OFFICE OF NAVAL RESEARCH14( ; c pGrant N00014-90-1-1193

TECHNICAL REPORT No. 41

C') Critical Behavior in Magnetic Superlattices

by

(N T. Hai, Z. Y. Li, D. L. Lin and Thomas F. George

SPrepared for publication

Journal of Magnetism and Magnetic Materials D T ICELECTEt~ JAN 3o0191

Departments of Chemistry and Physics JState University of New York at BuffaloBuffalo, New York 14260D

January 1991

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Arlington, Virginia 22217 r11. TITLE (include Secuffty CIa10fication)Critical Behavior in Magnetic Superlattices

12. PERSONAL AUTHOR(S)T. Hai, Z. Y. Li, D. L. Lin and Thomas F. Geor e

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17. COSATI CODES Is. SUBJECT TERMS (Continue on revers if neCessary and identi by bkock number)FIELD GROUP SUB-CROUP 0.=1FSQ 4 G .~--,j. QPJIELATIC

MAGNETIC SUPERLATTICES, Wee-- FERROMAGNETSA!5-I EFFECTIVE FIELD THEORY C.1T u9WM9f

19 A RACT (Contnu on reveri if necestay and dentrfy by block number)

Based on the effective field theory with correlations, the critical behavior of amagnetic superlattice consisting of two different ferromagnets is examined. A simplecubic Ising model with nearest-neighbor coupling is assumed. It is found that thereexists a critical value for the interface exchange coupling above which the interfacemagnetism appears. The critical temperature of the superlattice is studied as a functionof the thickness of the constituents in a unit cell and of the exchange coupling energies

in each material. A phase diagram is also given.420. DISTRIBUTION/4 AVAILABILITY OF ABSTRACT 121. ABSTRACT SECURITY CLASSIFICATION

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Journal of Magnetism and Magnetic Materials, in press

Critical behavior in magnetic superlattices

T. HaiP. 0. Box 1001-73

Zhengzhou 450002, P. R. China

Z. Y. Li , D. L. Lin and Thomas F. GeorgeDepartment of Physics and Astronomy

State University of New York at BuffaloBuffalo, New York 14260, USA

Abstract

Based on the effective field theory with correlations, the critical

behavior of a magnetic superlattice consisting of two different ferromagnets

is examined. A simple cubic Ising model with nearest-neighbor coupling is

assumed. It is found that there exists a critical value for the interface

exchange coupling above which the interface magnetism appears. The critical

temperature of the superlattice is studied as a function of the thickness of

the constituents in a unit cell and of the exchange coupling energies in each

material. A phase diagram is also given.

* On leave of absence from the Department of Physics, Suzhou University,

Suzhou 215006, P. R. China Acce-ioo For

NTIS CRMU

U: a.moi.2J LJ

B y ......................

Di-t ibutio, i

Avaiiabiiity Codes

Dlst Avail and or,f q Dist Special

2

I. Introduction

In recent years, there has been increasing interest in the nature of

spin waves as well as critical phenomena in various magnetic layered

structures and superlattices. Ma and Tsai1 have studied the variation with

modulation wavelength of the Curie temperature for a Heisenberg magnetic

superlattice. Their results agree qualitatively with experiments on the Cu/Ni

film.2 Superlattice structures composed of alternating ferromagnetic and

antiferromagnetic layers have been investigated by Hinchey and Mills,3 '4 using

a localized spin model. A sequence of spin-reorientation transitions are

found to be different for superlattices with the antiferromagnetic component

consisting of an even or odd number of spin layers. In addition, the spin-

wave spectrum and infrared absorption spectrum are calculated.

For a periodic multilayer system formed from two different ferromagnetic

materials, Fishman et al5 have discussed its statics and dynamics within the

framework of the Ginzburg-Landau formulation. They have computed the

transition temperature and spin-wave spectra. On the other hand, the Landau

formalism of Camley and Tilley 6 has been applied to calculate the critical

temperature in the same system.7 Compared to Ref. 5, the formalism of Ref. 6

appears to be more general because it allows for a wider range of boundary

conditions and includes the sign of exchange coupling across the interface.

For more complicated superlattices with arbitrary number of different

layers in an elementary unit, Barnas 8 has derived some general dispersion

equations for the bulk and surface magnetic polaritons. These equations are

then applied to magnetostatic modes and to retarded wave propagation in the

Voigt geometry.9

We study, in this article, the critical temperature in an infinite

superlattice consisting of two ferromagnetic materials with different bulk

A

3

properties. In particular, we consider the two constituents A and B with

(a) (b)different bulk transition temperatures as a simple model, i.e., T a o T .C C

The interface is in general different in nature from both bulks, even if the

bulk critical temperatures are the same. We use the effective-field theory

with correlations10 in the present work, as it is believed to be far superior

to the standard mean-field approximation.

Because of the periodicity of the superlattice structure, we restrict

our discussions to a unit cell which interacts with its nearest-neighbor cells

via interface couplings. Thus the system can be treated by extending the

method developed for a magnetic slab. Our major concerns are the dependence

of the transition temperature on the thickness of individual constituents in

the cell and the influence of the interface magnetic properties on the phase

transition temperature. These questions, to our knowledge, have not been

considered in the literature thus far.

In Sec. II. we outline the theory and derive the equation that

determines the transition temperature. Numerical results are discussed in

Sec. III where the existence of the interface magnetic phase transition is

discovered and the critical value of the interface coupling relative the bulk

coupling is determined. A brief conclusion is given in Sec. IV.

II. Theory

Consider an infinite superlattice consisting of two different

ferromagnetic materials A and B. For simplicity, we restrict our attention to

the case of simple cubic Isng-type structures. The periodic condition

suggests that we only have to consider one unit cell which interacts with its

nearest neighbors via the interface coupling. The situation is depicted in

Fig. 1. The coupling strength between nearest-neighboring spins in A(B) is

4

denoted by J a(Jb) while Jab stands for the exchange coupling between the

nearest-neighbor spins across the interface. The corresponding number of

atomic layers in A(B) is La(Lb), and the thickness of the cell is L - La + Lb.

The Hamiltonian of the system is given by

ij

where the sum is taken over all the nearest-neighbor pairs only once, S - ±1

is the usual Ising variable, and Jij stands for one of the three coupling

constants depending on where the spin pair is located.

To evaluate the mean spin <Si>, we start with the exact Callen

identity, 12

<Si> - <tanh Si )> (2)

where - l/kBT, <...> indicates the usual canonical ensemble average for a

given configuration of {Jij), and j runs over all nearest neighbors of site i.

We now introduce the differential operator D - and recall that theax

displacement operator exp(aD) satisfies the relation

exp(aD)f(x) - f(x+a) . (3)

Equation (2) can then be rewritten, with the help of (3), as

<Si> - <exp(D J iS )> (tanh(fX)lxJo

J

-<IR [cosh(DJ ) + S sinh(DJij)]> tanh(Ox)lx,0 (4)i

This equation represents a set of L coupled equations for the L spin layers in

the unit cell. Each layer is only coupled to its nearest-neighbor layers.

The multi-spin correlation function, however, must be decoupled before a

practical calculation can be made. We follow the standard procedure13 with

the decoupling approximation <xIx 2..xI> - <xI><X2> ... <xI>. Thus the average

atomic magnetization of each layer is given by

4m, - (cosh(DJab) + mosinh(DJab)][cosh(DJa) + mlsinh(DJa)]

x [cosh(DJ a) + m2sinh(DJa)] f(x)Ix_ 0 (5a)

m i - (cosh(DJ a) + m iaSinh(DJa )]cosh(DJa ) + misinh(DJa)]4

x (cosh(DJa) + mi+lsinh(DJa)]f()Ix, 0 2 < i La (5b)

i4

mL - (cosh(DJ ) + mL -1sinh(DJ a)]cosh(DJa) + mL sinh(DJa)]

a a a

x [cosh(DJab) + ML +lsinh(DJab )]f(x) Ix 0 (5c)a

mL a+1 (cosh(DJab) + mL sinh(DJab)][cosh(DJb) + mL +Isinh(DJb)]4

a a

x [cosh(DJb) + DLa+ 2sifh(Dab)]f(X)x. (5d)

6

Il - (cosh(DJa) + mj lsinh(DJb)][cosh(DJb) + mjsinh(DJb)]4

(c,)sh(DJb) + mj+lsinh(DJb)]f(x) IX-0 La - i : L-1 (5e)

mL - [cosh(DJb) + mL.isinh(DJb)]cosh(DJb) + mLsinh(DJb)]4

(cosh(DJab) + mL+isinh(DJab)]f(x) ix-O ' (5f)

where we have defined

f(x) - tanh(fx) (6)

on which the operator D applies. The variable x is set to zero at the end of

the calculation in each equation. It is observed that the periodic condition

of the superlattice is indeed satisfied, namely, m° - mL and mI - mL+i . As

the temperature becomes higher than the critical temperature, the whole system

becomes demagnetized. Thus, we can determine T from Eq. (5) by requiringc

that the mean atomic magnetization in every spin layer approaches zero.

Consequently, all terms of the order higher than linear in Eqs. (5) can be

neglected. This leads to a set of simultaneous equations

(1-4A1 )mI - A2m0 A1m 2 0

(l-4A 3)mi - A3(mil+mi+l) - 0

(1-4A 1 - AlmL . 1 - A2m +1 - 0a a a

7

(7)

(l- 4 A4)mL +1 AS mL -A 4ML +2 " 0a a a

(l-4A6 )mj - A6 (mj..m,+l) - 0

(1-4A4)X - A4 L. - ASL+l - 0

where the coefficients are given by

4A1 - cosh (DJ a)sinh(DJa )cosh(DJ ab)f(x)Ix-0

A2 - cosh (DJ a)sinh(DJab)f(X)Ix.O

A 3 - cosh (DJ a)sinh(DJ a)f(x)x-0

(8)

A4 - cosh (DJb)sinh(DJb)cosh(DJ ab)f(x) jx-0

A5 - cosh (DJb)sinh(DJab)f(x) x-0

A6 - cosh5(DJb)sinh(DJb)f(x)Ix-O

The secular equation of this set of coupled equations is

8

a1 -1 bI

-1 cI -1. ...1

aa bb* * * * -1 1

b a2 - - 0

2 2 0

: 2

b- a2

(9)

where we have defined

a1 - (1-4A1 )/A1 a2 - (l-4A4)/A 4

b I - -A2/A1 b2 - -A5/A4 (10)

cI - (1-4A 3)/A3 c2 - (1-4A 6)/A6

The L x L determinant in (9) can only be evaluated numerically for a given set

of coupling constants JiJ As all the coefficients A involve P through (6),

T can be obtained as a function of the 's. The results of this calculation

are discussed in the following section.

III. Results and discussion

Throughout this paper, we take Jb as the unit of energy, and the length

is measured in the unit of the lattice constant in our numerical work. We

first show the critical temperature as a function of the unit cell width L in

9

Fig. 1. Four cases of different La values are plotted for fixed J and Jab

We first note that T decreases with increasing Lat in general. This is

because we have assumed J < Jb in our calculation, and it is well known that

the bulk transition temperature is approximately proportional to the exchange

coupling if the spin and lattice structure are specified. For a given La, it

is observed that Tc increases steadily as L increases indefinitely. All the

curves approach the same limiting value Tc - 5.073 Jb/kB, which is simply

T(b). This is of course easily understood.C

To study the effects of the interface exchange coupling on the

transition temperature of the superlattice, Tc is calculated for various Jab

but fixed J and L . Figure 3 shows the results for J - J Curve ca a a

corresponds to the case Jab " Jb and appears like that of a slab of material

B. It is very interesting to see that there exists a critical interface

coupling Jcb - 1.46 Jb such that T remains constant for any Lb, as

represented by curve b. For Jc larger than this critical value, we find thatab

T is higher than both TCa) and T(b). Curve a illustrates one such particularc c C

case Jab b 2Jb" This suggests that there exists an interface magnetism in the

system. For Jab > ab the system may order in the interface layers before it

orders in other layers. The situation does not change for Ja J The

results for Ja - 0.5 Jb are plotted in Fig. 4, whose curves are very similar

except that the critical interface coupling jb in this case is bigger than in

Fig. 3. This is because the lower T (a ) needs a stronger, compensatingC

interface exchange coupling in order to reach the interface phase transition.

Figure 5 demonstrates that as long as Ja = b the critical coupling J c

- 1.46 Jb does not depend upon the thickness La . Curve d remains the same for

all three cases calculated, while for Jb - 2.0 Jb and 0.1 Jb' Tc behaves very

differently when L is varied. Curves (a,a'), (b,b') and (c,c') correspond to

10

La - 3, 5 and 10 respectively. For Jab > J ab when the interface magnetism

appears, Tc decreases as L increases. For the a and b cases, material B

dominates the system for L > 13. For the case c, L. - 10, and T drops

quickly at the beginning and approaches T(b) for L > 20. On the other hand,C

the transition temperature increases with L for Jab < cab The apparently

anomalous behavior of curve c' can be understood in the following manner. As

Jab " 0.1 Jb' T is mainly determined by material A(B) if La(Lb) is larger

than Lb(L a)' Hence, Tc changes little until L approaches 18, because La - 10

in this case.

Figure 6 shows the phase diagram of the system by plotting the critical

interface exchange coupling as a function of Ja . Three different thicknesses

L are plotted. It is interesting to note that the three curves meet whena

Ja - Jb' suggesting that the critical interface exchange is independent of the

relative thickness of the constituents in the unit cell as long as their

exchange couplings are the same. This is of course in agreement with the

above results in Fig. 5.

We have considered ferromagnetic interface coupling in our discussion.

The critical temperature of the system, however, remains the same if the

interface exchange coupling becomes antiferromagnetic. This is consistent

with the discussion in Ref. 7.

IV. Conclusions

We have investigated the criticality of a two-component ferromagnetic

superlattice by considering one unit cell, assuming nearest-neighbor spin

exchange couplings. The interface exchange energy is, in general, different

from either of the bulk ones. For the first time, the critical value J C isab

introduced. When Jab is larger than this critical value, the interface

Iab

magnetism appears, and T for the system is higher than either T or Tb . WhenC c c

J is smaller than the critical value, T < Ta , Tbab c c c

Finally, we remark that the method is general and can be used to treat

the ferromagnetic/antiferromagnet and antiferromagnet/antiferromagnet

superlattices. With modifications in spatial fluctuations of the coupling

J's, the method can easily be generalized to superlattices involving certain

type of amorphous magnetic materials.

AcknovLedg ents

This research was supported in part by the National Natural Science

Foundation of China and US Office of Naval Research.

12

Figure captions

1. Unit cell of the superlattice composed of ferromagnetic materials A and B,

where L La + is the thickness of the cell. The solid and dashed

lines represent the nearest-neighbor exchange coupling energies between

spins in one material and in the interface, respectively.

2. Dependence of T on the thickness L. The interface coupling is chosen asc

Jab " 0.9 Jb and Ja - 0.8 1 Number of spin layers of material A is (a)

3, (b) 4 and (c) 5, (d) 6.

3. Tc as a function of L for La - 3 and Ja - Jb but various interface

coupling strengths. (a) Jab - 2.0 Jb' (b) Jab - 1.46 and (c) Jab - b"

4. Same as Fig. 3 except that Ja - 0.5 1" (a) Jab - 3.0 Jb- (b) Jab - 2.51

3b and (c) Jab -1.9 Jb"

5. T as a function of L for Ja - but different J aand L b - 2.0cJa b bu ifrn ab aL- ab 0 b

for curves a, b and c, and Jab 0.1 Jb for a', b' and c'. The number of

spin layers L is 3 for a, a', 5 for b, b', and 10 for c, c'.a

6. Phase diagram in terms of the coupling Jab and Ja for La - (a) 3, (b) 4

and (c) 10.

13

References

1. H. R. Ma and C. H. Tasi, Solid State Commun. 5, 499 (1985).

2. J. Q. Zheng, J. B. Ketterson, C. M. Falco and I. K. Schuller, J. Appi.

Phys. U, 3150 (1982).

3. L. L. Hinchey and D. L. Mills, Phys. Rev. B 33, 3329 (1986).

4. L. L. Hinchey and D. L. Mills, Phys. Rev. B 34, 1689 (1986).

5. F. Fishman, F. Schwable and D. Schwenk, Phys. Lett. A In2, 192 (1987).

6. R. E. Camley and D. R. Tilley, Phys. Rev. B 37, 3413 (1988).

7. D. R. Tilley, Solid State Commun. 65, 657 (1988).

8. J. Barnat, J. Phys. C: Solid State Phys. 21, 1021, 4097 (1988).

9. J. Barnat, J. Phys.: Condensed Matter Z, 7173 (1990).

10. R. Monmura and T. Kaneyoshi, J. Phys. C: Solid State Phys. 12, 3979

(1979), Z. Phys. B 51, 307 (1984).

11. T. Hai and Z. Y. Li, Phys. Stat. Solldi (b) 156, 641 (1990).

12. H. B. Callen, Phys. Lett. 4, 161 (1963).

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