expectation value may be calculated by means of the boson method to yield [22 ]

__ 4n (j(x))=-2~52C(-iV) ( (A(x))- ehcV f ( x ) ) e

where C ( - iV) is a nonlocal kernel given by

,

(2.13)

O. Wonget al. / Mixedstatein bctErRh4B4

1( )2

C(k) = e x p - ~

,

37

(2.14)

and f(x) denotes the phase of the order parameter, introduced through the boson transformation. For a vortex lattice this takes the form f ( x ) = 1 ,~

O(x-~i),

(2.15)

where ~i denotes the position of the ith vortex of the lattice and 0 denotes the cylindrical angle the point x makes with respect to the cylindrical axis of the vortex core. The constant 2 L denotes the London penetration depth and is given by 2L ~-2 = 1 + f d ~ - ~a ( f ( E +

lz)+f(E-Iz)),

(2.16)

where 2Lo is defined as

8~e2v~N(O) 3hc2

2~o2 -

(2.17)

Equation ( 2.13 ) together with the Maxwell equation eq. (2.12 ) allows us to calculate the magnetic field ( B ( x ) ) for a specified vortex density and magnetisation fields (s~(x)): (V2-2~-2C(-iV) ) (B(x)) =

-

~h~C,~L-2C ( - - i V ) n ( x ) +4xVAVA ~

(s~(x)),

(2.18)

i

where we have defined the vortex density

n(x) as

n ( x ) = l v A vf(x) .

(2.19)

The solution to the above equation may be expressed as (B(x))

=

(~.L-2C ( - i V )

-V2)

-'

(2L2C(--iV)n(x)0--4=V 2 ~ (s,(x)) ~, \

where

i

/

(2.20)

fb=hc/2e. The magnetic field (H(x)) = (B(x)) - 4 n ( Z ~ s , ( x ) ) may be obtained from eq. (2.20) as

(H(x))=(2~2C(-iV)-V2)-I2~2C(-iV)(n(x)O-4~ ~ ( s , ( x ) ) ) .

(2.21)

The determination of the fields (B(x)) and (H(x)) therefore requires the simultaneous solution of either eq. (2.20) or eq. (2.21 ) together with the as yet unspecified constitutive equation for the magnetisation fields (s~(x)). In practice, of course, such an undertaking can only be carried out within the context of a particular approximation scheme. In the application of the boson method to the analysis of the mixed state in a ferromagnetic superconductor presented in refs. [ 20,21 ] it was assumed that the spatial variation of the magnetisation fields about the mean value need only be calculated to the lowest order in the magnetic field (H(x)). Extending this to the present analysis we may write V2 ~

(s, (x)) "ej= ~ zjkVZ(H(x) ) "ek k

= ~ Zjk(1 + k,I

4nZ)h'VZ(B(x) )'e~,

(2.22)

38

O. Wong et aL / Mixed state in bct ErRh,B,

where Zo represents the susceptibility tensor. If we restrict our attention to situations in which the vortices lie parallel to the easy axis of magnetisation, the ( B ( x ) ) and ( H ( x ) ) will both lie parallel to the direction of the net magnetisation field Y(s~(x)), which we defined earlier as lying in the z direction. We can therefore write that

(Si (x) >=M(x)e3 ,

(2.23)

i

( B(x) ) =B(x)e3

,

( tt(X) ) =H(x)e3 ,

(2.24) (2.25)

and hence eq. (2.22) reduces to give

V2M( x ) = z V 2 H ( x ) =X( 1 + 4 x y ) - lV2B(x)

(2.26)

which, together with the Maxwell equation, yields the following expression for B(x):

a(x) = ( - V 2 + ( 1 + 4xZ)2E-2 C ( - i V ) _t( I + 4~)C)2~-2 C( -iV)n(x)O,

(2.27)

where n(x) may be calculated from eq. (2.15) to give n(x)= ~ 8(x-~i).

(2.28)

i

We note that the above treatment of the magnetisation simply results in the renormalisation of the London penetration depth by the paramagnetic interactions of the rare earth ions [27]. It remains only to provide a means of calculating the spatial average of the magnetisation, M(x) together with the corresponding susceptibility ,~ in order to complete the above system of equations. Specifically once the magnetisation M has been calculated, for a given vortex density, together with the susceptibility Z then we can determine the parameter lt=IM and hence the BCS order parameter A, the C function and the London penetration depth 2 L may be calculated by means of eqs. (2.11 ), (2.14) and (2.16) respectively. From this we can determine the spatial distribution of the introduction field B (x) from eq. (2.27 ). With these quantities determined it is possible to obtain an expression for the Gibbs free energy for a particular value of the vortex density n [20,21 ]. The equilibrium vortex density for a given applied external field may then be determined by a minimisation of the Gibbs free energy with respect to n. Before going on to discuss the calculation of the spatial average of the magnetisation M(x) and the susceptibility X together with the determination of the Gibbs free energy, we wish to comment on the assumption introduced regarding the relationship between the spatial variation of the magnetic field M(x) and the magnetic field H(x) implicit in eq. (2.26). While the assumption that the spatial variation of the magnetisation field M(x) need only be calculated to the lowest order in the magnetic field, may be justified in the case of a ferromagnetic or purely antiferromagnetic system, the metamagnetic character of the system under consideration means that small variations in the induction field B(x) can induce significant variations in the magnetisation field M(x). Such effects are likely to arise when the mean value of the magnetic induction field is close to the critical field required to induce a spin reversal from one magnetic configuration to another [ 18,19 ]. Such effects are neglected in the above approximation. In section 6 we extend the present analysis to include the presence of a ferrimagnetic domain the vortex core and discuss its effect on the magnetisation curves and the upper critical field curves.

O. Wong et al. / M i x e d state in bct ErRh4B4

39

3. The magnetic free energy Metamagnetic systems, of which the bct phase of ErRh4B4 is an example, are characterised by a magnetic field induced transition from one magnetic configuration to another. The transition may be of first order and involve a discontinuous change in the magnetisation and susceptibility or second order, in which the magnetisation is continuous across the transition but the susceptibility changes discontinuously. Magnetic materials which exhibit field induced transformations may be divided into two classes depending on the degree of anisotropy. In the case of compounds with highly anisotropic magnetic interactions the transition occurs by means of a spin reversal, while in the case of a material for which the interactions are essentially isotropic the transition consists of a rotation of the local spin direction. The latter are generally referred to as spin flop transitions while we will refer to the former as a metamagnetic transition. It should be noted however that while the use of the term metamagnetic in this context is relatively common within the scientific literature it is somewhat more restrictive than that originally conceived by Kramers [ 11 ]. The distinction between the two classes is, however, by no means a clear cut, as many of the field induced transitions observed involve both a spin reversal and a rotation of the local spins. Theoretical studies show that an important characteristic of a metamagnetic system is that in addition to the strong anisotropy required to suppress the spin flop transition, there must exist a competition between "antiferromagnetic" and "ferromagnetic" exchange coupling. The structure of the bct phase of ErRh4B4 is shown schematically in fig. l together with the corresponding structure for the pt phase [ 13 ]. In the case of the bct phase, magnetisation measurements indicate that the c axis constitutes the hard magnetisation axis while the a axes constitute the easy axis of magnetisation. The

pt

c

a

bct

i: Fig. 1. Crystal structure ofpt and bct ErRh4B4.

40

O. Wong et al. /Mixed state in bct ErR h4B4

situation, while qualitatively similar to that observed in the pt phase of ErRh4B4 [ 9 ], differs in as much as the degree of anisotropy between the two axes is somewhat less pronounced. The discontinuous changes in the magnetisation, observed below the N6el temperature in the bct phase of ErRh4B4, may be interpreted as a transition from an antiferromagnetic state to a ferrimagnetic state followed by a transition from a ferrimagnetic state to a ferromagnetic state [ 13], which arises from the metamagnetic character of the rare earth ions. Noting that the structure of the rare earth ions corresponds approximately to that of an fcc lattice, with a unit cell comprising four Er ions, the observed behaviour is consistent with the mean field results [28 ] if we assume the nearest neighbour coupling J~ to be negative and the next nearest neighbour coupling J2 to he positive [ 19 ]. While it is possible to construct models, of varying degrees of complexity, which correctly represent the essential features discussed above, the nature of the present calculation is such that only the simplest representations are feasible. As such we will therefore represent the magnetic free energy by means of a simple Landau expansion. Specifically we identify the contribution to the Hamiltonian, given in eq. (2.1), arising from the magnetic moment of the rare earth ions, and we define the free energy per unit volume Fm~g as

f m a g : -- I i~j 21 f d 3 x f d 3 y ( l i ( x , J ( x - y ) $ j ( y ) ) - l j ~ ) f d 3 x < $ r ( x ) ' ( B ( x ) + l ~ P ( x ) ~ ( x ) ) > - T ' m ,

(3.1)

where Sm denotes the entropy per unit volume associated with the magnetic degrees of freedom. We simplify the present analysis by noting that, owing to the singlet nature of the superconducting condensate and the fact that the s-f interaction is weak, we can neglect the contribution of the conduction electrons to Fmagin the above expression. The above free energy, we assume, may be expanded in terms of the order parameter fields m, (x) defined as m, (x) = (s~ ( x ) ) ,

(3.2)

corresponding to the mean magnetisation on each of the rare earth ions in each unit cell. We further assume with regard to the magnetic properties that the unit cell of the bct lattice may be considered as two equivalent subcells each comprising four Er ions and that the effect of the anisotropy is to restrict the direction of the order parameter fields to the plane normal to the c axis. The order parameter fields may therefore be expressed as (3.3)

mi = rnr (cos0,e, - sin 0,e2) ,

where m, denotes the amplitude of the magnetic order parameter on the ith site, 0r denotes the angle with respect to the a axis and ¢1, e2 denote the unit vectors defining the two a axes and i = ( l, 2, 3, 4). For the present analysis it is sufficient to consider the following expansion for the magnetic free energy.

4 fmag = E (ao + a l sin20,)m~ +a2 m4 - m i B cos (O,-OB) +c ~ rnimj cos (Oi-Oj) , i= I

(3.4)

i~j

where ao, aj, a2 and c denote the Landau parameters, and B and 0B denote the magnitude and angle of the magnetic induction field respectively. The above free energy reflects the four-fold symmetry of the bct crystal structure with respect to the c axis. If we further assume that the Landau parameter at characterising the degree of the planar anisotropy, is sufficiently large then we need only consider configurations for which 0i-- {0, 7t/ 2, 7t, 3n/2}. The above assumptions correspond approximately to the situation which exists in the bct phase of ErRh4B4. The order parameters m, are determined from the free energy Fmag by the requirement that

0Fmag = 0 . 0mr From this we obtain a set of coupled non-linear equations for the order parameters mi:

(3.5)

O. Wong et al. / M i x e d state in bct ErRh~B4

2aomi+4a2m3+2c ~ mjcos (Oi-Oj)-Bcos (0/--0B)=0 •

41

(3.6)

Despite the apparent simplicity of the above set of equations there can exist, for a particular choice of Landau parameters, several distinct stable solutions, corresponding to the various local minima of the free energy given by eq. (3.4). The physically realised solution of course corresponds to the global minima, which is dependent upon the value of the magnetic induction field B. By changing the induction field we can induce a transformation from one magnetic configuration to another. The precise nature of the stable magnetic configurations and the fields required to induce a transformation from one to another depends on the particular choice of the Landau coefficients. Let us consider the particular case B = 0 . In this care we find that when ao-c<0

(3.7)

we obtain a solution for which 0~ = 0 2 = 0 and 03=04=7~ with m~ =m2=m3=m4=m, where m # 0 . This corresponds to an antiferromagnetic (AF) configuration. We therefore write

a o - c = a ( T - TA)

(3.8)

and identify TA as the N6el temperature. Similarly for ao+3C<0,

(3.9)

it is straightforward to show that we can obtain a solution for which 0~=02=03=04=0 with m t = m 2 = m 3 = m4= m, where rn ~ 0. This corresponds to a ferromagnetic configuration. We therefore write

ao + 3C=a( T - Tv)

(3.10)

and identify Tv as the Curie temperature. The Landau parameters may therefore be expressed as OL

C= ~ (TA--TF)

(3.11)

and a o = (T-- ~(3TA--TF)) .

(3.12)

For c> 0 we have that TA>TF

(3.13)

and Fmag(AF)
(3.14)

where g m a g ( F ) and Fmag(AF) represent the free energies associated with the ferromagnetic and antiferromagnetic states respectively. In addition to the antiferromagnetic (AF) and ferromagnetic (F) solutions referred to above there exist a number of other solutions which contain both an antiferromagnetic and a ferromagnetic component. These are generally referred to as ferrimagnetic states. In appendix B we consider three such solutions which we refer to as the intermediate 1 (I1), the intermediate 2 (I2) and the intermediate 3 (13), respectively. The spin configurations for each of the solutions is shown schematically in fig. 2. On the application of a homogeneous magnetic field the stable configuration changes and a transformation from one magnetic configuration to another occurs. In order to analyse the detailed nature of such transformations, we must specify the dependence of the Landau parameters on the induction field [ 12 ]. However, in the absence of a detailed microscopic model, to describe the complex magnetic interactions which must exist in the bct phase of ErRh4B4, this requires the introduction of additional parameters into our expression for

42

O. Wong et al. / Mixed state in bct ErRh4B4

$$

AF

m,

$$q"., Int.

1

m'1

Int.

2

>

m ,t

m'2

2

$$ Int.

m3

___)

3

> ~N ~N ~ F

m'

3

/~ m F

Fig. 2. Schematicrepresentationofthemagneticconfigurations consideredin the presentanalysis.

the magnetic free energy. Since the introduction of what are essentially experimentally inaccessable parameters adds little to the present analysis while adding substantially to the complexity calculations involved we treat the Landau parameters ao, a~, a2 and c as independent of the magnetic field within the region of interest. In appendix B we present a relatively detailed analysis of the phase behaviour obtained from the free energy given by eq. (3.4) together with the corresponding expression for the order parameters rnj given by eq. (3.6). For the particular case 0B=0 and TA> TF we find that for B B3 the ferromagnetic state has the lowest free energy. The resultant magnetisation curves for TF/TA = 0.2 are shown in fig. 3 for T/TA=O.O, 0.5 and 0.9. The curves clearly show the discontinuity in the magnetisation at the fields B~ and B3. In fig. 4 we p r e s ~ h showing the temperature dependence of the fields b~ = B l / 4a~_M3 and b2=B2/4a2M2o with Mo=x/aTA/2a2 which represents the magnetisation at zero temperature. In all our calculations we find that to a good approximation b,~(l-

TF'~ / l T TA]NI - T-~n '

(3.15)

b~(l- :r~~ T TA.] "~/1 / --~'n"

(3.16)

For the case 0h ~ 0 the phase behaviour is somewhat more complex since there now appear regions in which the other intermediate magnetic configurations are stable. Detailed phase diagrams for various choices of parameters are presented in fig. 5 which illustrates the fact that despite the somewhat simple structure of the free energy given by eq. (3.4) it nevertheless gives rise to a very complex phase behaviour. However, since in our

43

O. Wong et al. I Mixed state in bct ErRh4B,

40-

T= 0.0

~

T= 0.5

5.0

0.5 ~

M Mo

/ -

T=0.9 .0.4

i

20-

4OZM~o3~ 0.2

LO0.1

o,

02

0'.3 o!4 o'.~ o'.~ o'7 B 4o 2 M3o

0.1 02 0.3 0.4 05 06 0.? 08 0.9 1.0

o'.a

Fig. 4. The reduced fields bt and b 3 a s a function of the reduced temperature t = T / T A for the parameter choice t2=0.2 and 0a=0.0.

Fig. 3. The net magnetisation as a function of the applied field b = B~ (4a2M 3 ) for the parameter choice t2= 0.2 and 0B---0.0. The

discontinuous behaviour in the magnetisation at b= b~andb= b3, indicating the transition from the AF to the I 1 state and from the I l state to the AF state respectively.

0.8

004

F /

(0)

/

F

003

0.9

(b)

(c)

0.6 ¸

•

B

B 402 M~

11

402--='~ 0.02

0.01 .

Z3

0.7

B 4~ M~ Ores

0.4

0.2

0.3 AF

AF

AF

0.00

I

i

0

0.0

*

i

%

It

4

O,I

i

, K

Oe

Fig. 5. Phase diagrams obtained from the solution ofeqs. (B.2) to (B.21) for tF=0.9 and t---0.9,0.5 and 0.0. analysis of the mixed state we confine ourselves to the case 0B = 0 we need only consider the antiferromagnetic ( A F ) state, the intermediate 1 (I 1 ) state a n d the ferromagnetic ( F ) state. In the following sections we therefore refer to the intermediate (I 1 ) state simply as the ferrimagnetic state.

4. The free energy of the mixed state In this section we c o m b i n e the results presented in section 2, in which we discussed the electronic and electromagnetic properties of the superconducting mixed state with the results of the previous section, regarding the magnetic properties of the rare earth ions to obtain an expression for the Gibbs free energy for the mixed state as a function of the vortex density. In deriving our expression for the G i b b s free energy and in the subsequent analysis, in which we compute the equilibrium vortex density n as a function of the applied external

O. Wonget al. / Mixedstate in bct ErRh~B4

44

field, we follow the procedure outlined in refs. [20] and [21 ]. For T < TA we compute the equlibrium vortex density and the Gibbs free energy for each magnetic configuration. A comparison of the Gibbs free energy for a given applied external field then determines the physically realised configuration. In the analysis we restrict our considerations to the case in which vortex lattice lies parallel to the easy axis of magnetisation and hence B(x), H(x) and M,(x)= Y~i(si(x)) all lie parallel to the z axis. Following the procedure in ref. [20] we consider the Hamiltonian given in eq. (2.1) to consist of three parts, representing the contribution from the electronic, electromagnetic and magnetic degrees of freedom. These we denote by .~f~, "~'em and ~f,, respectively and are given by the following expressions:

.If,.,=g/t(x)e(--iV)q/(x)--VN{(x)~Nz(x)q/r(x)--½(l~si(x)--pBB(x))gt*(x)~r~(x),

(4.1)

A(X)-

(4.2)

"~m= ~

(IB(x)12+ IE(x)12)+/(x) "

Vf(x)

--½(s,(x)--ttB~u*(x)~rc/(x))B(x),

.Ygm= ~,-- ½~ f d3y s, (x)J(x-y)sj (y) - ~ ~ s, (x) (B(x) +I~/*(x)~,(x) ),

(4.3 )

where ~'cr denotes the effect of the crystal fields on the rare earth ions. We do not specify o~¢~fexplicitly but include its presence to account for the anisotropy of the magnetic response discussed in the previous section. We define U~, Ucm and Um as Ucl = f d3x(.~cl(x) ) ,

(4.4)

U~m = f d3x(J~m(X) } ,

(4.5)

(. Um = J d3x(.~m(X) ) ,

(4.6)

d

where V represents the volume. By means of the propagator compute Ue~ to give

If(

No,- (27t)3

d3k

S ( x - y ) , defined in eqs. (2.9) and (2.8), we can

e-E+ -~ (1-f(E+lt)-f(E-it))

+ (E-tt)f(E-/~) + ( E + tt)f(E + tt) + ~(f(E-/z) - f ( E ( E + tt) ) ) ,

(4.7)

while the entropy associated with the electronic degrees of freedom is given by

S~,- (2~) 1 3 f d3k(f(E+p)ln(f(E+l~))+(l_f(E+#))ln(l_f(E+lt)) +f(E--#)ln(f(E-lt) ) + ( 1 - f ( E - I t ) )In( 1 - f ( E - I t ) ) ).

(4.8)

Combining these two terms we obtain the following expression for the free energy of the superconducting electrons F~l = Ucj-fl-IScl:

if(

FS,= (27t)3

+ fl-'In(1

d3k

e-E+ ~-E(1-f(E+lt)-f(E-It))

- f ( E - I t ) ) +fl-'ln(1 - f i E + p ) ) + ~Z ( f ( E - p ) - f ( E + # ) ) ) .}

(4.9)

O. Wonget al. /Mixedstatein bctErRh4B4

45

We define the condensation energy as H i / 8 n as Hi N S 8n =Fel - F e t ,

(4.10)

where FeN denotes the free energy of the electrons in the normal state (i.e. A=0). In the evaluation of the electromagnetic free energy we must include the self-energy of the photon calculated in the superconducting state. This is outlined in some detail in ref. [22 ] and yields the following result Uem --

8~Vf d3xn(x)n(x)'

(4.11)

where n (x) denotes the vortex density and is defined by eq. (2.28) and tion (4.11 ) simplifies to give

H(x)

denotes the magnetic field. Equa-

nO Uem= ~-~ H ( 0 ) ,

(4.12)

where H ( 0 ) represents the magnetic field at the vortex core. If we write the fields

H(x)

and

B(x)

as

H(x)= E

H(KI) eix''x

(4.13)

B(x)= ~

B(Kz)e ~x''x ,

(4.14)

/

/

where Kz denote the reciprocal lattice vectors associated with the vortex lattice, then from eq. (2.27) as

B(K~) may be calculated

1 + 4nX)2~-2 C(K~) B(K/)=

IK~I2+(I+4~Z)2~2C(Kt ) n0,

where for

H(K/)

(4.15)

we obtain that 2ff2C(K/)

H(K/)= IKII2+(l+4nZ)2£2C(K~ ) n~.

(4.16)

The field at the centre of the vortex core, H ( 0 ) , may then be evaluated as H(0)= ~

H(K/)=nO-4rc ~ mi+h'(O),

/

(4.17)

i

where we have defined h' (0) as

J.c2C(KI)

h' ( 0 ) = n 0 ~ 2+ J,o IKzl (1 +4=X)2~-2C(KI) "

(4.18)

The contribution to the free energy from the magnetic degrees of freedom may be obtained in the manner discussed in the preceding section. Defining FM as

FM=Um-TSm,

(4.19)

where SM denotes the entropy associated with the magnetic degrees of freedom, we obtain FM = 1 ~ d 3 x ( ( ~ r ) -

~m,(x)B(x)cos(O,))

-- 12V Ei.j~ d32 ; d3ym'(x)J(x--y)mj(Y)cos(Oi--Oj)+ 2~ ~ d3x ~ mi(x)B(x)cos (Oi)- TSm .

(4.20)

O. Wonget al. / Mixed state in bct ErRh4B~

46

Comparing this expression with eq. (3.1) we may write FM as 1

FM =Fmag + ~ ~

m, nO.

(4.21)

The value of Fr~ag may be calculated in the manner described in appendix B for a given value of n and T. For T< TA we require that this be calculated for each magnetic configuration namely the antiferromagnetic, the intermediate and the ferromagnetic configuration. To complete our calculation of the free energy it remains only to calculate the effect of the vortices on the condensation energy. This is referred to as the core energy and may be understood as follows. The presence of a vortex will give rise to an effective potential which acts on the electrons in the vicinity of the vortex core. Roughly speaking we may imagine the potential as arising from the spatial variation of the order parameter Ll(x). This will modify the electron wave function and the corresponding energy spectra and will in general result in an increase in the electronic contribution to the free energy, namely the condensation energy. While the exact calculation of the change in the condensation energy induced by the boson transformation is extremely difficult it is nevertheless possible to make a reliable estimate of the core energy as a function of the vortex density based on the following functional form [20,23,24]: E ..... = 8~2~

~-e2bim(n)

,

(4.22)

where the first term represents the core energy of a single vortex while the second term represents the effect of multiple vortices on the condensation energy. The field bi,t(n) represents the magnetic induction field at the vortex core due to the presence of the other vortices and may be computed from the solution of the Maxwell equation to give (~1 +4~Z)2£2 C(K') ~ bi,, ( n ) = nO ( 1 + ~ / IKII-+(I+4nZ)2c2C(Kz),}

O fd2k (2/I;) 2

(l+47tZ)2C2C(k) k2+(l+41tZ)2cZC(KI) '

(4.23)

where Z~ denotes the sum over the reciprocal lattice vectors Kz of the vortex lattice. The coefficient E2appearing in eq. (4.22), and which determines the overall scale of the multiple vortex effect on the condensation energy, is determined by the requirement that there exist a continuous transition from the mixed state to the normal state. We will discuss the determination of ez in more detail later. Combining terms we obtain the following expression for the free energy of the mixed state as a function of the vortex density n:

FS(n)--H~-

8---~+nO_~n(no+h,(O))+ 8__~Ln02(

1~-f:2bim(n))+Fmag(nO) ,

(4.24)

where h' (0) is given by eq. (4.18). In order to obtain the equilibrium vortex density for a given applied external field Hex, we define the Gibbs free energy for the mixed state to be

GS(n)=FS(n) - ~ Hcx,

(4.25)

and the equilibrium vortex density for the mixed state is then obtained from the requirement that dGS(n) - 0 . dn

(4.26)

o. Wong et aL / Mixed state in bct ErRh4B4

47

The solution to eq. (4.26) serves to determine the applied external field required to stabilise a vortex lattice with vortex density n. This may be written as (4.27)

H ( n ) =Hex~.

The net magnetisation M may then be calculated as a function of the vortex density as 4n=nfb-H(

n) .

(4.28)

Alternatively the net magnetisation may also be expressed as a function of the applied external field by means ofeqs. (4.27) and (4.28). It remains however to determine the value of the parameter E2. This is determined by the requirement that there exists a continuous transition from the mixed state to the normal state at some value of the vortex density nc corresponding to an applied external field H*2. Thus we require that ncO=H*2 + 4n~7/(H*2 )

(4.29)

and that GN(H*2) = GS(ncq~) ,

(4.30)

respectively, where ~Q(Hex,) denotes the net magnetisation in the normal state for a given applied external magnetic field Hex, and G N(Hex,) denotes the corresponding value of the Gibbs free energy of the normal state. This is given by G N = _ H~x, +FM(Hex,) • 8n

(4.31)

The solution of the above set of equations permits the simultaneous determination of H*2 and e2. Above the N6el temperature this serves to determine the upper critical field He2 since it is at this point that the system makes the transition to the normal state. Below the N6el temperature the situation is complicated by the fact that the system can exist in one of three magnetic configurations, we assume therefore that for each magnetic configuration there exists a continuous transition from the mixed state to the normal state at some value of the vortex density no. By means of eqs. (4.29) and (4.30) it is possible to determine H*2 and ~2 for each magnetic configuration and hence determine the Gibbs free energy for each magnetic configuration by means of eq. (4.25). In this region the upper critical field is obtained by comparing the Gibbs free energy of the mixed state for each of the three magnetic configurations with that of the normal state for each of the three magnetic configurations. The transition from the mixed state to the normal state then occurs when the Gibbs free energy of the most stable configuration in the mixed state is equal to the Gibbs free energy of the most stable configuration in the normal state. This serves to determine the upper critical field He2. If the stable magnetic configuration in the normal and the mixed state is the same at H~x,=Hc2 then the transition to the normal state is simply given by the value of He*2 calculated for that particular configuration and the transition is second order. If however the stable magnetic configuration in the normal and the mixed state are different, at Hext= He2, then the transition to the normal state will be first order and will be accompanied by a spin reversal. Examples of both are in the contained numerical results presented in the following section. Similar considerations apply to the determination of the lower critical field H~,. At the lower critical field H2 we require that GM=GS(nc,)

,

(4.32)

where the critical vortex density nd is determined from the solution of eq. (4.27) as He, = H ( n c , ) and G M denotes the free energy of the Meissner state which is given as

(4.33)

48

GM - -

O. Wong et al. / Mixed state in bct ErRh4B4

H~(n=0) +FM(n=0)

(4.34)

8~

since it is at this point that the applied external field is large enough to stabilise vortices within the superconductor and consequently the system will make the transition from the Meissner state to the mixed state. In the case of non-magnetic superconductors it is well known [29,23 ] that, if the vortex-vortex interaction is everywhere repulsive then eqs. (4.33) and (4.34) are only satisfied for n = 0 and H = H ( 0 ) . In such a situation the transition from the Meissner state to the mixed is second order and superconductors manifesting this behaviour are generally referred to as type I I / I I superconductors. If on the other hand the vortex-vortex interaction is attractive in some region, then the situation is somewhat more complicated since, in addition to the solution at n = 0 and H = H ( 0 ) , there exists another solution at n = n*# 0 with

H(n*)
(4.35)

Furthermore one finds that the solution of eqs. (4.33) and (4.34) are unstable in the region 0 < n < n*. This arises because the attractive nature of the vortex-vortex interaction means that the free energy will have a minimum at some finite value of the intervortex distance. The attractive part of the vortex-vortex interaction, which is typically electromagnetic in origin, serves to counteract the generally repulsive nature of the core energy in the neighbourhood of Hc~ [29,23 ]. The presence of such a solution indicates a first order transition at Hc~ which in non-magnetic superconductors is characteristic of a type I I / I superconductor. It is worth noting that while the distinction between a type I I / I I and type I I / I superconductor is quite straightforward in the case of a non-magnetic superconductor the situation in a magnetic superconductor is somewhat more complicated since the renormalisation of the London penetration depth by the susceptibility appearing in eq. (2.27) means that the order of the transition at Hcl can change from second order at high temperature to first order at low temperature [27]. Indeed detailed calculations indicate that this is indeed the situation that arises in pt ErRh4B4, although experimental confirmation of this behavior is difficult due to flux pinning at Hc~. In the case of a metamagnetic system, such as that presently under consideration, the nature of the transition at Hd is further complicated by the fact that, below the N6el temperature, the transition can be accompanied by a spin reversal. Such complications can be accommodated within the present analysis since the determination of Hc~ follows simply from a comparison from the Gibbs free energies G M, for-the Meissner state, and G s, for the mixed state, calculated for the most stable magnetic configuration. For computational purposes the variables appearing in the above equations are normalised by means of parameter Ao and//co, which are defined as do = 2¢ODexp ( ~ )

(4.36)

and H~o = 4rtA2N(0).

(4.37)

By means of the above parameters we can rewrite the Gibbs free energy in terms of the dimensionless variables listed in table I. The dimensionless Gibbs free energy for the mixed state is given as 3/¢2 2 Gs(~)= ~(~g(ti)- ~4 hc(~))+fmag,

(4.38)

where g ( a ) is given by 1

g(n) q- h-' (n) -k- ~ - ~ ( ~

1

- ~2~nt ( n ) ) ,

(4.39)

O. Wong et al. / Mixed state in bct ErRh4B4

49

Table I List of dimensionless variables.

fi= 1~/3o hc(rT) =Hc/Hco fi = n2 ~,, t= T /T~ t, = T~/T¢ t,= T~IT~

~Tu(~) : Zu/AL<> c=4x/a2M~ u = 4a,_M3o/02 ~? [=lMo/Ao fi' (~) =h' (0)10~? 6,,,(a) = b,°,/¢~~?

GS(rT) =GS(n)/O22~ 4 G M= GM/fb~-2E4 GN(fl) = G N ( n ) / 0 2 2 ~ , '1 h~, = H~x,/02E,,-"

fmag=Fm~g/022C? ~_,= ~2q~2~,,-' t~,=mi/mo

where X=~L0/~0 and g,o= h vr/nAo denotes the superconducting coherence length. The dimensionless ratio x is closely related to the Ginzburg-Landau parameter [23] and serves to characterise the superconducting prop° erties in the mixed state. The value of x is generally determined in order to provide a satisfactory fit with experiment. The quantities/t,/2, c and u, defined in table I, characterise the magnetic interactions and properties of the Er ions in the material. These are chosen in accordance with the data presented for the bct phase of ErRh4B4. Finally the p a r a m e t e r / s e r v e s to determine the strength of the exchange coupling between the conduction electrons and the Er ions. These parameters essentially provide a complete description of the system, within the context to the boson method and the approximations developed in the previous section, from which quantities such as Hct may be computed over the entire temperature range in the manner outlined. In the subsequent sections we present the results for three values of x and I w h i c h serve to illustrate the various possible behaviours. The quantities hc and ;t may be calculated from the following expressions

hc( ~ ) = ~ f ' ( t, fl) ,

(4.40)

)?(~) = 5e(t, f i ) ,

(4.41)

where the functions ~ ( t , / 2 ) and LP(t,/i) are given in Appendix C. The magnetic contribution to the free energy, fmas, appearing in eq. (4.38), together with the parameter/i and the susceptibility Z may be calculated in terms of the dimensionless variables for each of the three magnetic configurations as follows. i) The antiferromagnetic s/a/e: The antiferromagnetic state may characterised by two order parameters fi and f i ' with

fi, = f i 2 = f i 01=02=0

and and

DT/3=/'Y/4=/~/' ,

03=04=x.

The magnetic free energy fmag may then be expressed as fm,~(AF) = -cu2[~ i - - ~ ~ (m-4 + f i t " ) + ~ ( t _ t , ) ( f i 2 + f i , 2

) + ~ ( 1t ~ - t , _ ) ( f i - f i ' ) 2 - 2 n ~ O u ( m - f i ' ) ) ,

(4.42)

where fi and f i ' are obtained from the solution of fi3+ fi,3+

(/-t,)fi+ ~

(/--t,)fi'--

2-~(/,--/2)(fi--fi')-

~ 1( / ~ - / 2 ) ( f i - - f i ' ) +

riO=Ou rTO=O " u

(4.43) (4.44)

50

O. Wonget al. /Mixed statein bct ErRh4B4

The parameter fl, expressed in terms of the dimensionless variables fi and fi', is given by

I~mi /2= j - - ~

=2Bm-m')

.

(4.45)

ii) The intermediate 1 state: The intermediate state may also be characterised in terms of two order parameters fi and fi' are defined as FF/l=F/7/2=gn3~--~-Fn and 0,=02=03=0

and

/'/~3=/~/' ,

03=n.

The magnetic free energy fmag may be written in terms of fi and fit' as

cu2:

fmag(I1)= 1 6 n k , ( 3 f i 4 + f i ' 4 ) +

(t-t,)(3fi2+fi'2)+ -g-~(t,-t2)(3fi-fi')2-2

)

)

(4.46)

where fi and fi' are obtained from

(t-fi)m+-~ (t'-tz)(3~-fi')-

rn3+ 1

afb=Ou'

1

fi,3+ ~ ( t - t , ) m ' - --~(t, - t 2 ) ( 3 f i - fi' ) + tt~=O " u

(4.47) (4.48)

The parameter/2, expressed in terms of the dimensionless variables fi and f i ' , is given by

IEm, fl=

Ao

--2I(3m--m').

(4.49)

iii) The ferromagnetic state: The ferromagnetic state may be characterised in terms of one order parameter fi, which is defined as t/7/I = / ' ~ 2 =/~/3 = / ~ 4 =/'~/ , O I - ~ 0 2 ~---O3 = 8 4 = 0 .

The magnetic free energy fmag is given by fmag(F)= ~

m4q- t7

(4.50)

where fi is obtained from fi3+ ~ ( t - - t 2 ) f i - - nO=0.u

(4.51)

The parameter fl, expressed in terms of the dimensionless variable fi is given as

l~mi ~=

i ,40

=4fro .

(4.52)

In an analogous fashion the Gibbs free energy for the Meissner state and the normal state may be expressed in terms of the dimensionless variables as

O. Wong et al. / Mixed state in bct ErRh4B4

51

(~M(~)_~._ h2(/~=0) ..[..fraag(/,~=O)

(4.53)

8u

and

(~ N(/,~) =

h 2xt(/z) 8g "kfmag( hext) .

(4.54)

In this section we have shown how the free energy of the mixed state may be evaluated by means of the boson method, the resultant expression is given by eq. (4.24). From this the magnetisation may be calculated for a given applied external field from the Gibbs free energy by means of eq. (4.26). The phase boundaries may be obtained by a comparison of the Gibbs free energies for the mixed state, the Meissner state and the normal state given by eqs. (4.25), (4.31) and (4.34) respectively, which must be computed for each of the three magnetic configurations considered. In deriving the above expansion we have introduced a number of simplifying assumptions. In particular we assumed that the contribution of the magnetic degrees of freedom of the rare earth ions to the free energy may be expressed as a Landau expansion in the other parameter fields m r ( x ) , which are defined as the mean magnetisation of the ith rare earth ion in the unit cell ( i = 1...4). The temperature dependence of the Landau parameters was carefully chosen in order to reproduce, qualitatively, the metamagnetic character of ErRh4B4. Furthermore it was assumed that the spatial variation of the magnetisation was linearly related to the spatial variation of the magnetic induction field B ( x ) . It is to be hoped that despite the rather simple representation of the magnetic properties we nevertheless still retain many of the essential characteristics of a metamagnetic system. In the next section we examine the results obtained from the analysis of this section for various choices of parameters.

5. T h e r e s u l t s

In this section we present the results of our calculations based on the formalism described in the previous section. In particular we present results based on three parameter sets which show a variety of different types of behaviour. The parameter sets are listed in table II. The procedure is then as follows, for each of the potentially stable configurations below the N6el temperature, we calculate the corresponding value of ~2 and HoE by means of eqs. (4.29) and (4.30), from this we can compute the Gibbs free energy GS(n) of each configuration as a function of the applied external field Hext(n) and from this we can find the most stable configuration for a given value of the applied field and hence compute the magnetisation curves and critical field curves. For each parameter set we present the upper critical field curve H¢2 as well as a more detailed phase Table II Parameter set used for figs. 6-8 (column a), figs. (9-11 ) (column (b))and figs. (12-14) (column (c)).

x [ c VN(O)

u tA tv

a

b

c

3.3 0.23 0.5 0.3 2.0 0.0833 0.0166

2.0 0.29 0.5 0.3 2.0 0.0833 0.0166

1.5 0.2 0,5 0,3 2.0 0,0833 0.0166

52

O. Wong et al. / M i x e d state in bct ErRh~B4

diagram for the temperature region below TA. Magnetisation curves for three temperatures below the N6el temperature are also presented. The effect of the metamagnetic transition is clearly seen on each of the curves.

5.1. x=3.3 a n d [ = 0 . 2 3 The free energy curves obtained for the parameter set given in column a of table II indicate a field induced transition in the magnetic structure at fields Hi and/-/3. We find that for all temperatures below the N6el temperature//3 < He2 and consequently He2 corresponds to the field H*2, defined in the previous section, calculated for the ferromagnetic state. The sequence of phase transitions, with increasing applied external field is therefore given as at H~l antiferro. Meissner state

~ antiferro.mixed state,

at Hi antiferro, mixed state

--, ferrimagnetic mixed state,

at

nc3 ferrimagnetic mixed state ~ ferromagnetic mixed state,

at n c 2 ferromagnetic mixed state --, ferromagnetic normal state. The upper critical field curve is shown in fig. 6 and since//3 < H~2 we find that the curves obtained are qualitatively similar to those obtained in the case of a ferromagnetic system. A somewhat more detailed phase dia-

2.0,

H

,o

2.0"

_V

Hc2 1,0-

o 0.2

0.4

0.6

%

0.8

1.0

Fig. 6. The upper critical field He2 as a function of the reduced temperature calculated from the parameters given in column a of table If.

o & o G3 o

,,oas o T/Tc

o

o

o

I

I

o,

Fig. 7. The phase behaviour for T< TA calculated from the parameters given in column a of table It. Region (i): Meissner state; Region (ii): Antiferromagnetic Meissner state; Region (iii): Antiferromagnetic mixed state; Region (iv): Intermediate mixed state; Region (v): Ferromagnetic mixed state; Region (vi): Ferromagnetic normal state.

O. Wong et al. I M i x e d state in bct ErRh~B4

53

gram for T< TA is shown in fig. 7. The corresponding magnetisation curves for t=0.06, 0.03 and 0.01 are shown in fig. 8 each of which show the effect of a field induced transition very clearly. 5.2. x = 2 a n d [ = 0 . 2 9

For the parameter set given in column b of table II we find that immediately below the N6el temperature the sequence of phase transitions is similar to that given above. However with decreasing temperature the upper critical field decreases while the field H3 increases. At a particular temperature they cross and we obtain the following sequence of transitions which persists down to T = 0: at Hc~ antiferro. Meissner state --, antiferro mixed state, at HI antiferro, mixed state

-, ferrimagnetic mixed state,

at He2 ferrimagnetic mixed state -, ferromagnetic mixed state. The resultant upper critical field curve displays a cusp at the point at which the upper critical field He2 and He3 intersect; this is seen in fig. 9. Below this point the transition to the normal state is first order in that the magnetisation changes discontinuously in going from the mixed state to the normal state. The rise in the upper critical field curve below the N6el temperature reflects the fact that the transition to the normal state occurs as a result of the transition from the ferrimagnetic to the ferromagnetic state, since the field required to induce this transition increases with decreasing temperature we find that the upper critical field rises with decreasing temperature. In fig. 10 we present a more detailed phase diagram for the temperature range T_< TA. The magnetisation curves for t--0.06, 0.03 and 0.01 are shown in fig. 11 and again we see the effect of the metamagnetic character on the magnetisation curves at all three temperatures. 5.3. x = l.5 a n d [ = 0 . 2

For the parameter set given in column c of table II we find that immediately below the N6el temperature the sequence of transitions is equivalent to that given in section 5.1. At t~ 0.03 however the upper critical intersects the field Ha and the phase boundary shows a cusp. Immediately below this temperature the sequence of phase transitions is similar to that given in section 5.2. At t ~ 0.02 the transition to the normal state occurs below Ha and hence coincides with the field H*2 calculated for the intermediate state, and hence we obtain the following sequence of transitions:

Hc2

Hext 0.0, -02 _

4'n" M

-0.4

,/1

-

4,/X~o °~

~,,'x%

~

Hext

Idlxt

_ 4TM

-0.2

4,/x~o-o.4

0.2

•2 0.4 016 0!70'.8019

0.0 "0.2" _ 4"wM

2 0!4

0:6

0:8

I:0

-04-

-0.6

- 06

-0.8

-OB

•

-0.6 -0.8 - LO

-I.0

- 1.2

I0

-12

Fig. 8. The net magnetisation as a function of the applied external field calculated using the parameters given in column a of table II, for t= T/Tc=O.O1, 0.03 and 0.06.

54

O. Wong et al. I M i x e d state in bct ErRh4B4 1.0

,2I

0.9

1.0

0.8 0.7

Hc2

0.8"

0.6

¢>IXo

H

¢,/x~" 05 0.4.

0.4-

IV

2

0.3-

0.2-

0.2

0.2 0.4 0.6 0.8 1.0

0.0

0.1 , ,

T,Tc

, l , II ,

IT

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

T/T c Fig. 10. The phase behaviour for T< T A calculated from the parameters given in column b of table II. Region (i): Meissner state; Region (ii): Antiferromagnetic Meissner state; Region (iii): Antiferromagnetic mixed state; Region (iv): Intermediate mixed state; Region (v): Ferromagnetic mixed state; Region (vi): Ferromagnetic normal state.

Fig. 9. The upper critical field He2 as a function of the reduced temperature calculated from the parameters given in column b of table If.

0.2

O0

-

4.rM -02

H==i

oi,

o12 0[3 03

°~~a)

--

" ~ o'.2

0.2 •~1 4 o.o

ds

Hilt!

Hemt

0.2

o14

L

O.

-0.4

o'.~ (b)

o.o -oz

'

~X~o o!~ (el

4.q,/X~o-O.6 eM - 0.4

-0.6 -08

-0.6

-0.8 - IO

-0.8

-

1.0

Fig. I I. The net magnetisation as a function of the applied external field calculated using the parameters given in column b of table II, for t = T/Tc=O.Ol, 0.03 and 0.06.

at Hcl antiferro. Meissner state

- , antiferro mixed state,

at H, antiferro, mixed state

--} ferrimagnetic mixed state,

at He2 ferrimagnetic mixed state --, ferrimagnetic normal state, at//ca ferrimagnetic normal state --, ferromagnetic normal state.

55

O. Wong et al. /Mixed state in bct ErRh4B4

The above b e h a v i o u r gives rise to the second cusp in the upper critical field curve, as shown in fig. 12. The corresponding phase diagram for T < TA is shown in fig. 13. The resultant magnetisation diagrams for t = 0.06, 0.03 and 0.01 are shown in fig. 14. In s u m m a r y we have presented within the a p p r o x i m a t i o n discussed in the previous section the upper critical field curves and magnetisation curves for a four sub-lattice antiferromagnetic superconductor. Below the N6el temperature the metamagnetic character of the magnetic ions gives rise to structure in both the magnetisation

0,5

°6 I 0.4

0.50.4"

0.3

Hc2

H

, / XZLO0.30.2

0.20.1 J

0.1 0.0 0.0

rr

0,2

0.4

0.6

%

0.8

1.0

O0 -0.2 -

4TM

-0.4

Hext

I

I

0 0.01 0.020.030.04 0.0.50.060.07 0.080.09 0.1 T/T c

Fig. 12. The upper critical field H¢2 as a function of the reduced temperature calculated from the parameters given in column c of table I1.

02

m

0.2

Fig. 13. The phase behaviour for T< TAcalculated from the parameters given in column (c) of table II. Region (i): Meissner state; Region (ii): AntiferromagneticMeissnerstate; Region (iii): Antiferromagneticmixed state; Region (iv): Intermediate mixed state; Region (v): Ferromagnetic mixed state; Region (vi): Ferromagneticnormal state; Region (vii); Intermediatenormal state.

02~,L

He~t

Ht=I ~,'X~.o

o'3 0'4 o'~ o'6

o.o

o:~

o'.3

0.0

o'.4 oL~

-0.2 4T....MM

(b)

o.i~,~[

(c)°"3

41rM

,/,/X~o -02

" ~X~o-o.4 -0~

- 0,6

-0.8' - 0.8

1

-0.4" -0.6

Fig. 14. The net magnetisation as a function of the applied external field calculated using the parameters given in column c of table II, for t=T/Tc=O.Ol, 0.03 and 0.06.

56

o. Wonget al. / Mixed state in bct ErRh4B4

curves and the upper critical field curve. In particular the sequence of the transitions can, for a different parameter choice, change with decreasing temperature giving rise to discontinuities in the slope of the upper critical field curve. The results obtained for the particular parameter choice x = 3.3 and [ = 0.23 are qualitatively similar to the experimental results obtained for the bct phase of ErRhaB4. We note however that the magnetisation curves show no sign of the anomaly observed in these measurements and discussed in section 2.

6. The magnetic domain structure in the mixed state

The analysis of the magnetic field distribution, for the vortex lattice presented in section 3, was based on the assumption that we need only evaluate the spatial distribution of the magnetisation fields m i ( x ) to the lowest order in the induction field. As we have pointed out, the metamagnetic character of the system we are considering means that such an assumption may not be appropriate, as there exists the possibility that small variations in the induction field could give rise to substantial variations in the magnetisation fields. This may be of particular importance as the applied external field approaches the value at which a transition from one magnetic configuration to another occurs; This possibility has been discussed by a number of workers [ 18,19 ] and it has been proposed that such an effect may account for the anomalous decrease in the magnetisation observed in the recent experiments, although the published work differs with regard to the precise nature of the mechanism involved. It is therefore interesting to extend our present calculation to include this possibility. We therefore modify the calculation of the antiferromagnetic mixed state described in section 3 to include the presence of a cylindrical ferrimagnetic domain of radius R around the vortex core. Within each domain we assume that the spatial variation of the magnetisation is negligible and need only be treated to lowest order. We therefore divide the unit cell of the vortex lattice up into two domains, domain 1 (r < R) and domain 2 ( r > R) and approximate the magnetisation in each domain as M(x)=M(BI)=Mj

r
M(x)=M(B2)=M2

r
(6.1)

where Bj and B2 represent the spatial average of the magnetic induction fields in domains 1 and 2 respectively. As in section 3 we expand the fields H ( x ) , B ( x ) and M ( x ) as H ( x ) = ~ H ( K t ) e ix~'x ,

(6.2)

/

B ( x ) = ~. B ( K ~ ) ¢ x''x ,

(6.3)

/

M ( x ) = ~ M ( K ~ ) ¢ x''x ,

(6.4)

I

where K~ represents the reciprocal lattice vectors associated with the vortex lattice. From eqs. (6.1) we obtain that 2nR 2 Jl ( KtR ) M ( K , ) =M26co q- - - - - ~ (Ml --)142) - K~R '

(6.5)

where/2 denotes the area of the unit cell of the vortex lattice and Jl is the first order Bessel function. The function B ( K D may be calculated from eq. (2.20) as 1 B(KI ) = K~ +2L-2 C(K/) n(2ff2C(K/)fO+8n2(M~ - M 2 ) K t R J, ( K t R ) )

(6.6)

O. Wong et al. / M i x e d state in bct ErRh4B¢

57

and since H (x) = B (x) - 4riM(x) we obtain the following expression for

H(KI) = K~)]'LZC(K') +,,].E-2C(K/) n ( q~-8neR2(M~

-3'/2) ~ J'

)

_ 4~M2~;,,0.

(6.7)

The second term appearing in the brackets represents the effect of the ferrimagnetic core on the spatial distribution of the magnetic field. We note that the above solution for the induction field B(x) satisfies the requirement of flux quantisation since

f d2xB(x)=n f d2xB(x)=ng2B(K,=o)=nfb.

(6.8)

12

The spatial averaged fields, Bt and

if d2xB(x) B~_= ~_if d2xB(x) B~ = ~

=2 ~

B2

may be calculated as

B(KI) JI(KIR_._____~) K~R

.Qn

=¢-2

~

B(KI) JI(KIR_..__~)KIR'

(6.9)

(6.10)

122

where g2~ and g22 denote the area of domain 1 and 2 respectively. Equations (6.9) and (6.10) for B l and B 2 together with eqs. (6.1) for Mt and M2 constitute a set self-consistent equations which may be solved to yield Bj, B2, M~ and M2 for a given choice of n and R. Once M~ and ME are obtained then we can evaluate the magnetic field distribution at the vortex core H ( 0 )

as

H(O)= ~H(KI)=nO+h'(O)-4n(~-M,+(I-ff-~-)M2),

(6.11)

where h' (0) is given by V ';t L-zC(K') h' (O) =n /~oKT +2~2C(K~) (k~-8~2R2(MI -M2) Jt(K~R)'~ K,R ]"

(6.12)

Similarly the core energy may be written as E ....

= 8~:~,~ ~ -e2bint(rt)

)'

(6.13)

where the field b~,t(n), representing the magnetic induction field at the vortex core due to presence of the other vortices, may be computed from the solution of the Maxwell equation to give

bim(n)=n ( q)+ i~ 7-0b(Ki) ) - fd2Pb(P)4rc--5 where we have defined

b(p)- B(p) n

1

b(p)

-- p Z + 2 ~ Z C ( p )

(6.14)

as

().E2C(p)O+8nZ(Mj-Mz)pRJ,(pR) ) .

(6.15)

Combining the various terms the free energy of the mixed state may be calculated, for a given choice of n and R, as

58

O. Wong et al. / M i x e d state in bct ErRh~B~

FS(n,R)=- ~H~ +

+

~R2/

\/

\

nR2 ~-tFm~g(AF)+ IB, M, ) +t l---~-)(Fmag(II )+½B2Mz)+E.... (n,R)

nO+h'(O)-4nt---~-M,+ 1-

M2

,

(6.16)

where Fmag(II ) and Fma~(AF) represent the free energies in domains 1 and 2 respectively and which may be calculated from B~ and B 2 by means of eq. (4.42) and eq. (4.46) respectively. The parameter e2 is obtained from the calculation described in the previous section. From this we therefore obtain the Gibbs free energy as a function of the vortex density n and the radius of ferrimagnetic core R from the relationship

GS(n, R) = F S ( n ,

(6.17)

R) - ~-~¢Hex,,

where as in our previous discussion Hext represents the applied external magnetic field. The equilibrium vortex density and the radius R are then determined from the requirement that

OGS(n'R) = 0

(6.18)

On and

OGS(n'R) = 0

(6.19)

OR with R > 0. The existence of the ferrlmagnetic core requires that the solution to the above set of equations yield R ¢ 0. We have performed calculations for various parameters. The results for the particular data set x = 1.5 [ = 0.2 for t = 0.01 are presented in fig. 15. The calculations clearly show the effect of the ferrimagnetic domain around the vortex core. As the applied external field increases the radius of the ferrimagnetic core increases, as one would expect, giving rise to the increase in the magnetisation observed in the calculated magnetisation curves. At some value of Hext the Gibbs free energy of the ferrimagnetic mixed state equals that of the anti-

0.2J ql~/X~O0"O (

Hext 011 "~

i

0.2

-O.I

-0.2

Fig. 15. The magnetisation calculated as a function of the applied external field including the presence of the ferrimagnetic domain around the vortex core (solid line) and without (dashed line). The parameters used are those given in column c of table II.

O. Wong et al. / M i x e d state in bct ErRh4B~

59

ferromagnetic mixed state and a first order transition to the mixed state occurs. At this point the area of the ferrimagnetic core has grown to approximately 5% of the area of the unit cell. As seen from the calculated magnetisation curves presented in figs. 15 while the appearance of the ferrimagnetic core modifies the magnetisation curve somewhat the external field required to induce the transition is not much modified.

7. Summary and discussion In the preceding analysis we have outlined how the boson method of superconductivity can be extended to consider the case of magnetic systems which manifest a metamagnetic behaviour. In the analysis the contribution to the free energy, arising from the magnetic interactions associated with the rare earth ions, was expressed as a simple Landau expansion in four magnetic order parameter fields, representing the magnetisation associated with the four rare earth ions in each sub-cell. The direction of the magnetisation was constrained and the nature of the Landau coefficients chosen so as to reproduce, at least qualitatively, the basic features of the magnetic properties of bct ErRh4B4. This was then incorporated into the free energy of the superconductivity mixed state together with the contribution to the free energy arising from electronic degrees of freedom and the electromagnetic contribution of the vortex lattice. The method and the approximations employed followed closely those presented in refs. [ 20] and [21 ]. In particular it was assumed that the spatial variation of the magnetisation need only be calculated to linear order in the magnetic field. The results presented in section 5 for the parameter set presented in column b of table II are qualitatively similar to the easy axis magnetisation curves and critical field behaviour reported in ref. [ 13 ]. In particular the calculated magnetisation curves show the two distinct discontinuities in the magnetisation with increasing field, below the N6el temperature. The magnitude of the discontinuities and the field at which they occur both increase with decreasing temperature. The corresponding curve for the upper critical field shows a minimum just above the N6el temperature corresponding to the coincidence of the upper critical field with the field at which the transition from the ferrimagnetic to the ferromagnetic state. The results from the other data sets show how by varying the various parameters, notably x, one can remove the minima, as seen in fig. 6 or induce both a minimum and a maximum as seen in fig. 12. One characteristic feature of the results reported in ref. [ 13 ] for which we cannot provide satisfactory explanation is the anomalous decrease in the magnetisation observed for increasing field in the region below the transition from the antiferromagnetic to the ferrimagnetic state. It has been suggested that such an effect may arise as a consequence of the formation of ferrimagnetic domains around the vortex core [ 18,19 ]. Extending our present analysis of the antiferromagnetic mixed state to include the possible formation of a ferrimagnetic domain around the vortex core, does not result in any significant increase in the vortex-vortex interaction, consequently we find that the net effect of the domain formation is simply to increase the magnetisation with increasing magnetic field due to the growth of the ferrimagnetic domains around the vortex core; a perfectly reasonable result for the choice of parameters and approximations employed in the present analysis. Based on the present analysis we are currently examining various mechanisms which could account for the observed magnetisation anomaly referred to above which, due to the assumptions we have introduced, is absent in the present work. In particular we are extending the present studies to include a more satisfactory and detailed treatment of the spatial variation of the magnetisation in the vortex lattice, as well as a more detailed treatment regarding the effect of the magnetic field generated by the vortex lattice on the screening of the dipole fields generated by the rare earth ions by the superconducting electrons. The present analysis can also be generalised to consider the modification to the vortex lattice introduced by the presence of a free surface. To conclude we have presented a relatively detailed analysis of the mixed state of a metamagnetic system, which includes the evaluation of the magnetisation curves and the phase boundaries seperating the magnetic and superconducting phases for various parameter sets. The results of the calculation are in reasonable agreement with the observed results of bct ErRh4B4. We have extended the analysis of the antiferromagnetic mixed

60

O. Wonget al. /Mixed state in bct ErRh4B4

state to include the possible formation of a ferrimagnetie domain around thea,-ortex coi-e: This restdted~in a n increase in the magnetisation close to the transition to the ferrimagnetic phase. Further study is required in order to properly understand the mechanism giving rise to the anomaly observed in the magnetisation. Several extensions to the present analysis were outlined.

Acknowledgements This work was supported by grants from the Natural Sciences and Engineering Research Council of Canada and the Dean of the Faculty of Science, University of Alberta. The authors would like to thank Mr. R. Teshima for his fine numerical work and Dr. J.A. Tuszynski for a number of valuable discussions regarding the properties and theory of metamagnetic systems. One of the authors (J.P.W.) would like to thank Dr. K. Rogacki for a number of useful discussions.

Appendix A The 4 X 4 matrices t7, and z, introduced in eq. (2.5) are given by

5 and

o--(;

2

where #, denote the two dimensional Pauli matrices and I denotes the two dimensional unit matrix.

Appendix B In this appendix we consider in some detail the expression for the magnetic free energy presented in section 3. In particular we consider how the equations for the magnetisation fields given in eq. (3.6) simplify for the various magnetic configurations shown schematically in fig. 2. The corresponding expression for the magnetic free energy is also presented and we complete the appendix by presenting several phase diagrams for various choices of parameters. We begin by defining the reduced variables

T t=~-A;

Tr. tV=TA,

t,--

TA+Tv 2~'

m, rh,=--mo

and

B b=--.4azm 3

(B.1)

i) The antiferromagnetic state In the antiferromagnetic state the order parameters r~i, defined above, together with the angles 0i, may be expressed as

r~ = m : = m 0j=02=0

and and

r/~3~-/7~4=ff/' ,

03=04=~.

(B.2)

O. Wong et al. / Mixed state in bct ErRh~B~

61

The quantities rh and fit' are obtained from the solution of eq. (3.6), which m a y be rewritten in terms of the reduced variables as fit3+ ( t - 1 )fit+ ½( 1 - t v ) (fit-- fit' ) - - b ( f i t - f i t ' ) cos (0B) = 0 ,

(B.3)

fit,3+ ( t - 1 )fit' - ½( 1 - t v ) ( f i t - f i t ' ) + b ( f i t - f i t ' ) cos (0 B ) = 0 .

(B.4)

The magnetic free energy Fmag given by eq. (3.4) reduces in the antiferromagnetic state to give

Fm~g(AF)=4a~m~(½(fit4+~t~4)+(t-~)(~t2+~t~2)+½(~-t~)(~t-~t~)2-2b(~t-~t~)c~s(~)).

(B.5)

ii) The intermediate 1 state In the intermediate 1 (I 1 ) state the order parameters fiti defined above, together with the angles 0~, may be expressed as fitl = f i t 2 = f i t 3 =if'/

0~=02=03=0

and

fit3=fit ' ,

and 03=n,

(B.6)

where m and fit' are obtained from eq. (3.6), which may be expressed in terms of the reduced variables as fit3+ ½( t - 1 )fit+ ¼( 1 - t v ) (3fit--fit') - b cos (0a) = 0 ,

(B.7)

fit, 3+ ½( t - 1 )fit' - ¼( 1 - t v ) ( 3 f i t - f i t ' ) + b cos (0B) = 0 .

(B.8)

The magnetic free energy Fmag given by eq. (3.4) may also be rewritten, for the intermediate 1 (I1) state, in terms of the reduced variables as Fma~ (I 1 ) = 4 a , m4( ~ ( 3 m 4 + fit,4) _ ½( t - 1 ) (3fit 2 + fit,2)

+ ~ ( 1 -tF) ( 3 f i t - f i t ' ) 2 - b ( 3 f i t - f i t ' )

cos (0B)).

(B.9)

iiO The intermediate 2 state In the intermediate 2 (I2) state the order parameter fit~ defined above, together with the angles 0i, may be expressed as F/~ I ~---~/qlV/2=/'JV/ ,

0t=02=0,

ff/3=PJ~

03=n/2

'

and and

r/~4=/~"

,

04=3n/2,

(B.10)

where fit, fit' and fit" are obtained from eq. (3.6), which may be expressed in terms of the reduced variables as

(B.11)

fit3+(t--1)fit+½(1--tF)fit--bcos (0a) = 0 , fit'3+(t-t)fit'+l(1-tv)(fit'-fit")-bsin

(0B) = 0 ,

fit,,3+ ( t - 1 )fit" - ¼( 1 - t F ) (fit' - f i t " ) + b sin (0B) = 0 .

(B.12) (B.13)

The magnetic free energy Fmag given by eq. (3.4) may also be rewritten, for the intermediate 2 (I2) state, in terms of the reduced variables as

62

(9. Wonget aL /Mixed state in bct ErRh4B~

F,,a, (I2) = 4 a l m4(~ (fi,4 + fi,,4) - - 1 (l - - t ) ( f i , 2 + fi,,2)

+ ~ ( 1 --tv) (fi' --fi" ) 2 - - b ( f i ' - fi" ) sin (0B) +½fi4-(1-t)fiz+½(1-tv)fi2-Zbcos

(OB))

(B.14)

iv) The intermediate 3 state In the intermediate 3 (I3) state the order parameters fii defined above, together with the angles 0,, may be expressed as tfTt = f i ~ = 0

Ot=02=O

and and

fi3=fi4=fi', 03=04=rt/2,

(B.15)

where fit and f i ' are obtained from eq. (3.6), which may be expressed in terms of the reduced variables as m3+ ( t - f l ) f i - b c o s

fi'3+(t-t~)fi'-bsin

(0B) = 0 ,

(B.16)

(08)=0.

(B.17)

The magnetic free energy Fro,g, given by eq. (3.4) may also be rewritten, for the intermediate 3 (I3) state, in terms of the reduced variables as Fm,g(I3) = 4 a , m4( ½( f i 4 .[_ f i , 4 ) _[_ (t-- tl) ( f i 2 + f i , 2 ) - 2 b ( f i cos (0a) +fit sin (0B)) ) •

(B.18)

v) The ferromagnetic state In the ferromagnetic (F) state the order parameters fi, defined above, together with the angles Oi, may be expressed as

01 =02 =03 =04 = 0 ,

(B.19)

where fi is obtained from eq. (3.6), which may be expressed in terms of the reduced variables as fi3+ (t-tv)fi-b

cos (0B) = 0 .

(B.20)

The magnetic free energy Fmag, given by eq. (3.4) may also be rewritten, for the ferromagnetic state, in terms of the reduced variables as Fmag(F) =4a~ m ~ ( f i a + 2 ( t - t v ) f i 2 - 4 b f i

cos (0B)) •

(B.21)

The various order parameters for each spin configuration may then be calculated for a given value of temperature and applied field by means of the above expressions for each of the various spin configurations and the corresponding free energy calculated. The configuration with the lowest free energy is then identified to be the most stable configuration. For t> 1 all of the above states reduce to the paramagnetic state and hence it is only for t < 1 that we must consider the transition between the different spin configurations. The resultant phase diagrams illustrating the various phases and their dependence upon the magnitude and the angle of the induction field b are shown in fig. 5 for tv=0.9 at several values of the reduced temperature t. For 0B=0 we note that only the intermediate 1 (I 1 ) state appears to be stable between the fields b~ and b3. A graph for showing the dependence of b~ and b3 is given in fig. 4 for 0B=0.

O. Wong et al. /Mixed state in bct ErRh4B4

Appendix C T h e f u n c t i o n s ~'ffand L # d e f i n e d in eqs. ( 4 . 4 1 ) a n d ( 4 . 4 0 ) are g i v e n by [20] -~'

~,~2( t, l~)= ~ ( t, lZ)-- 2 ~ 2 ( t 2

3 \ h e -~'

- 347t-~4

~(t, # ) , - - 7 - #

~e-;'

)

3 ~ 2 ( t ' ~)ti~3( r c e - ~ ' - - - 7 - - - 7 - n e - ~ ' ) n2 ~(t,#), /~

))

'

w h e r e ~ ( t , # ) is g i v e n by the s o l u t i o n o f the self-consistent e q u a t i o n :

64

O. Wong et aL / Mixed state in bct ErRh,B4

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