Chapter 1 Magnetism in artificial metallic superlattices of rare earth metals

chapter 1 MAGNETISM IN ARTIFICIAL METALLIC SUPERLATTICES OF RARE EARTH METALS

J.J. RHYNE Research Reactor Center and Department of Physics University of Missouri-Columbia Columbia, Missouri 65211 U.S.A.

and

R.W. ERWlN Materials Science and Engineering Laboratory National Institute of Standards and Technology Gaithersburg, Maryland 20899 U.S.A.

Handbook of Magnetic Materials, Vol. 8 Edited by K. H.J. Buschow ©1995 Elsevier Science B.V. All rights reserved

CONTENTS 1. Introduction to superlattices and rare ealths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Neutron scattering and artificial metallic superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Magnetic scattering, structure, and coherenee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Superlattices with e-axis growth directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Superlattices with a basal plane growth direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Interlayer magnetic coupling in superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Effect of applied magnetic fields on interlayer coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Incommensurate magnetic periodicity - interplanar turn angles, spin slips, and their temperature dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Magnetoelasticity in lare earth superlattices and films and the suppression of ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Theory of magnetoelasticity in superlattices and films . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Magnetoelastie energies in Dy superlattices and films . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Coherent magnetic m o m e n t s and disorder at interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Residual m o m e n t effects in superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. S u m m a r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 6 11 11 26 30 32 35 41 43 44 48 50 54 55 55

1. Introduction to superlattices and rare earths Intense interest has been generated over the past several years in the growth and properties of layered magnetic materials, both from a fundamental point of view and for applications. Layered structures have been prepared by a variety of techniques such as sputtering, electro-deposition, and evaporation, and include semi-conducting, metallic, and insulating materials. These systems can consist of crystalline layers of one element or compound interleaved with layers of a different element or compound, or alternatively may be built of amorphous layers or amorphous layers alternated with crystalline layers. Depending on the materials and growth techniques, these multilayers may be produced (a) with no uniform crystallographic alignment or coherence from layer to layer, (b) with alignment of one specific crystallographic axis direction along the growth (stacking) direction, or (c) with true three-dimensional atomic order (epitaxy) in which there is multilayer atomic registry both along the growth axis and also within the growth planes. For the purposes of this review, the term artificial metallic superlattice will be reserved for this latter category of true threedimensionally coherent layered structures, while the term multilayer will be used for layered structures in which coherence is present in less than three dimensions. Refinements in computer-controlled molecular beam epitaxy (MBE) techniques for the growth of single crystal artificial superlattices of two or more distinct compositions have unearthed vast possibilities for the production of tailor-made artificial superlattices with controlled film thicknesses down to atomic dimensions and with highly reproducible stacking sequences. This has provided previously unavailable opportunities to examine problems of interaction ranges, tunneling distances, and other coherent phenomena which are dependent on the superlattice periodicity. The lanthanide elements and their alloys with non-magnetic but chemically and electronically similar elements such as yttrium, scandium, and lutecium (collectively known as the rare earths), have long provided a fertile area for the study of longrange indirect exchange interactions, crystal field anisotropy, and magnetostrictive effects. The elements have weak exchange compared to the 3d transition elements as illustrated by their low ordering temperatures (e.g., Tb, TN = 230 K) and have anomalously large crystal field and magnetoelastic interactions that arise from the strong spin orbit coupling and highly non-spherical 4f charge distribution. Below their magnetic ordering temperatures, many of the lanthanide elements exhibit one or more forms of periodic incommensurate spin structures such as helices or longitudinal spin density waves, etc., and transitions among them as the temperature is varied. The occurrence of these periodic magnetic orderings below their Nrel

4

J.J. RHYNEand R.W. ERWIN

temperatures in the heavy lanthanides (except for the S state ion Gd) is a consequence of electronic effects, in particular the occurrence of nearly two-dimensional parallel sections (nesting) on the hole Fermi surface that are spanned by a Q-vector whose magnitude determines the initial stable periodicity of the magnetic ordering. At lower temperatures, the free energy associated with the magnetostrictive and anisotropy interactions becomes significant and strongly perturbs the basic periodic magnetic orderings leading to phase transitions to a ferromagnetic state in Tb, Dy, Ho, and Er. These transitions are largely driven by a lowering of the magnetoelastic energy arising from a coupling of the local moments to the hexagonally symmetric lattice strains. It has been suggested by Larsen, Jensen, and Mackintosh (1987) that dipole-dipole interactions of extremely long range may be also responsible for perturbing the e-axis modulated moment states found in Ho and probably Er. The development (Nigh 1963) of techniques for producing high-quality large single crystals of the heavy rare earth elements and alloys provided an opportunity to study the anisotropy in the magnetic properties of the rare earths that led to much of our current insight into the fundamental orderings and interactions in the rare earth metals. For details and references on these works, one is referred to the many review works available, including, but certainly not limited to, Elliott (1972), Gschneidner and Eyring (1979), Coqblin (1977), and the extensive treatise by Jensen and Mackintosh (1991). A major breakthrough in rare earth materials occurred in 1984 with the development (Durbin, Cunningham, and Flynn 1982, Kwo et al. 1985a, 1985b) of MBE growth procedures for single crystal rare earth metal superlattices. These exotic materials have made possible prototypical tests for verifying many of the theoretical concepts of magnetic exchange, anisotropy, and magnetostrictive effects in the rare earths that could not previously be examined in as controlled a way using conventional bulk materials. Superlattices consisting of magnetically concentrated layers (e.g., Dy) interleaved in a controlled fashion with magnetically 'dead' layers (e.g., Y) offer a near-ideal opportunity to investigate these basic interactions. It should be noted that such a system is unique and can never be simulated by bulk dilute alloys because of the attendant reduction in the average exchange interaction with the decreased density of magnetic ions and the probability of some nearest neighbors even in very dilute samples. Y and Lu have similar physical and electronic properties to the magnetic heavy rare earths and good epitaxial growth is achieved because of the relatively small mismatch between the basal plane lattice parameters (e.g., 1.6% for Dy and Y). The key to the growth (see fig. 1) of rare earth superlattices lies in the use of a [110] Nb buffer layer evaporated onto a [1120] sapphire substrate beneath the rare earth metals. This buffer layer prevents a chemical reaction between the sapphire and the rare earths during growth. A strain-relieving Y overlayer is placed between the Nb and the rare earth bilayers. Y and Nb have a nearly perfect 3:4 atomic registration sequence that allows good epitaxial growth in spite of the 33% lattice parameter mismatch. Lattice parameter mismatch is a moderately serious constraint for MBE-produced materials, since this mismatch must generally be taken up by lattice dislocations. As shown in the figure for a [DyIY] superlattice, the thick Y layer is followed by a constant bilayer repeat sequence of I atomic planes of Dy and

MAGNETISM IN ARTIFICIAL METALLIC SUPERLA'ITICES

5

RARE EARTH SUPERLATTICE STRUCTURE bilayer / / ."

Y 100021 Nb [110]

,-

~

/ :- d I ( A / p l a n e )

Y ~ 2oJ e (rlldlans/plane)

2~1

221 B

lCb:'.,277.oo,

soo i - i 5 0 0 ,~

." repeat N times

sapphire

II1~o1

substrate Fig. 1. Schematic drawing of a rare earth artificial metallic superlattice structure [NAINB] consisting of N bilayers each with NA atomic planes of element A and NB atomic planes of a dissimilar element B. The expanded view of a bilayer, of thickness L, lists the physical parameters appropriate to A (and B) layers. (See text.)

m atomic planes of Y. This is designated [DytlYm]N where N is the total number of bilayers. Rare earth superlattices have been produced by similar procedures in the [GdlY] (Kwo et al. 1986, Majkrzak et al. 1986), [DylY] (Durbin, Culmingham, and Flynn 1982, Rhyne et al. 1987, 1989 and references therein, Salamon et al. 1986, Erwin et al. 1987, Borchers et al. 1987, and Majkrzak et al. 1988), [ErlY] (Erwin et al. 1988a, 1988b, Borchers et al. 1989a, 1989b, 1991a, 1991b, Rhyne et al. 1991), [HolY] (Majkrzak et al. 1988), and [DyIGd] (Majkrzak et al. 1986, 1988) systems. This chapter concentrates primarily on experimental and theoretical studies of [Dy[Y] and [ErIY] superlattices and the Sc and Lu counterparts of Y. A review of rare earth superlattice systems with emphasis on [Gd[YI, [HolY], and [GdlDy] materials is found in Majkrzak et al. (1991). The [DyIY], [Er[Y], and counterpart Lu and Sc superlattices were grown at the University of Illinois. In the procedures developed there (Dnrbin et al. 1982), the chamber pressure during evaporation was maintained in the 10 - 9 Tort range to minimize oxidation of the rare earths which were evaporated at a rate of only about 0.5-1.0 A per second with a substrate temperature of approximately 300°C. These conditions were chosen to minimize interdiffusion between the lanthanide layers. Analysis of the neutron or X-ray diffraction data (Botchers 1989b) for the intensity of the bilayer harmonics (see next section) confirms that this interdiffusion is quite minimal and that the composition reaches 85% of the pure element (Y or Er, Dy) within about two planes on either side of the interface. (See fig. 2.) The intensity analysis of the same [Erz3.5[Y19]10o superlattice also confmns that the d-spacing variation is quite abrupt as shown in fig. 2, reflecting the relatively large (2.5%) lattice mismatch between Er and Y.

6

J.J. RHYNE and R.W. ERWIN

© O

t.O

.-4 v

0.6 0 ~3

fl) o

0.2

0 C3 #3

i

-0.2 0.0

I

r

I

20.0

i

I

40.0

I

I

60.0

1

80.0

ALomic P l a n e I n d e x 2.92

o<

i

i

i

i

2.88

v

0

2.84

\

O'3

I q~ 2.80

2.76

! 0.0

I 20.0

,

I 40.0

,

I 60.0

,

I BOO

At,o m i e P l a n e I n d e x

Fig. 2. The composition profile (top) calculated from the X-ray diffraction intensities for an [Er23.5] Y19]100 superlatticeshowing that interdiffusion is limited to approximately5 total planes. At the bottom is shown the d-spacing variation for the same superlatticeillustrating the ratherabrupttransitionresulting from the relatively large Er-Y lattice mismatch. 2. Neutron scattering and artificial metallic superlattices Neutron scattering has played a seminal role in elucidating the spin structures that occur in the lanthanide metals and in detailing the perturbations of these structures at temperatures below the N6el point and those induced by the application of external fields. Much of this work was done by W.C. Koehler (who once described the multiplicity of spin states as a 'panoply of exotic spin configurations') and his associates at Oak Ridge more than 30 years ago. (See Elliott 1972.) More recently, highly precise synchrotron radiation studies, conducted at very close temperature intervals, have s h o w n additional details of the magnetic ordering in Ho (Gibbs, Moncton, and

MAGNETISM IN ARTIFICIALMETALLICSUPERLATrlCES

7

D'Amico 1985) and Er (Gibbs et al. 1986) including the occurrence of lock-in transitions in which the periodic moment wave-vector passes through a series of lattice-commensurate phases as the temperature is lowered, according to a model of spin-slip states. (See above references and later discussion.) The resulting steplike changes contrast to the more monotonic linear variation reported in the early neutron measurements taken with much coarser temperature intervals and have been confirmed also by neutron scattering studies. For artificial metallic superlattices, neutron diffraction provides atomic structural information complementary to that obtained from X-ray diffraction. In addition, the magnetic interaction of the neutron with ordered spins in the superlattice provides the opportunity to study the magnetic spin configurations and the range of coherence of the magnetism. In particular, it can be used as a probe of the propagation of magnetic phase information from a magnetically ordered layer (e.g., Dy) across an intervening non-magnetic layer to additional ordered layers. Neutron diffraction studies on supeflattices are typically performed using a triple axis spectrometer with the sample in reflection geometry. An analyzer crystal set for elastic scattering is used to suppress background scattering. Both conventional unpolarized neutron beams and beams polarized by reflection from a magnetic crystal have been used for studying superlattice properties. The polarization technique offers the advantage of extracting only the magnetic component of the scattering (see later discussion), however, with a significant reduction in beam intensity. Astute choices of polarization techniques and also selection of optimal resolution conditions have been the keys to successful diffraction experiments on superlattices. The trade-off between intensity and enhanced resolution is intensified by the available sample volumes of these superlattices that typically range from 0.01 to 0.05 mm 3. Even considering the high crystal perfection, this represents a major challenge for neutron scattering experiments with presently available fluxes. Most of the superlattice studies reviewed here were performed at the National Bureau of Standards Reactor at the National Institute of Standards and Technology. For a typical geometry in which the growth axis (perpendicular to the layer planes) is oriented along the neutron scattering vector, the basic unit cell of an A-B superlattice is one bilayer (e.g., one Y + one Dy layer). If the growth axis is along the c crystal direction [0001] then, considered apart from the effects of intralayer atomic ordering, the bilayer sequence would give rise to a set of peaks along the c* [000l] direction of reciprocal space corresponding to diffraction from the bilayer cell at evenly spaced values of AQ = 2~r/L where L is the bilayer thickness. The Bragg diffraction conditions can be satisfied in two ways: (a) the above diffraction from the bilayer unit cells that can be directly observed at low angle experiments (e.g., [Mo]Si] artificial superlattices, Fernandez et al. 1987) and (b) conventional diffraction from the atomic order within the bilayers seen at large angles. In the small angle case N - 2 subsidiary maxima, due to the finite size of the superlattice, are present between the principal bilayer diffraction peaks, where N is the total number of bilayers. These subsidiary maxima are only observable if the bilayer repeat sequence is nearly perfect and the spectrometer resolution is at least comparable to 2~r/NL. In the large angle regime, illustrated by the X-ray diffraction pattern for

8

' J.J. RHYNE and R.W. ERWIN

10 7

~o~

,- i0 ~ ~0 C.

10 ~

10~

2.0

2.1

2.2

(i-')

2.3

2.4

2.5

Fig. 3. Room temperature X-ray (0001 scan) diffractionpattern for an [Er23.s[Y19]10o superlattice showing the presence of harmonicsup to fifth order. The arrow marks the position of the central (0002) Bragg peak. (Note K and Q are used interchangeablyto denote the wave vector in inverseangstroms.) an [Er23.51Y19]lOO superlattice in fig. 3, the atomic order within a bilayer modulates the intensity of the bilayer Bragg peaks and has the effect of singling out sets of predominant (000/) peaks from the infinite set of bilayer reflections. If the lattice parameter mismatch is reasonably small, these groups of peaks occur at multiples of Q = 2~r/(d) where (d> corresponds to a weighted average d-spacing of the A and B layers of the bilayer. Other peaks (bilayer harmonics) in the neighborhood of the principal ones fall off in intensity as controlled by the relative nuclear scattering amplitudes bA,s, layer thicknesses, and the lattice parameter mismatch. The last quantity is additionally responsible for intensity asymmetry, which can occur for the harmonics on opposite sides of the fundamental peak. In general, the number of harmonics present is an indication of the crystal perfection of the superlattice and the sharpness of the interface boundary. In the extreme limit of a sine wave composition modulation, only one harmonic exists. The addition of incommensurate magnetic order, such as found in Dy and Er, produces additional principal scattering peaks of solely magnetic origin at Q's dictated by the magnitude crf the magnetic propagation vector. The principal satellite peaks occur along the c* direction on opposite sides of the nuclear (000l) peaks (for the helical structures) and are designated Q+ and Q - . They are again individually part

MAGNETISMIN ARTIFICIALMETALLICSUPERLATFICES

9

of an infinite repeating set of magnetic harmonics with amplitudes controlled by factors analogous to the nuclear case - the magnetic scattering amplitudes PA,B (PB = 0 for Y) and the magnetic propagation vectors ~A,B in both A and B layers. Note that even though Y has no 4f magnetic electrons and is only paramagnetic, a spin density wave may be formed in the conduction band with characteristic propagation vector ~B as discussed later. In Er supeflattices, the principal moment component is parallel to the c-axis and thus gives rise to satellites along c* about (h, k, h + k, l) {h, k # 0} class nuclear reflections, but not about the (000l) peaks. (See eq. (3).) As discussed later, at low temperature the transverse moment components order into a helix that then gives rise to weak [000l] ± satellites as well. The neutron scattering intensity for the c* direction (z) for a superlattice can be written as the sum of the coherent nuclear and magnetic scattering terms as:

I(Q) -- Snuc(q) + Smag(+Q) + Smag(-q),

(1)

where the nuclear structure factor has the form:

N_ l bm eiQ.z,, ' 2

s°oo(q) =

(2)

m----O

and the magnetic structure factor can be written:

Smag(+Q) =

N-lprn' ~ l'eiQ'z"~:t=i¢"~m=O

2,

(3)

bm is the nuclear scattering amplitude of atoms in the m atomic plane, PmJ. is the coherent magnetic scattering amplitude proportional to the moment component perpendicular to the scattering vector, and Cm is the cumulative layer magnetic phase shift of the propagating magnetization wave. In the case of an idealized [nAlns]u superlattice with perfectly sharp interfaces, the nuclear scattering intensity is:

Snuc(Q)-

sin2(NqL/2) sin2(nAqdA/2) b2A+ sin2(nBqdB/2) bZB+ sin2(QL/2) { sin2(QdA/2) sin2(QdB/2) + 2bAbB cos (QL/2)

sin (nAQdA/2) sin (nBqdB/2) ~ sin (qdB/2)

}'

(4)

where N is the total number of bilayers each of thickness L = n A d A + nBdB, and d is the atomic d-spacing of the respective A and B layers The same functional form as eq. (4) also describes the X-ray diffraction profile. For a superlattice in which only the A layer contains magnetic atoms (e.g., PB = 0) and for which the total magnetic phase shift across a bilayer is given by # = nA~CAdA + nBXBdB, the corresponding magnetic scattering is:

10

J.J. RHYNE and R.W. ERWlN

sin2[N(QL q- ~ ) / 2 ] s i n 2 [ n A ( Q 4. ~¢A)dA/2] "1 p 2 Smag(q-Q) =

s i n 2 [ ( Q L 4- ~ ) / 2 ]

sin2[(Q 4- tCA)dA/2]

~

2

(5)

Details of the calculation of these structure factors are given by Erwin et al. (1987). Each of the above structure factors consists of a product of terms - the first produces the infinite set of bilayer peaks and the second is a much broader envelope function (i.e., generally nadA << NL) which, for the nuclear case, has its maximum on one of the bilayer Bragg peaks only if there is no lattice mismatch. The product of these two terms gives the intensity modulation of the diffracted intensity peaks as shown in fig. 3 for the atomic structure and in fig. 4 for the magnetic structure.

/ [l~/ls IY=o]. T - Dependence Tc = 167K

I /~ 250

, 160 K

i

~200 t--

.~ 1so n-

=>, loo 50

o/ 1.80

2.00

2.20

2.40

2.60

Oz (A"1) Fig. 4. Neutron diffraction scans along (0001) in a [DY16[Y2o]89 superlattice for temperatures below TN. Note the temperature independence of the (0002) peak at Q z = 2.215 .~- 1 . The small peak to the right is a bilayer harmonic. The fundamental and two bilayer harmonics are shown for both Q - (~ 2.02 A - l ) and for Q+ (~, 2.42 ,~-]) magnetic satellites. Tc denotes the helical ordering temperature TN = 167 K.

MAGNETISMIN ARTIFICIALMETALLICSUPERLATTICES

11

3. Magnetic scattering, structure, and coherence One of the intriguing concepts in the study of artificial metallic superlattices is the magnetic exchange coupling of spins in one atomic layer to those in adjacent magnetic layers separated by non-magnetic intervening layers. For the rare earth systems, different superlattices have been examined to determine the dependence of the magnetic coherence on (a) the direction of the spin alignment (e.g., basal plane or c-axis), (b) the type of spin structure (ferromagnetic or periodic antiferromagnetic), (c) the direction of the superlattice growth (basal plane or c-axis), and (d) different non-magnetic interlayers (e.g., Y, Lu, or Sc). The first two case studies involve varying the magnetic rare earth element and uniformly result in coherent multilayer coupling. The Dy superlattices discussed in subsection 3.1.1 are examples of basal plane helical spin structures with spins perpendicular to the propagation vector along the e* axis, while [ErlY] superlattices discussed in subsection 3.1.2 are an example of an Ising-like incommensurate antiferromagnetic system with spins principally parallel to the c-direction. [GdiY] superlattices (subsection 3.1.3) are examples of a system with pure ferromagnetic coupling of spins within the layers for which either longrange ferromagnetic or antiferromagnetic interlayer coupling is found, depending on the Y thickness as shown by Majkrzak et al. 1988. The definitive case of rotating the growth axis relative to the spin propagation direction does in fact destroy the longrange magnetic coherence in [DylY] and [Gdly ] superlattices as will be reviewed in subsection 3.2. Changing the non-magnetic interlayer has the effect of enhancing or suppressing transitions between spin configurations as given in subsections 3.1.1.1 and 3.1.1.2, and of suppressing long-range order in the case of Sc (subsection 3.1.1.3).

3.1. Superlattices with c-axis growth directions As discussed above, rare earth superlattices are typically grown with layers stacked up along the c-axis crystallographic direction. Within this category are three distinct classes of magnetic structures. These are (a) magnetic layers in which the spins lie perpendicular to the growth direction and are ferromagnetically aligned in each atomic plane with the alignment direction precessing in a helical fashion from one plane to the next, (b) layers in which the order parameter is aligned along the caxis and is modulated from one atomic plane to the next, and (c) layers for which spins in all planes are ferromagnetically aligned and parallel, but are coupled either ferromagnetically or antiferromagnetically to spins in adjacent magnetic layers. Other more complex structures have been envisioned that are combinations of these three basic types.

3.1.1. Basal plane helical spin configurations 3.1.1.1. Superlattices with Y interlayers. Figure 4 shows the results of a Qz = (O00l) scan around the (0002) principal Bragg peak for a [Dy16]Y20189 superlattice for several temperatures near and below TN = 167 K. Note that there is one temperatureindependent central peak at Q = 27r/(d) ofstructural origin with one harmonic visible and displaced by AQ = 27r/L 27r/(102 A) = 0.0308 A -1. Higher order harmonics

12

J.J. RHYNE and R.W. ERWlN

are not visible on this scale. The magnetic scattering arising from the incommensurate spin structure produces the temperature-dependent Q- and Q+ satellite peaks on either side of the nuclear structure peaks. As discussed later, the ferromagnetic transition that occurs at 85 K in bulk Dy is suppressed in the superlattices. This would have been manifested by additional intensity appearing at low temperatures on the nuclear peaks. The presence of the fully resolved bilayer harmonics on either side of the Q+ and Q - satellites as shown in fig. 4 is dramatic evidence that the magnetic structure is coherent over many supedattice periods and is n o t interrupted at the layer boundaries or by the intervening Y layers. Also important is the fact that the chirality of the helix is maintained across the bilayer boundary. If the magnetic order were confined within single Dy layers, the Q width of the central magnetic satellite peak would encompass the bilayer harmonic positions rendering them unresolved. This is the case found for a superlattice of [DY14]Y34174 (see fig. 5 and later discussion) in which the coherence range has dropped to less than one bilayer due to the increased number of intervening layers of Y. From the scattering data, the coherence ranges of the atomic (nuclear) and magnetic order can be calculated from the intrinsic Q-width of the respective nuclear and magnetic peaks after deconvolution of the instrumental resolution width. The nuclear peaks typically give a coherence range of 500-700 ~. The resulting magnetic coherence distance, after correcting for the nuclear coherence, ranges from 580 ~ in a sample of 46 ~ of Dy separated by 26.5 ,~ of Y, down to 245 ,~ in a 46 ~ Dy, 56 ~ ' Y sample, and finally to 80 ~k in a 40 ~ Dy, 95 ~k Y specimen. All values but the last correspond to the propagation of magnetic phase coherence across multiple bilayer cells. o

[DYl4 ]Y34 ]74 ~(XI-

.'tZ- .,n

%

1 1.80

1.90

2.00

2.10

2.20

2.50

2.40

2.50

Qz Fig. 5. Broadened magnetic scattering peaks from a [Dy141Y34174 superlattice showing the loss of magnetic coherence at larger Y layer thicknesses.

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATTICES

13

r(~,) 2 0 0 I00

600

i

i

50

50

i

~

#

r-' Dependence of Coher

25 12I

500 A

-r 4 0 0 I-Z W

-J 3 0 0 bJ U Z W

~ 200 "Io I00

,,~.. > 1 4 0 A

,~f" 0

0.00

I

0.01

single Dy layer I

I

0.02 0.03 r-I=(y THICKNESS)-I(~, -I )

0.04

Fig. 6. The magnetic coherence length for a series of [DyIY] superlattiees as a function of the reciprocal of the thickness of the Y layer. The number of Dy planes was held approximately constant at 15 4- 1 planes (~ 43 ,~). The plot illustrates the 1/r falloff of the coherence, and shows that the layers would become non-interacting for Y thickness greater than approximately 140 ,~.

The decrease in the correlation length with increasing Y thickness has been examined in a series of [DyJY] superlattices with essentially fixed Dy layer thicknesses of 15 4- 1 atomic planes (~ 43 ~) and varying Y layer thicknesses as shown in fig. 6. The coherence range dependence quite accurately reflects a 1/rv falloff where rv is the thickness of the intervening planes (~, 2.87 ,~ per plane). As discussed later, the exchange coupling in the c-direction is of finite range, and thus this behavior suggests the presence of a decorrelation mechanism (viz., Dy basal plane anisotropy, thermal disorder) that competes with the exchange and leads to the loss of coherence for increasing Y thickness. The coherence length extrapolates to the non-interacting single layer limit of one Dy layer thickness (43 ~ ) at approximately 140 ~ of Y. Superlattices consisting of alternating layers of Ho and Y have also been fabricated and studied by neutron diffraction by Bohr et al. (1989) and by magnetic X-ray scattering by Gibbs et al. (see Majkrzak et al. 1991). The results are similar to those in Dy[Y superlattices and exhibit long-range coherence of the Ho moment across multiple bilayers. At low temperatures additional fifth and possibly seventh harmonics of the magnetic structure were observed that, as in bulk holmium, signal a spin-slip structure or equivalent bunching of the moments induced by the large 6-fold basal plane crystal field anisotropy. The presence of higher harmonics was also confirmed in more recent results on HolY superlattices by Jehan et al. (1993).

14

J.J. RHYNEand R.W. ERWIN

They also deduced from the width of the higher harmonics relative to the first order peaks that there is no spatial coherence between the spin-slip blocks across multiple layers, even though long-range coherence of the helical chirality and average turn angle exists from layer to layer similar to that in DyIY superlattices. The light rare earths (La-Eu) form a series of magnetic elements (except for La) where the magnetic exchange is considerably weaker than in the heavy series and becomes comparable in magnitude to the crystal field anisotropy energies. In addition departures from hexagonal close packed crystal symmetry are found based on an ABAC stacking of hexagonal layers instead of the ABAB sequence for conventional hexagonal close packed structures. This can be described in terms of two distinct atomic sites, one of hexagonal symmetry and one of cubic symmetry. In bulk Nd the ordering is of a multi-q incommensurate structure (see Jensen and Macintosh 1991, and Moon and Nicklow 1991, and references therein) in which the hexagonal site moment orders at 19.9 K with spins along the b crystal direction in an incommensurate modulation. At 8.2 K the cubic site moments order with a distinct incommensurate wave vector. When fabricated as one component of an [Nd[Y] supeflattice, the effects of epitaxial strain produce ordering characteristics in Nd (Everitt et al. 1994) that are quite distinct from those of the bulk element. The Nrel temperature is enhanced by almost 30% in the superlattice and the hexagonal sites order as ferromagnetic sheets in the basal plane with a small sinusoidal component superposed. Even though grown along the c crystal axis, magnetic order is not propagated between layers of Nd in contrast to the heavy rare earth Y superlattices. Below about 6 K order of the cubic site moment was also observed for thin Y interlayer superlattices.

3.1.1.2. Superlattices with Lu interlayers. Superlattices grown with Lu as the nonmagnetic spacer layer [Beach et al. 1993] show similar long-range coupling and loss of coherence as the spacer layer thickness is increased. In dramatic contrast to the [DyIY] superlattices, the ferromagnetic transition is enhanced by approximately a factor of two in [DylLu] superlattices as shown in fig. 7. (Also see plot of turn angles in fig. 25 of section 6.) This effect on the magnetic structure is discussed in section 7. A second difference observed in the [DylLu] superlattices compared to the [DyIY] superlattices is that the long-range coupling through the Lu has a different character that apparently depends on the fact that the Dy layers become ferromagnetic. When the Lu spacer layers are thin the coupling between layers is ferromagnetic, whereas for Lu layer thicknesses greater than approximately 10 atomic planes, antiparallel alignment of the Dy layers is always observed. Also the coherence of the helimagnetic state is reduced to a single layer for a Lu layer thickness of 80 ,~. However, the coherence of the low temperature antiparallel state remains at about two bilayers for this sample. These effects can be explained by a model for the interlayer coupling that includes a competition between indirect exchange and dipolar energies. As shown in section 3.2, the dipolar coupling of 300 ~ domains can approach several kilogauss. This is the approximate size of the field required to align the Dy layers in the antiparallel state, a value which was shown to be nearly independent of Lu layer thickness, as would be the case

MAGNETISM IN ARTIFICIALMETALLICSUPERLATIqCES I

10

I

I

I

t

150 K 170 K

lLu

8-

I

15

Q O o ___

o

4

1.8

2.0

2.2

2.4-

2.6

ez Fig. 7. Diffractionscans for a [DY21lLul0] superlattice show that below 160 K the helical magnetism of the Dy layers is transformed into ferromagnetismwith antiparallel alignment of the layers. The peak widths demonstrate that the helimagneticstate is of shorter range than the ferromagneticstate, and there is some remanence of the helix in the ferromagnetic state. The centroids of the magnetic scattering at 170 K and Q ,,~ 2.05 ,~-1 and Q ,,~ 2.4 ~-1 determine the turn angle in the Dy layers. for dipolar coupling. The domain size has been confirmed from X-ray diffraction peak widths of basal plane reflections in the ferromagnetic state. This domain size is limited by epitaxial strain energies as the lattice obtains a local orthorhombic distortion to minimize the magnetoelastic energy. If the domain size becomes too large, the energy minimizing distortion would force a large number of energetically costly atomic dislocations. The modification of the magnetic structures in the superlattices compared to the bulk magnetic structures provides mfique insights into the physics that controls these magnetic structures. The ferromagnetic transitions are altered as well as the temperature dependence of the turn angles that describe the magnetic structures. These magnetic structures are summarized for Dy-based materials in section 6. The classical theory of the ferromagnetic transition in Dy will be modified to account for epitaxial strain in section 7. Theories of the temperature dependence of the turn angle have concentrated on superzone-gap effects. The data for [DylY ] and [DylLu ] superlattices indicate a direct relation of the turn angle to the strains in the system.

16

J.J. RHYNE and R.W. ERWIN

While strain effects might be included in the band theories of the temperature dependence of the turn angle, it is interesting that the superlattice data can be described by a simple phenomenological theory. It is also significant that strain produces strong effects on the turn angle at the initial ordering temperature. 3.1.1.3. Superlattices with Sc interlayers. Scandium, like Y and Lu, is a non-magnetic hcp metal with electronic and physical properties similar to the heavy rare earths, and Sc forms solid solution alloys with them. However, in marked contrast to Y (e.g., Gotaas et al. 1988), dilute alloys of the heavy metals with Sc do not show longrange magnetic order (Child and Koehler 1968) below concentrations of 20-30% of the magnetic rare earth. The lower concentration alloys instead exhibit spin glass behavior. Superlattices of Dy layers alternated with Sc layers have been prepared and studied by magnetization and neutron diffraction by Tsui et al. (1993). The 8% basal plane lattice mismatch between Dy and Sc presented a major obstacle to production of highquality superlattices that was solved by reducing the deposition temperature and rate. A superlattice of [DY25 AIScao /~]66 was produced with crystal coherence of about 500 ~ along the c growth axis. Neutron diffraction scans along c* are shown in fig. 8 8000

I

o Sc (0002)

l ~ 400C

.0

2.3

2.6

Qc~(~-I) Fig. 8. Neutron diffraction scans along the [0002] direction from a [Dyz5 i]Sc4o A]66 superlattice: (a) Nuclear intensity at 160 K showing five structural superlattice sidebands and a (0002) reflection from the Sc buffer layer, (b) a zero field scan at 10 K showing the short ranged ferromagnetic order along the growth direction as indicated by the thick line underneath the structural superlattice peaks which remains unchanged, and (c) zero-field-cooled scan at 10 K with a 60 kOe field applied along the a-axis showing the magnetic superlattice intensities on top of the structural peaks. This latter result indicates a coherent ferromagnetic order with vanishing short-range order. Scans (b) and (c) are displaced by 2000 and 4000 counts for clarity. Lines through the data points are Gaussian fits, and arrows indicate superlattice reflections.

MAGNETISM IN ARTIFICIALMETALLICSUPERLA'ITICES

17

at 160 K (a), which is above the magnetic ordering temperature, and at 10 K (b). The observation of superlattice sidebands in (a) attests to the quality of the superlattice; however, there is no significant change in or additional peaks on cooling to 10 K (b) which would signal long-range magnetic ordering. Rather, as shown by the heavy line in (b), the magnetic scattering appears as a broad diffuse background about the nuclear peak positions, characteristic of short-range ferromagnetic correlations. The width of the peak corresponds to a coherence range of 24 ± 3 ~, which is close to the Dy layer thickness of 25 ~ indicating that the Dy layers individually order ferromagnetically but with no interlayer coupling. The ferromagnetic peak intensity decreases at elevated temperatures and becomes broader. The onset of irreversibility in the bulk magnetization data indicates a Te of 147 K considerably above the To of bulk Dy, reflecting strain effects as discussed in section 7. Figure 8c shows the diffraction results of application of a 60 kOe field in the basal plane to align the spins in different Dy layers. The structural peaks show markedly enhanced intensity and the broad diffuse peak is essentially removed, both corresponding to the long-range coherence induced by application of the field. The lack of long-range order in the c-axis Dy-Sc superlattices, as well as in the dilute alloys of Dy and Sc, suggests that the features of the generalized susceptibility x(q, ~) in Sc may be very different from those in Y or Lu.

3.1.2. c-axis modulated spin systems Superlattices consisting of alternate layers of Er and Y represent a more complex magnetic ordering than the [DyIY] helical and [GdIY] antiferromagnetic or ferromagnetic configurations. The c-axis is the favored moment direction in Er due to a change in sign of the 4f electron quadrupole moment compared to Dy. Thus the spin alignment is parallel to the stacking direction of the superlattice and also parallel to the c-axis magnetic propagation vector. In bulk elemental form (Cable et al. 1965, Habenschuss et al. 1974), Er initially orders at 84 K into a c-axis modulated moment structure in which the parallel (c) component of the magnetization has a sine-wave amplitude modulation with a period of approximately seven atomic layers. The wave vector of the modulated structure progresses through a series of lattice-commensurate lock-in states with decreasing temperature as derived by Gibbs et al. (1986) and discussed in section 6. The transverse moment components in Er are disordered down to 56 K and were originally considered (Cable et al. 1965) to exhibit helical ordering at lower temperatures. More recent neutron scattering data, supplemented by detailed calculations of exchange and two-ion couplings (Jensen and Cowley 1993), show that the intermediate temperature state (18 K < T < 56 K) is described by a wobbling cycloidal ordering in which there is a b-axis moment perpendicular to an a--c plane cycloid that oscillates with a different period from the basic structure. Below 18 K the c-axis order becomes ferromagnetic, resulting in a conical moment state with apex angle about 280 and with a bunching of moments around the a-axis directioxts. The scattering geometry for the Er-Y superlattice studies (Erwin et al. 1988a, 1988b, Borchers 1989a, 1989b, 1991a) is somewhat more complex, in that predominately basal plane reflections, for example [10i0], must be used to detect the c-axis

18

1 I RHYNE and R.W. ERWIN

moment components, since the neutron scattering is sensitive only to moment components perpendicular to Q. Basal plane ordering was detected as previously by re-orienting the sample with [0002] parallel to the scattering vector. Figure 9 shows temperature-dependent scans along c* at (1010) and along c* at (0002) (fig. 10) from a [Er32[Y21] superlattice demonstrating that well-defined nuclear peaks and harmonics as well as magnetic satellite peaks and harmonics are observed. The magnetic peak widths again confirm that the magnetic order is long range. The reduced intensity of the basal plane nuclear satellites results from the near equality of Er and Y scattering lengths and possible interface defects resulting from the nearly 3% lattice mismatch. The TN for the superlattice, marked by the onset

/

!

""

70K

"

65K

250 "~

200

C

--

150

.~

ioo

c

__c

,

5O

,20K

,6K

-0.5

-0.5 -0.1

0.1

0.5

0.5

K z ( A -I )

Fig. 9. Diffraction scans along the c* direction through (10i0) for [Er32]Y21] at several different temperatures showing the development of a linear spin density wave with principal moment component along the c-axis. At lower temperatures the ordering becomes more complex as indicated by the appearance of higher-order harmonics [e.g., (1010)4"3 harmonic of the (1010) reflection.

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATI'ICES

19

70

60

5O C

~- 4 0 ~ 3O 55 K

20

I0

'

/ - - - - " - ' - '

/

20K

61< 1,8

2.0

2.2

2.4

2.6

Fig. 10. Diffraction scan through (0002) for [Er321Y21]. Below about 30 K, the basal plane component of the moment exhibits periodic order as indicated by the satellites of (0002) 4-.

of magnetic satellites in the (10i0) scans, is 78 4- 1 K which is 7% lower than for bulk Er and similar to the reduction seen in Dy superlattices. The initial ordering is nearly sinusoidal for the c-axis moment components with the transverse components disordered. Below 35 K, additional magnetic scattering appears which can be indexed as higher harmonics [e.g., (1010) +3 and (1012) -3,-5] of the fundamental magnetic satellites (101 l) ±. These reflect the presence of a more complex intermediate spin state as seen in bulk Er, but at a correspondingly lower temperature than in the bulk. Below about 30 K, the basal plane moment components order into a periodic state; however, no conical ferromagnetic ordering is observed down to 5 K in contrast to bulk Er. The magnetic coherence range derived from the magnetic peak width, after

20

J.J. RHYNE and R.W. ERWIN

,-... 600 i Z

~ 400 W 0 Z W W -r

o o 200 Z

[Dy, I Y,]

0;

I

[] I

I

I

I

10 20 Y THICKNESS (plones)

i

30

Fig. 11. Coherence length for [Er321Y21 ] and [ErlaIY26 ] superlattices derived from both basal plane and c-axis moment component satellites. The basal plane values from the [DyIY ] superlattices are shown for comparison, along with 1/,rv and 1/r 2 fits to the [DyIY] data.

deconvolution of the instrumental resolution, is shown in fig. 11 as a function of the Y thickness. It is noted that the coherence range derived from the separate widths of basal plane and c-axis moment components is different. The figure shows the coherence range for [DyIY ] superlattices for comparison. The basal plane coherence length is comparable to that found for [DyIY ] superlattices, while the c-axis values are relatively larger. It is noted that the Er layer thicknesses were not held constant as was the case for the [DyIY ] superlattices in fig. 6.

3.1.3. Ferromagnetism and antiferromagnetism in Gd-Y superlattices 3.1.3.1. Polarized neutron scattering. Superlattices consisting of alternate layers of Gd and Y have been studied extensively with neutron scattering by Majkrzak et al. (1986, 1988, 1991). Gadolinium is an S-state ion and does not possess the Fermi surface nesting features that lead to an incommensurate periodic magnetic structure and is thus ferromagnetic in bulk form. A superlattice of alternating Gd and Y layers would then be expected to show a collinear alignment of Gd spins on opposite sides of the Y layer, assuming long-range order to be developed. In contrast to superlattices of the other heavy lanthanides with yttrium in which the magnetic and nuclear scattering are well separated in Q due to the incommensurate structure, the ferromagnetic coupling expected for some Gd-Y multilayers virtually necessitates the use of polarized beam techniques such as used in reference (Majkrzak et al. 1986) and also in the studies of Gd-Dy superlattices described in reference (Majkrzak et al. 1988). In the polarization analysis, four structure factors can be

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATTICES

21

derived, corresponding to the combinations of neutron spin-flip and non-spin-flip scattering as follows (Majkrzak et al. 1986): (A) non-spin-flip scattering: N

F+±(Q) = - y'~(-bj 4- pj cos ¢)e iqu~,

(6)

j=l

and (B) the spin-flip scattering: N

F±a:(Q) = - y~(pj sin ¢)eiQ~'j,

(7)

j=l

where ¢ is the angle between the spin direction and the neutron polarization which is parallel to the applied guide field. The variable uj is the position of the jth atomic plane along the c* stacking direction. The non-spin-flip scattering thus contains information about the alignment of the spins along the applied field (polarization) axis, while the spin-flip scattering contains information about only spin components perpendicular to the field direction. Performing the lattice sums for the multilayer results in expressions for the spin-flip and non-spin-flip cross sections analogous to eqs (4) and (5) that then contain terms separating out the magnetic only scattering amplitudes. Using the same reflection geometry described above for the Dy-Y multilayers, Majkrzak et al. 1986 obtained the intensity distribution in fig. 12 (plotted on a logarithmic scale) for a [Gd10[Y101225 superlattice. Non-spin-flip magnetic harmonic peaks were observed at spacings AQ = 27r/L (L = bilayer thickness) corresponding to long-range ferromagnetic alignment of spins in the Gd layers and are designated by the even numbers in the figure. However, additional peaks arising from spinflip scattering were observed at half this Q separation and thus correspond to an antiferromagnetic alignment of spins on opposite sides of the Y interface. The spin alignment is then an antiphase domain structure as shown in fig. 13 where the canting angle (90 o - e ) is induced by the 150 Oe field applied as shown at 75 K. In contrast to these results for repeated bilayers of 10 planes each of Gd and Y, other Y thicknesses, in particular 6 and 20 planes, showed no evidence for the antiferromagnetic coupling of alternate Gd layers. Rather they exhibited a ferromagnetic coupling of Gd spins on opposite sides of the Y layer at all temperatures below Te. Further confirmation of the unique ferromagnetic-antiferromagneticexchange coupling oscillation in the [GdlY] superlattices was provided by studies of a [GdsIY~IGdsIY10] quadlayer as discussed by Majkrzak et al. (1991). Neutron diffraction studies indeed showed reflections confirming the parallel coupling of spins between Gd layers separated by 5 atomic planes of Y and antiferromagnetic coupling of spins between Gd layers separated by 10 atomic planes of Y. Thus the exchange oscillations found in discrete single-period superlattices were confirmed in a superlattice of quadlayers for which the magnetic unit cell was found to be twice the length of the chemical unit cell.

22

J.J. RHYNE and R.W. ERWIN

r

r

!

(0002)

40 I- HSLR(III)- 801-S-20 / k,=2.67 ~-' il T= 150K H= 150 Oe ] 104

-s t£)

103

t6 ~3

g 8 $-I H 4I I H

g -

J 5

!

102 i

1.600

,.900

2.200

2.500

0(~.-') Fig. 12. Polarized neutron scattering data from a [Gda0JY101225 superlattice. The odd numbered peaks represent spin flip scattering and correspond to antiferromagnetic coupling of the Gd spins across the Y layers. The even numbered peaks are non-spin-flip scattering and represent the ferromagnetic coupling of the spins within a Gcl layer (after Majkrzak et al. 1986).

Yafet et al. (1988) have calculated the exchange coupling across Y layers of various thicknesses using an indirect exchange model that is based on the generalized conduction electron susceptibility calculated by Liu, Gupta, and Sinha (1971). This model correctly predicts the crossover from ferromagnetic to antiferromagnetic spin configurations with Y thickness as shown in the inset to fig. 13. 3.1.3.2. Bulk magnetization. The bulk magnetic response of the Gd-Y superlattices has been studied by SQUID magnetometry. As shown in fig. 14, the magnetization of a [GdsJYs]80 superlattice (Kwo et al. 1985b) with the field applied parallel to the film plane shows a square loop behavior, while that with the field applied perpendicular to the film plane shows a near linear field dependence. This is the behavior expected for a thin film with relatively small anisotropy. The loop feature at low field is not explained and was not observed in other superlattices. The large slope of the Hparallel curves above technical saturation Hs reflects the induced magnetization of disordered spins in the interface region as discussed in section 8. The spontaneous magnetization or(0) determined from the post-saturation magnetization extrapolated back to H = 0 is significantly lower than the 250 emu/g value expected for ferromagnetic Gd +++, again the result of the interface disorder. The spontaneous moment is observed to

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATIqCES

(+)

-,

~

. . . . . . . . .

0 ...........

(-~

~i

S .

.

.

23

.

.

.

~ .

.

~ .

.

.

.

.

J

. . . . . . . . . 4. . . . . . . . . . .8. . .

12

t6

Ny Gd

XSL

Y +~

Q'II C-AXIS

Fig. 13. Schematic representation of the antiphase domain configuration of Gd moments in the [Gd101Y101225 superlattice at 75 IC The angle c is approximately 80*, and its departure from 90 o is a consequence of the 150 Oe applied field. The inset at the top illustrates the oscillatory ferromagnetic and antiferromagnetic spin couplings across the Y layer depending on the thickness of Y. The solid line is the result of an indirect exchange calculation of Yafet et al. (1988) (after Majkrzak et al. 1986).

depend linearly on the inverse of the number of Gd planes in each bilayer of the superlattice as shown in fig. 15a, which includes data for a Gd fihn. In addition to the increase in the spontaneous moment toward the bulk value for thicker Gd layers, there is a concomitant reduction in the induced moment A~r = ~r(15) - cr(0), where tr(15) is the measured moment at 15 kOe. The quantity A~r is also proportional to the inverse number of Gd planes per bilayer as shown in fig. 15b and is essentially 0 for a pure Gd fihn. The oscillation in the interlayer exchange coupling with increasing number of Y intervening planes in [GdIY ] superlattices was observed definitively in the neutron diffraction studies described above. As shown by Kwo et al. (1986), the property is also manifested in the bulk magnetization, particularly the remanent magnetization err and the field Hs required for moment saturation defined as the field at which the

24

J.J. RHYNE and R.W. ERWIN

I

I

I

l-

I

T

ZOO

r

.

r -----

i

---t

HII

I00

.[ o

I

i

,1

I 6

12

q.) v

b -100

-200 I

a

I

I

t

H

J

I

t

I

(kOe)

Fig. 14. Magnetization data for a [GdsIY518o superlattice for field applied parallel to the film plane HII and perpendicular to the plane Ha. (after Kwo et al. 1985b).

300 i 0

l

----_____

t

I

--.-_____

200

E Q; b

I00

I

I

0

0.1

(O)

I

0.2

t/NGd Fig. 15. (a) Linear dependence of the spontaneous moment 0-(0) and (b) the excess moment A~r = o'(15) - ~(0) on the inverse of the number of Gd planes in a bilayer of superlattices with 5, 10, and 20 Gd planes. Data for a Gd film are also shown. The Curie temperatures of the superlattlces deviate very little from the bulk value of 293 K except for the 5 plane sample that shows a lower Tc possibly from alloying effects (after Kwo et al. 1985b).

MAGNETISM IN ARTIFICIAL METALLIC SUPERLA'ITICES

25

Y THICKNESS (,~)

0

20

40

60

8O t

.od=4

,

.od--lo+-t

"I,O-

s

!÷i I I

I I

/5 ÷

i i

I I

%'

e

1

/ \ /~ ,

5

i

0

I

I

i

:,,

p

t

t0 20 N y ( ATOM IC LAYERS)

30

Fig. 16. The oscillations in [top] the normalized remanent magnetization ~rr/O'(0), and [bottom] the field for technical saturation Hs plotted as a function of the number of intervening Y layers for [GdIY ] superlattices with both 4 and 10 atomic planes of Gd. The out-of-phase oscillations reflect the oscillations in the interlayer exchange coupling as illustrated in the inset of fig. 13 (after Kwo et al. 1985b).

rapid initial rise in the moment for O'parallelis complete. The definitions of Hs and o"r can be visualized from the HII curve of fig. 14. In fig. 16 the remanent magnetization ~r/~r(0) (normalized by the spontaneous magnetization) and the saturation field Hs are plotted for a series of [GdlY] superlattices as a function of the thickness of the intervening Y layer, Ny. Remarkably, these two quantities show out-of-phase oscillations with Ny and have identical values for superlattices with both 4 and 10 atomic planes of Gd. A comparison of these curves with the coupling oscillations shown in the inset of fig. 13, for which the solid line gives the result of the calculation by Yafet et al. (1988) and the solid points are experimental results from the neutron diffraction, shows a consistent behavior with the number of Y atomic planes.

3.1.4. Superlattices with two magnetic layers The helical ordering of magnetic layers and the long-range interlayer phase coherence in D y - Y superlattices are well confirmed as reviewed in previous sections. Likewise, in Gd-Y superlattices, the occurrence of ferromagnetic layers and an oscillation of

26

J.J. RHYNEand R.W. ERWIN

the interlayer coupling is fully established. Superlattices made up of alternate layers of Gd and Dy would then be expected to show a highly complex ordering as found by Majkrzak et al. (1988, 1991) from neutron diffraction studies. The bulk magnetization results show that as the temperature is lowered, the Gd layers first become ferromagnetically aligned (To ~ 250 K for [GdsIDys], reduced from the 293 K of bulk Gd by interface alloying). At about 200 K the Dy layers begin to order non-colinearly. As shown from the neutron diffraction (Majkrzak 1988), the turn angle of the Dy is significantly perturbed from the bulk values, showing a tendency toward ferromagnetic alignment for planes adjacent to the Gd layers with a gradually increasing magnitude toward the center of the Dy layers. For only 5 Dy planes, the distorted ordering is essentially a fan structure or incomplete helix. Gd-Dy superlattices with thicker Dy layers show a somewhat distinct ordering described as a 'superspiral.' Neutron data of Majkrzak et al. (1988) confirm that the Gd planes of a [GdsIDYlo] superlattice initially order ferromagnetically, and then below 200 K the onset of Dy ordering induces a change in the direction of adjacent Gd layers. Initially parallel spins in adjacent Gd planes are found to rotate by about 900 (angle specific to this superlattice) as a result of the influence of the intrinsic Dy helical ordering. The turn angle of the Dy helix is in turn perturbed by coupling to the Gd spins, and is again found to vary from near 0 (ferromagnetic) for spins near the Gd-Dy interface to essentially the bulk Dy value in the center of the Dy layer. Superlattices of alternate layers of Dy and Er present the opportunity to study the interacting order of two systems that intrinsically both show incommensurate magnetic structures of different types. A series of c-axis [DylEr ] superlattices has been studied by magnetization and by X-ray (Dumesnil 1994a) and neutron diffraction (Lee 1994 and Dumesnil 1994b). In these results distinct ordering temperatures were found for the two dissimilar magnetic layers. The initial ordering temperature is close to TN of bulk Dy and involves helical ordering of the Dy spins with no involvement of the Er 4f spins. Spiral coherence is still propagated across multiple bilayers via the ostensibly paramagnetic Er layers in a manner similar to Y superlattices. At temperatures below 100 K the Dy helical order is gradually supplanted by a ferromagnetic intralayer moment structure. The coupling between layers is antiferromagnetic, except for very thin Er layer thicknesses (~< about 15 ~) in which the interlayer coupling is ferromagnetic. At temperatures below the TN of bulk Er, incommensurate ordering of the c-axis component of the Er spin structure was observed, and at 10 K evidence was found of basal plane moment components also ordering in an incommensurate structure.

3.2. Superlattices with a basal plane growth direction 3.2.1. Helical spin configurations As discussed in the previous subsection, the occurrence of long-range coherent magnetism in e-axis rare earth superlattices is independent of the primary orientation of the spins and independent of whether the spin order is ferromagnetic or incommensurate antiferromagnetic. All of these systems have the common feature that the superlattice growth axis is along the c* propagation direction of the spin system.

MAGNETISM IN ARTIFICIALMETALLICSUPERLATTICES

27

In order to test the dependence of the ordering on the superlattice growth direction relative to the c* magnetic propagation vector, superlattices were prepared by Flynn et al. (1989) with the growth axis along a basal plane direction. This required new preparation procedures due to the unavailability of conventional substrate materials with an appropriate lattice matching to the a-c plane of the hexagonal rare earths. Highly polished single crystal slabs of Y were used as a substrate material with the superlattice of alternate Dy and Y layers grown on top, following a thick Y buffer layer. In the neutron scans along the a* (growth axis) direction, the near coincidence of the Y lattice parameters and that of the superlattice prevented the direct observation of the (0002) structural peak. Instead, as illustrated in fig. 17 for a [Dy261Yg]s2 superlattice, the bilayer harmonics of the (0002) (enhanced by applying a magnetic field which produced ferromagnetic ordering) were scanned and their width confinned the quality and chemical order coherence of the superlattice as shown by the (Q, O, 2.221) nuclear harmonics of near resolution width. Below 170 K, helical magnetic ordering was found in the basal plane of the Dy layers; however, as shown in I

60-

160

09

I

I

I

[DY291Y1°]

I

(qb 0 2.221)

120

CO

Qu 0 1.99!

80 O C)

40

=

I

-0.12

27t/~

!

=

0.078

A -t !

r

-0.04

0.[)4

Q (A -I)

0.12

Fig. 17. Magnetic and structural peaks of the b-axis superlattice [Dy26[Yg]82scanned along the a* direction. The narrow resolution-limitedpeaks are the bilayer harmonics of the (0002), and broad peak is the helimagnetic satellite. The width of this peak corresponds to a coherence range nearly identical with the Dy layer thickness showing the loss of long-range magnetic coupling.

28

J.J. RHYNE and R.W. ERWIN

the figure the (0, 0, 1.995) Q - magnetic satellite scanned along a* is quite broad and there are no resolved bilayer harmonics of the magnetic ordering. Indeed the intrinsic width of this peak, after deconvolution of the resolution broadening, is ~ = 80 ~. This is approximately the 7 3 / ~ thickness of the Dy layers and is compelling evidence that the coherence between the helices in adjacent Dy layers is destroyed in these basal plane superlattices. This result thus established that the multiple layer coherence indeed depends on the magnetic propagation vector (c') being parallel to the growth axis. For the b-axis growth superlattices, the helical order which develops within each Dy layer also is observed to have finite range. Transverse scans show that the width of the magnetic satellite peaks along the (11"~0) direction is essentially resolution limited (~ 1> 500 ~), indicating that the ferromagnetic sheets in the basal planes of each layer are uniformly ordered and probably limited in extent only by the sample size (along a) and by the layer thickness (along b). However, the coherence range measured along the c* in-plane direction is significantly smaller and decreases with temperature to about 250 ~ below 100 K. Thus the helical stacking of the a--b ferromagnetic sheets within each layer is within a finite and temperature-dependent domain size. Similar results for the range of magnetic coherence were obtained for a [DyTIY25167 basal plane superlattice. I I I l - 6 0 0 - ~) Cd(43A)/Y(52A) ,

-

2oo

I

•%1~1200

I

I

b) (~d(OOA)/V(ZGA)

4OO

Q× (~-') Fig. 18. (a) Neutron diffraction scans at 80 K (circles) and at 315 K (triangles) along the b* direction for a [Gd43 AIYs2 :,]85 superlattice with a b-axis growth direction. At 80 K peaks develop with twice the bilayer periodicity indicating that the Gd layer moments are anti-aligned. (b) Scans along the b* direction for a [Gd6o :,1Y26 :,]so superlattice at 90 K (circles) and at 315 K (triangles). The extra intensity evident at the superlattice satellite positions at 90 K indicates that the Gd layer moments are ferromagnetically aligned.

MAGNETISM IN ARTIFICIALMETALLICSUPERLATTICES

29

3.2.2. Dipole-dipole coupling in Gd-Y superlattices The coupling in Gd-Y superlattices with a c-axis growth direction has been observed to oscillate with the thickness of the Y layer with a period of approximately 25 A as reviewed in section 3.1.3. The behavior is consistent with a simple RKKY indirect exchange model of the interlayer coupling. This coupling should decay quite rapidly along the basal plane directions as was observed for b-axis Dy-Y superlattices (see section 3.2.1). Neutron diffraction studies of two Gd-Y superlattices grown along the b crystallographic direction are shown in fig. 18. Scans along the a* direction above Tc for both [Gd43.~IY52/~]85 and [Gd_60.~lY26/~]80 show the presence of structural harmonic peaks separated from a (1010) dominant central peak arising from the Y substrate by AQ = 21r/L where L is the bilayer thickness. Below Tc, the superlattice with the thicker (52 .~) Y interlayer shows new peaks with a periodicity of 2x AQ reflecting an antiparallel alignment of Gd moments across the Y interlayer. In contrast, 80 K data (below Tc) on the superlattice with the thinner 26 ,~ Y interlayer show extra intensity developing on the structural harmonics, indicating a parallel alignment of Gd moments between layers. At elevated temperatures (> 120 K) the structure reverts to antiferromagnetic alignment. Studies of the external magnetic fields required to flip the antiferromagnetic spins into parallel alignment yield a coupling strength of 80 and 20 G for the thick and thin Y interlayer superlattices, respectively, which is significantly smaller than the critical fields measured in c-axis Gd-Y systems (~ 1 kG) (Majkrzak et al. 1986, Kwo et al. 1986). This sharp contrast in flipping fields for b-axis and c-axis samples again reflects the anisotropic nature of the exchange interaction discussed in section 4, which suggests that the range of the coupling along basal plane directions is smaller by a factor of at least 10 compared to the c-axis direction. The fact that b-axis Gd-Y superlattices do retain an antiferromagnetic interlayer coupling while b-axis Dy-Y superlattices show no interlayer coupling requires additional explanation. The basal plane helical order of Dy with propagation direction along c* leads to a complex spin configuration with spins at various angles to the layer interface. Such a structure has potential spin frustration and boundary effects that may adversely affect spin ordering compared to the relatively simple ferromagnetic or antiferromagnetic spin configurations found in the Gd-Y superlattices where the spins are all parallel to the layer interface. The Gd-Y case has a clear analog with antiparallel dipole coupling between two ferromagnetic sheets. The critical coupling field for such a dipole configuration is: He = 4M W In

tGa +

ty

'

(8)

where M is the magnitude of the Gd moment, W is the lateral width of the Gd layers, and tC,d and ~y are the Gd and Y layer thicknesses. The fields required to flip the antiparallel layer spins into ferromagnetic alignment as calculated from this expression agree quite well with the experimentally determined fields given above assuming intralayer domains of approximately 4000 .~ dimensions. The transition from parallel to antiparallel interlayer coupling with Y interlayer thickness in Gd-Y

30

J.J. RHYNE and R.W. ERWIN

b-axis superlattices can then be rationalized as arising from a relatively short-range ferromagnetic RKKY exchange contribution and a longer ranged antiferromagnetic dipole coupling favoring antiferromagnetism. In simplest form, both these interactions should exhibit an r -3 fall off; however, the RKKY mechanism may be subject to more complex electronic and mean free path damping effects which can further limit the range of the exchange coupling, but would have no effect on the dipole coupling. 4.

Interlayer magnetic coupling in superlattices

The above results illustrate that for periodic moment systems, long-range interlayer magnetic coupling is present in superlattices for which the stacking direction is parallel to the c-axis propagation direction of the periodic magnetic system and that such coupling is destroyed for stacking sequences along basal plane directions. The interlayer exchange coupling is not simply ferromagnetic or antiferromagnetic (except for Gd and Lu superlattices) but is of a more complex form as can be demonstrated from the effective phase shift of the magnetic ordering across the Y or Lu layers. This phase shift can be calculated from the asymmetry in the intensities 6.00-

5.50 Y layer [DY16 1Y2o ]sg 5.00

4.50

LAYER TOTAL PHASE SHIFT

tl.

o ¢n 4.00

3.50

3.00

2.50

2.00

-

~

Y layer [DYt6 I Y9 1100

T

40

-w

80 T(K)

,

,

120

160

Fig. 19. The total phase shift of the spin density wave across Y and Dy layers as a function of temperature for [DYl6]Y20]89 and for [Dy161Yg]loo. Note that the Y phase shift is not a multiple of lr and is independent of temperature. The variation in the Dy shift reflects the T-dependence of the turn angle. (See text.)

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATFICES

31

of the magnetic satellites Q - and Q+ using eq. (5). This has been done for the [DyIY] superlattices, and as shown in fig. 19, the phase shift is not a multiple of ~r (as for purely ferromagnetic or antiferromagnetic interactions), but is completely prescribed by the number of interleaving Y atomic planes. The Y layeroPhase shift corresponds to an effective turn angle of 51-520 per Y layer (~ = 0.31 A-I). The theory by Yafet et al. (1988), invoking only the RKKY interaction between Dy layers, successfully predicts chiral coherency and the correct order of magnitude for the interlayer interactions (as measured, for example, by the magnitude of the applied field required to break down helical order). However, in this calculation the Y turn angle is not a constant independent of Y thickness. The experimental results suggest that the magnetic structures are determined by the separate conduction electron susceptibilities of each layer material. The complexity of the ordering in [Dy[Y] and [ErIY] superlattices and its dependence on propagation direction has led to the suggestion (Rhyne et al. 1987) that the mechanism behind the long-range spin coupling is the stabilization of a spin density wave (SDW) in the Y and 4f lanthanide conduction bands via RKKY coupling to the 4f local moments in the lanthanide. In linear response theory, the real space exchange coupling J(R) can be expressed (see Elliott 1972) in terms of a Fourier transform of a q-dependent exchange j(q): qmax

J(R) = ~ j(q)e -iq'n,

(9)

q=O

where j(q) is in turn proportional to a conduction electron generalized susceptibility x(q) through an exchange matrix element Jsf(q) in the following form:

j(q)-= Ij~f(q)12x(q)/2, x(q) has

(10)

been calculated for Dy and Y from the band structure (Gupta and Freeman 1976, Liu, Gupta, and Sinha 1971). This function exhibits strong positive maxima along the c* direction at q = qmax ~ 0 (q = reduced wave vector), where qmax is only minimally different for Dy and Y and is in general prescribed by 'nesting' features of parallel sheets of the Fermi surface (Keeton and Loucks 1968). In Dy and other lanthanide elements this calculated wave vector qmax is also very close to the measured magnitude of the helical wave vector at the initial ordering temperature. In the a* and b* directions the peaked behavior at q 5~ 0 is not observed and j(q) falls off in a manner that reflects the core size and can be approximated by a Gaussian of width 0.63 ]k-1. Flynn et al. (1989) derived the three-dimensional representation for j(q) shown in fig. 20a. by using the Gaussian form in the basal plane direction and the expression for x(q) calculated by Liu, Gupta, and Loucks (1971) for the c* direction. The real space exchange J(R) calculated from eq. (9) is also shown in fig. 20b, along with the overall envelope function for the exchange. The salient feature of this calculation is that the range of the real space exchange coupling is

32

J.J. RHYNE and R.W. ERWIN

j (F)

Fig. 20. (a) Schematic representation of j(q) as discussed in the text for which the width along a* reflects mainly the exchange matrix element while that along c* reflects x(q). (b) The envelope function for the Fourier transformreal space exchange, showing the highly anisotropic spatial range. The oscillatorycurve is the actual transformfunction. highly anisotropic and extends beyond 130 ,~ with significant amplitude in the cdirection, but falls rapidly to negligible values within about 12 ~ along the basal plane direction. This result clearly explains the existence of long-range coupling through Y layers exceeding 100 ,~ in thickness (e.g., fig. 6) in superlattices grown along the c-direction and also accounts for the lack of such coupling through Y layers as thin as 26 ,~ in [Dy[Y] superlattices with growth axis along b (e.g., fig. 17).

5. Effect of applied magnetic fields on interlayer coherence The response of the superlattice systems to an external applied field has been studied by both magnetization (SQUID magnetometry) and by neutron diffraction. The results of magnetization studies on [DyIY] superlattices can be found in Borchers et al. (1987), and on [ErIY] in Erwin et al. (1988b), and Borchers et al. (1989a, 1989b, 1991a, 1991b). In elemental lanthanide periodic moment structures (e.g., Dy, Ho, Er) the effect of applying an external field in the plane containing the spins is to effect a transition to a ferromagnetic state. This transition can either be of first order in which a discontinuous jump from a near zero net moment helical state to a saturated ferromagnetic state is measured by a magnetometer at a critical applied field, or the transition may proceed through a series of intermediate or 'fan-type' moment states occurring before final ferromagnetic alignment. The details of the magnetization process in bulk crystals reflect a free energy balance between the Zeeman energy and the exchange, crystal field anisotropy, and magnetostriction contributions (see Elliott 1972, and Jensen and Macintosh 1991).

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATI'ICES

33

In superlattices, magnetometry results are complicated by the large addenda correction to the data for the paramagnetism of the substrate and buffer layers. Although highly precise qualitative moment values are difficult to obtain, critical field and saturation effects are conveniently measured. Neutron diffraction data taken as a function of applied field provide straightforward insight into the magnetization process, including the critical fields, intermediate moment states, magnitude of the ferromagnetic and helical moments, and details of the transfer of the moment components from a helical configuration to ferromagnetism. This transfer is signaled by a loss of scattered intensity in the satellite reflections at Q- and Q+ and a concomitant increase in the intensity on the fundamental structural reflection and harmonic peaks (e.g., (0002)). In addition, the results continuously monitor the effect of the applied field on the long-range interlayer spin coherence. As an example, fig. 21 and fig. 22 show the comparative effects of a field applied in the basal plane of a [Dy14]Y14164 superlattice at 10 K (0.059Tc) and at 130 K (0.77Tc), respectively. At 10 K , the intensity of the Q - satellites was observed to decrease above about 3 kOe and to essentially collapse in fields of 10 kOe and !

B

~2

7

[Dy14Y14164 1OK ~-8 (0002)

E

24

20

c

.25 kOe Ferro

= O

% 16 1 0 kOe_ m r-

$

12

c

-

H:O

8-

1.8

1.9

2.0

Oz (A -~)

2.1

2.1

2.2

2.3

2.4

Oz (A -1)

Fig. 21. (Left) Applied field effect at 10 K on the Q - magnetic satellite reflection in [DYI41Y]4164 showing the continuous transition in intensity from the (0002)- incommensurate structure reflections into added ferromagnet intensity on the (0002) structural peak. (Right) The original helical phase is not restored after application of the field until the sample is annealed.

34

J.J. R H Y N E

5

Magnetic , Satellites ?

,~

and R.W. ERWIN

[DY14Y14164 130 K

20i [DY14Y14164 130 K 16~- (0002) .~25 kOe

Ferro

H=O, "<10 kOe 81t.O C

4b

1

0i

.8

1.19

2:0 Qz (A'I)

2:1 Qz (/~-1)

Fig. 22. (Left) Field dependence of the Q- magnetic satellite on the same superlattice as in fig. 15 but at the elevated temperature of 130 K. The satellite and magnetic bilayer harmonics are observed to broaden in low fields reflecting a loss of coherence. (Right) Only at fields higher than 10 kOe is the intra-layer helimagnetic structure gradually removed by the field and the resulting ferromagnetic component appears as intensity on the (0002). higher. This corresponds to the destruction of the helix by the applied field and its conversion to ferromagnetic order. This is the same as in bulk Dy, although without the low temperature discontinuous transition. The intensity removed from the Q± helix peaks reappears as ferromagnetic intensity added to the (0002) peak and its harmonics. In contrast, at 130 K the Q - satellites are observed to first broaden increasingly with field. This corresponds to a decrease in the magnetic coherence length and at the higher fields the coherence is lost to a degree that merges the three satellite peaks into one broad peak. This initial loss in coherence occurs before an appreciable ferromagnetic moment component is developed as shown in fig. 22 where additional (000l) intensity is evident only above 10 kOe. Erwin (1987) suggested that this loss of coherence may arise from a Zeeman coupling of the applied field to the uncompensated moment, which is the vector sum of the moments in a Dy layer. It will be in general non-zero because of an incomplete helical period at the layer boundary. As shown in fig. 23, the net uncompensated ferromagnetic moment can be quite large and varies with the Dy layer thickness and also with temperature because of the variation of both the turu angle and the order parameter. At temperatures near TN it is suggested that the coupling of the external field to the helically distorted (superspiral) uncompensated moment in each layer acts like a random field and breaks the weak interlayer exchange coupling of the Dy spins. This results in independent layer helical magnetism as is reflected in a single broad magnetic satellite at the centroid of the three original zero field satellites. At low temperatures the decreased thermal spin disorder and strongly increased anisotropy

MAGNETISM IN ARTIFICIALMETALLICSUPERLATFICES

35

16

[DYI61Yg]Io0

14 12[ o

0

o

0 30 60 T(K) 90 120 150 180

Fig. 23. Uncompensatednet ferromagnetic moment resulting from the incomplete helices in the Dy layers. The coupling of the applied field to this net moment at temperaturesapproachingTN is suggested to destroy the long-range interlayer coupling. and magnetostriction interactions result in a lower ferromagnetic transition field and apparently preclude the loss of coherence before the helical order is eliminated. This explanation is also consistent with the behavior of a [DYl61Y9]too sample that has relatively less Y and a larger zero field coherence length. Due to the thinner Y layer this sample is expected to have a stronger interlayer exchange coupling, and relatively little broadening of the Q - satellite was found at 130 K (or at lower T) on application of a field. This suggests that the applied field Zeeman coupling to the net moment is insufficient to break the relatively stronger c-axis exchange coupling in this sample. The onset transition fields (< 1 kOe at 10 K) for the helical to ferromagnetic conversion were lower than in the thicker Y sample, reflecting again the larger magnetostriction and anisotropy energy densities.

6. Incommensurate magnetic periodicity - interplanar turn angles, spin slips, and their temperature dependence The fit of eq. (5) to the magnetic diffraction data for the superlattices yields values of the propagation vector t~A for the magnetic layer and a total phase shift across the bilayer, #i. From these, the layer-to-layer turn angle ~A in the Dy (or Er) layers can be determined: 180 co ~A - - ~A. (11) ~" 2

36

J.J. RHYNE and R.W. ERWlN

Similarly, an effective turn angle can be determined for the spin density wave in the non-magnetic Y or Lu that represents the turn angle to which the 4f spins would lock if present. These turn angles can be temperature dependent with the initial (high T) value prescribed by the nesting wave vector coupling parallel sheets of the Fermi surface. For example, in bulk Dy at Trq, w (Wilkinson et al. 1961) has the value 43.20 , which decreases to 26.50 just above the first-order transition to ferromagnetism at Te. The variation of w with temperature has been suggested to arise from the development of superzone gaps in the Fermi surface coupled to the order parameter (Elliott and Wedgwood 1963), or from the effects of the magnetostrictive interactions (Evenson and Liu 1969). In the superlattices (or bulk crystals) the temperature dependence of the turn angle is reflected in a shift of the Q-position of the magnetic satellites. Figure 24 shows this effect for [Dy16[Y20189 for which the satellite position (marked with the vertical "

,h,

DY161Y2o] 89 120 / Q'Mag. Satellites Ill 108/Tc= 167K

~t

96

84 r-

= 72

zILl p.~ 48

36 24 12 0 1.76 1.86 1.96 2.06 2.16 Q, (A") Fig. 24. Expanded detail of the temperature dependence of the (0002)- satellites in a [Dy16]Y20189 superlattiee illustrating the temperature dependence of the turn angle (w). As shown by the vertical line, the magnetic peaks move to lower Qz (larger w) as the temperature is increased.

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATrlCES

I

60

I

oo

o

I

37

I

o

yttrium ~'40 co k_ •

-El

20 DylLu

0

I

0

I

80 T(K)

160

Fig. 25. The zero field magnetic structures in Dy superlattices described by the temperature dependence of the interplanar turn angle w. The low temperature ferromagnetic transitions and high temperature turn angles, compared to bulk Dy, are dramatically shifted in opposite directions for Y and Lu superlattice interlayers due to epitaxial constraints. (See text.)

lines) moves to lower Q (larger w) as the temperature in increased. The turn angles derived from the scattering data are shown in fig. 25 for several Dy superlattices, and the turn angles for the c-axis modulated spin structure of the Er superlattices are shown in fig. 26. It is noted that w in the Y remains constant at about 5152 degrees/Y layer (x -- 0.31 /~-I), corresponding closely to the peak q position (Gupta and Freeman 1976, Liu, Gupta, and Sinha 1971, Child et al. 1965) in x(q). The turn angle between the Dy planes varies with temperature as in bulk Dy (Wilkinson et al. 1961) but, for Y superlattices, without the first order drop to w = 0 at 85 K which signals the onset of the ferromagnetic state. The suppression of Tc in Y superlattices and its enhancement in Lu superlattices compared to bulk values is a consequence of epitaxial clamping of the magnetostrictive modes as discussed later. In the superlattice [DY161Yg]10o the turn angle is observed to lock in to 300 below about 50 K, presumably driven by the increase in the sixfold basal plane single ion anisotropy energy. For Er superlattices, using an analysis including the effects of strain modulation, distinct turn angles were obtained for the c-axis and the basal plane moment components which also order at different temperatures. However, measurements at low scattering angles, for which strain modulation effects are negligible, produced identical values for the turn angles of both components as in bulk Er. As shown in fig. 26, the c-axis turn angle is about 500 (~-, 27r/7) per layer corresponding to the high temperature initial state of bulk Er and shows relatively little temperature dependence compared to bulk Er which is indicated by the solid

38

J.J. RHYNE and R.W. ERWIN

60

C-axis basal [Er23[Y19]

A

[En3~%]

A

•

[]

layers

~o Q)

t~ t

50

| ..... s~/1~ 417 --

12/2~

bulk Er C-axis-basal

1/2

10/21

40

• ,

0

I Q

I

I I Y layers

I

40 T(K)

I

55

8045

Fig. 26. The temperature dependence of the turn angles in the Er and Y layers for three [Er[Y] superlattices compared to bulk Er. The superlattice Er layer to is 'clamped' near the high-temperature lock-in value of bulk Er (27r/7). The basal plane to appearing at low temperature has a somewhat smaller value than the c-axis to. The to in the Y layers is near the 50* value prescribed by the electronic structure. The fractions give (two times) the commensurate lock-in state values as developed by Gibbs et al. (1985, 1986) here referenced to the magnetic unit cell.

line. The plateaus in the bulk turn angle (indexed by the fractions) correspond to stable commensurate moment configurations (a 'spin-slip' structure) as derived by Gibbs et al. (1986) from an analysis of the magnetic X-ray scattering using synchrotron radiation. The denominator of the fraction gives the number of atomic planes required for a complete period of the spin structure and thus depends on the number of slips in the structure. The effect of the slip structure is to produce a small net moment to the configuration (e.g., a complete period in the 2/7 slip structure in Er consists of 3 spins 'up' followed by 4 spins 'down' for a net moment of 1 per 7 atomic planes). This is observed in the e-axis magnetization data on the superlattices (Borchers et al. 1991a, 1991b) as shown in fig. 27 for [Er23.5[Y1911oo. The nearplateau below 18 kOe corresponds to a moment approximately 1/7 of the saturation moment as consistent with the 2/7 spin-slip configuration. The abrupt transition fields of 18 kOe and above are those required to convert the c-axis periodic moment structure to a ferromagnetic spin alignment. Moment data taken at constant applied field show inflections corresponding to the development of spin slips as illustrated in fig. 28 for [Er13.5]Y25h00. The solid curve is for field cooled, the dashed line is for zero field cooled magnetization, and the lines marked B and D represent the

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATIqCES

250 0

q-

,

~

39

,

1OK __--l--t~-~lp~_%~ 2000

e ~ - o

_

20 30

K K

tl S, 60K tq

1000

X

(1(}

,

1. . . . . . . . .

__

0 0

~ ........

10 (1

L _

~

.

.

.

.

20 0

t

_

:~O 0

,10 0

lld,er'nal Fiohl (kOu) Fig. 27. G-axis magnetization data corrected for demagnetization effects taken with a SQUID magnetometer for the superlattice [Er23.slY1911o0 at several temperatures. The near-plateau below 18 kOe reflects the net moment of approximately 1/7 saturation produced by the 2/7 spin-slip structure. The transition fields reflect the energy required to uncouple the periodic moment structure and produce ferromagnetic spin alignment along the c-axis.

D B

20.0

¢ E

15.0

o-t" 1"-, X,

0 10.0 tq

5.0

I

0.0 0.0

,I

,

,

,

20.0

400

60.0

0o.o

loo.o

Temperakure (K) Fig. 28. Field cooled (solid line) and zero-field-cooled (dashed line) isofleld magnetization data taken on the same [Er23.5 [Y19]10o superlattice. The vertical lines marked B and D mark the temperatures where the 2/7 and 3/11 spin-slip states, respectively, develop.

40

J.J. RHYNE and R.W. E R W l N

I

I

!

I

(a) 120 eTr/7 state

80

51.4 °

413

600

~.a~

I

I

zero [

I

I

I (b)

T = 40K

°

480

H 11 too2 [Er13~26]

x, = 2.45 A CoUimation

"o 360

k remanence

-

40v 50h 30~

14°h

~ 240

120

-~.6

"0.2

q -

0.2

7.0

016

(A-b

Fig. 29. Neutron diffraction scans along (117.~) for [Er131Y26 ] at 10 K with a c-axis applied field showing the shift in ~v to the 21r/7 commensurate state. (b) At 40 K the zero field w = 2 / 7 and there is no shift at 8 kOe. Higher fields induce fan states that are also observed as inflections in the SQUID magnetization data. The zero field remanence curve indicates that the helical state is reformed with slightly reduced coherence after removal of the applied field. The quantities 40h, 40v, 50h, and 30v indicate the horizontal and vertical instrumental collimation in minutes of arc before and after the monochromator and analyzer, respectively.

MAGNETISM IN ARTIFICIALMETALLICSUPERLATTICES

41

temperatures at which the 2/7 and 3/11 commensurate spin-slip states are assumed to develop. The neutron diffraction results on the [Er13.slY2511oo superlattice clearly demonstrate the lock in effect to the 2/7 stable commensurate spin-slip state under the application of an applied field. At low temperatures the turn angle in zero field was found to be 50.3 °. As shown in the left portion of fig. 29(a) on application of a c-axis field (H = 7 kOe internal), the turn angle shifts to the commensurate value w = 27r/7 = 51.40 with no loss in the long-range magnetic coherence. The relatively poor resolution of the principal and superlattice peaks is a consequence of the outof-scattering plane (112.~) scan. At higher fields no further shift in q is observed, indicating that the 27r/7 state is the stable configuration. At 40 K (fig. 29b, left), the zero field state already has the stable w = 27r/7 value and is thus not shifted by application of an 8 kOe field. However, larger fields strongly shift the q-centroid and produce broadening, indicating a loss of coherence. This shift corresponds to the development of linear fan states with an average w ~ 35 °, which is significantly lower than any turn angle in bulk Er. The fan states also appear as higher field pseudo-plateaus in the SQUID magnetization data (Borchers 1991a) for temperatures of 20 K and higher. A similar remanent fan state shifted to longer wavelength is seen in bulk Ho (Koehler et al. 1967), presumably due to the dependence of the exchange interaction on lattice parameter as discussed in section 7.

7. Magnetoelasticity in rare earth superlattices and films and the suppression of ferromagnetism The Hamiltonian used to describe heavy rare earths is of the general form H = Hex + Hme + Hel + Her,

(12)

where the terms are exchange, magnetoelastic, elastic, and crystal field components. Magnetoelastic energies refer to the general dependence of magnetic energies upon the spacing between atoms, so that the magnetoelastic term contains the explicit dependence of the exchange and crystal field energies on atomic spacing. It has usually been found sufficient to consider only the first derivatives of the exchange and crystal field energies in the magnetoelastic term; however, the second-order derivatives must be included in the elastic energy term. These second-order derivatives of the exchange and crystal field energies could possibly be linked to the anomalous behavior of the measured elastic constants in the bulk rare earths. The most important effect of the crystal field term in the heavy rare earths is to determine the easy directions for the magnetic moments. At low temperatures the crystal field anisotropy energies can contribute to the stabilization of a ferromagnetic phase. Although there is a continuing interest in looking for band structure effects in superlattices, the experimental data suggest that it is the elastic constraints on rare earth superlattices and fihns grown by MBE that strongly perturb their magnetic structure compared to the bulk materials, as shown by studies of [Dy]Y], [ErIY ], [DylLu], and [DylSc] superlattices and Dy and Er thin films by neutron scattering

42

J.J. RHYNE and R.W. ERWIN

and SQUID magnetometry. The strains induced by epitaxy could in fact perturb the band structure. These studies have provided new insight into the interplay of magnetoelastic and exchange interactions in rare earths. The most striking result of the epitaxial clamping to the substrate and intervening non-magnetic layers is the suppression or enhancement of the ferromagnetic transitions found in bulk Dy and Er. The helical and linear spin density wave orderings are also significantly perturbed. For example, in the [Dy]Y] superlattices the temperature dependence of the interlayer turn angle is weakened as the elastic clamping is increased, while in [DylLu ] and [DylSc] superlattices the initial turn angle near the Nrel temperature is much smaller than in bulk Dy. This temperature dependence is completely suppressed in the Er superlattices where the turn angle is typically fixed at the commensurate lock-in value 21r/7. In the thin films, however, the turn angles are closer to the bulk values. There is also a 50% reduction of the basal plane ordering temperatures compared to bulk Er. These dramatic effects are also evident in quantitative calculations of the magnetoelastic energies and equilibrium spin configurations. In the bulk elements the first-order ferromagnetic phase transition is driven by a reduction in the magnetoelastic energy associated with the ferromagnetic state compared to the helical (or c-axis modulated) configuration (Cooper 1967, 1968, Evenson and Liu 1969). In Dy for example, the discontinuity in the magnetoelastic energy at the ferromagnetic transition is 3.6 K/atom as calculated from the bulk elastic constants and measured anomalous strains (Rosen and Klimer 1970 and Erwin et al. 1989). The total magnetoelastic energy reaches nearly - 6 K/atom just below the ferromagnetic transition. It is presumed that this energy is just sufficient to overcome the part of the exchange energy that is independent of atomic spacing. Elliot (1972) has also calculated the driving energy for ferromagnetism using the measured critical fields. That formalism assumes that the driving energy becomes negligible, compared to the exchange energy near the Nrel transition. While this is certainly true for the single-ion part of the driving energy, the two-ion exchange part will have approximately the same temperature dependence as the exchange, so that it is not clear that it can be neglected. Proceeding in this way, Elliot obtains good agreement between the calculated single-ion driving energy and the value extracted from the critical field measurements of about 1 K/atom at the ferromagnetic transition. The magnetoelastic energy can arise from two principal strain modes: (1) an anisotropic 7-mode strain of 0.5% at Te (Rhyne and Legvold 1965, Callen 1965, 1968) that corresponds to an orthorhombic symmetry distortion of the basal plane and that is effectively 'damped' in the helical state and (2) an c~-mode c-axis expansion of 0.25% and an isotropic basal plane dilation (both partially clamped in the helical state). In bulk Er the ferromagnetic alignment is along the c-axis so that there is no "r-mode distortion. The magnetoelastic driving energy at Te (20 K) is found to be 2.3 K/atom (Rosen and Klimer 1970). There is also a decrease of the magnetoelastic energy associated with the ordering of the basal plane spin components in Er below 55 K. In the superlattices there is further clamping by the presence of the non-magnetostrictive epitaxial Y layers attached on either side of the magnetic layers. In Dy supeflattices, this clamping strongly inhibits the 7-mode strain and is relatively weaker

MAGNETISM IN ARTIFICIAL METALLIC SUPERLA'ITICES

43

for the a-modes. The a-mode energy (calculated to be ~ 0.6 K/atom at 85 K by Erwin et al. 1987) is insufficient alone to drive the transition, and the clamping of the 7-mode strain forces the Dy in the superlattice to remain helical down to low T. In Er superlattices and films, the situation is more complicated, but again the absence of the conical ferromagnetic state and the weaker temperature dependencies of the turn angles compared to bulk Er are also ascribed to magnetostrictive effects as will be discussed in the next section. The effects of the perturbed magnetoelastic energies in the superlattice can be seen from the magnetization data of fig. 27 in which an external field of 18 kOe is required to induce ferromagnetism at 10 K, a regime in which bulk Er is intrinsically ferromagnetic.

7.1. Theory of magnetoelasticity in superlattices and films This section focuses on the role of magnetoelastic energetics in determining the magnetic structure of rare earth superlattices and films. While the ideas presented here are applicable to all magnetic thin film materials, the giant magnetostriction of the rare earths emphasizes the magnetoelastic effects. Dy and Er superlattices and films, which have received the most attention with regards to magnetoelastic effects, will be discussed here. The magnetoelastic free energy in Cartesian coordinates can be expressed as 1

fine = "~ E C#u~'u#£uu -- E K#~.##. ,u,u #

(13)

The magnetoelastic coefficients, Ku, appear in terms linear in the strains as they arise from derivatives of the exchange and single-ion energies with respect to the strains. The equilibrium strains satisfy the equation

E cu~-=Ku

(14)

and the equilibrium magnetoelastic free energy is

-

1

fme = - - - ~ E C , u u g # # ' u v #,u

= -- 21 E Ku'(uu"

(15)

#

When a magnetic material is grown with a non-magnetic material in a fihn or multilayer additional elastic terms are added to the free energy, 1

re1 = "~r,~ E 5uu(?uu - ~'°u)(2"~ - ?'0u),

(16)

#u

where r~ represents the ratio of non-magnetic to magnetic material and a labels the growth direction. The ~'o~ are the strains in the free non-magnetic material measured with respect to the non-magnetic strains in the free magnetic material. In the case

44

J.J. RHYNE and R.W. ERWIN

that the strains in the growth plane are coherent (guu ~ euu for/.t # c~) the additional elastic energy is 1 iel ---- ~ 7"or E

(C,av

- -

-

-

G).

(17)

This adds terms to free that are both quadratic and linear in the strains so that the effective elastic coefficients for #, v # c~ are

(18) and the effective magnetoelastic coefficients for # # c~ are K . = K . + r~ ~

(19)

ro~(~.v - ~ . o 6 ~ / ~ ) .

7.2. Magnetoelastic energies in Dy superlattices and films A first attempt to treat magnetoelastic effects in multilayers and fihns assumes that the basic exchange interactions and anisotropy energies can be transferred directly from the bulk magnetic materials. This assumption is likely to be incorrect at the interfaces, yet it provides reasonable explanations for many of the experimental results. The magnetoelastic coefficients of the bulk materials are to be applied in eqs (13)(14). Care must be taken to correct for the magnetic structure dependence of the coefficients, since the structures are typically different in the layered materials compared to the bulk. For example, using eq. (13) and the symmetry arguments for the spin correlation functions (Callen and Callen 1965), the magnetoelastic coefficients in the basal plane helimagnetic structure of Dy at T = 0 K are i

Kx,y(T = 0) = +0.686~,0 - 0.06(3 + cos w)/4

Kz(T = 0) = 1.4 cos w

(103 K/atom),

(103 K/atom),

(20) (21)

where 6,o,0 selects the large constant term 0.68 only if the structure is ferromagnetic = 0). Consider the case that the layered rare earth material is grown along the hcp eaxis. If the elastic constants are equal in the magnetic and non-magnetic material then eq. (13) for the equilibrium anomalous strains simplifies to ~'xx + gyy = ( g b + 2r'zCb~'0x,y)/((1 "[- rz)Cb),

(22)

gxx - "~yy= (Kx - Ky)/((1 + rz)(C11 -- e12)),

(23)

Fzz = (Kz -- Cl3('~xx + ~yy))/c33,

(24)

where Cb = Cll + c12 - 2c23/C33 and Kb = Kx + Ky - 2KzC13/c33.

MAGNETISM IN ARTIFICIALMETALLICSUPERLATFICES

45

Assuming for the moment that these expressions for the equilibrium strains are correct, then the magnetoelastic driving energy for the ferromagnetic transition, A = fm~(W = 0 °) - fme(W) can be calculated. If the low temperature helimagnetic state has w = 30 °, then Az = -1.25 - (3.95 - 91.0 rze'0x,y)/(1 + rz)

(K/atom).

(25)

This magnetoelastic driving energy is plotted in fig. 30 for rz ranging from zero to infinity as a function of ~'0x,y. It is believed that the exchange energy barrier to ferromagnetism is equal to the discontinuity in magnetoelastic energy found at Tc (there is about 5 K of temperature hysteresis in bulk Dy at Tc), which is 3.6 K/atom at 85 K as shown in fig. 31. This number may become slightly larger at 0 K. Figure 30 suggests qualitatively that ferromagnetism is obtained for ~"~ 1.5% (as applies for Dy on yttrium) in the weak clamping limit, rz < 1, which should apply in sufficiently thick films. Note also that the driving energy for weak clamping becomes stronger than the bulk driving energy when ~" x< -4.4%. For example, the lattice mismatch of Dy on Lu is only - 2 . 5 % at room temperature. The data for a 4000 ,~ Dy fihn shown in fig. 32 are quite similar to those obtained for the bulk material (Wilkinson et al. 1961). The helimagnetic structure collapses in a first-order phase transition to a ferromagnetic state just below Tc = 80 K (compared to 85 K for the bulk). However, the temperature hysteresis for this transition is about 15 K (compared to 5 K for bulk Dy), and on cooling the transition may require as long as one hour for completion. The temperature dependence of the turn angle is identical to that of the bulk except in a 10 K wide region near the Curie temperature, where the angle is slightly larger than in the bulk. Although all of the magnetic intensity collapses onto the nuclear (0002) Bragg peak below 80 K, the width of this magnetic scattering is not resolution limited. The peak shape is no longer Gaussian (as is the resolution function), and the width is estimated to correspond to a magnetic correlation range of 200 ~. No uniform -y-mode magnetostrictive distortion was observed. In the ferromagnetic state when a saturating field of 25 kOe is applied along the easy a-axis direction, the (0002) Bragg peak sharpens to a resolution limited width, consistent with the formation of a single domain. The resulting anomalous b-axis strain from just above Tc in zero field to below Tc with the field applied was found to be only about 1/3 of the discontinuity at Tc in bulk Dy (Rhyne and Legvold 1965). This is a peculiar result since eq. (21) then suggests that the effective clamping parameter, rz, is 2, and the magnetoelastic driving energy should be considerably reduced as in fig. 31. Furthermore it was found that the Nb buffer layer undergoes no anomalous strains at all. This indicates that inelastic processes are involved, probably at the interface between the rare earths and the Nb. It can be concluded that the results for the Dy film are not fully explained, and will require a more careful investigation of all of the relevant strains. The diffraction from a superlattice consisting of bilayers of 16 atomic planes of Dy alternated with 9 atomic planes of Y is compared to the film in fig., 32b. Although the initial ordering temperature, TN, is within 7 K of the

46

J.J. RHYNE and R.W. ERWIN

2O

0 _x¢

15

o

10

i -0

i r r l l l . , , t

. . . .

Dy (a)

Er (b)

5

/

bY_ 0

(2 o

o ~

r=O

-5

0 -10 -

0

1

-1

0

1

sfrain ¢ (10 -2) Fig. 30. Critical fields for Dy (a) and for Er (b) c-axis superlattices calculated as a function of the lattice mismatch e, based on a perfect epitaxy model. The experimental critical field values are shown by the triangles for comparison. The parameter r represents the ratio of non-magnetic to magnetic material. The model predicts that the growth of Er on a substrate with a smaller basal plane lattice parameter than Er would produce a ferromagnetic transition temperature higher than in the bulk.

o

I

I

I J

I

I f

o co - 4 -6 ©

( b ) I~r ,

0

100 W

(K)

i

,

50

7 - - - - - -

100

Fig. 31. Magnetoelastie energy as a function of temperature for (a) bulk Dy and (b) bulk Er.

bulk and film values (185 K), there is evidently no collapse to the ferromagnetic state down to 4 K. Note, however, that additional weak magnetic peaks develop near the (0002) nuclear peak at low temperature. These peaks are separated by 2~r/L (where L is the bilayer thickness) from the helimagnetic peaks, so that they represent the development of local ferromagnetic order within the helimagnetic framework. As in the case of the Dy film, this low temperature ferromagnetic-like state is extremely time dependent and temperature hysteretic. In fact this feature was not found in the

MAGNETISM IN ARTIFICIAL METALLIC SUPERLAqTICES

100

I

I

I

I

I

I

(o) [Dy,olVg]

Dy film

47

(b)

80 .~

60

4K

qO

-E 4.0

~

I

72K

20-

lsoK,AI,, A

0 2.0

2.2

2.4

L,- T

2.0

2,2

2.4

Qz Fig. 32. Diffraction scans along (000~) for a Dy film (a) and for a [DylY] superlattice (b). The film has a ferromagnetic transition where the magnetic intensity collapses onto the nuclear Bragg peak, whereas the multilayer does not. The helimagnetic intensity is shaded.

lO- ~ ) 5

~E

o

[Erz3IYI9 ]

-

(d) ~

5

0

"~

0

__i

[DY16~Yg] ''''''' 20 40 20 0 Infernal Field (kOe) ,

,

i

I

r

,

t

I

Fig. 33. Magnetization plotted versus internal field at 10 K for two [DyIY] superlattices (a, b), and for two [Er]Y] superlattices (c, d). The critical field for ferromagnetism is determined qualitatively as the point where the magnetization first increases sharply toward saturation.

other D y - Y superlattices where the ratio of Dy to Y was smaller. The energy difference between the exchange barrier and the magnetoelastic driving energy for growth on Y (e ,~ 1.5%) is calculated to range from 1.2 K/atom at rz ~ 1 to 2.0 K/atom at rz = 2 to 3.8 K/atom at rz = ~ . These differences correspond to critical fields of 2.0, 3.3, and 6.6 kOe on 10#B. Magnetization measurements for two of the DylY superlattices are shown in fig. 33. These data were obtained from the diffraction intensity of the (0002) Bragg peak as a function of the magnetic field applied

48

J.J. RHYNE and R.W. ERWIN

along the easy a-axis direction. In agreement with the above calculations, the low temperature critical fields are below 5 kOe. The calculation of critical fields is complicated by the contribution of the sixfold basal plane anisotropy. Also the effective clamping parameters and non-magnetic strains, ~'0, may depend sensitively on the growth conditions. There are not yet data for the anomalous growth plane strains for Dy multilayers where rz is expected to be of order 1 or greater. From clamping considerations the anomalous strain along the c-axis should be ezz = Kz/c33. The measured c-axis anomalous strain in Dy superlattices (Erwin et al. 1987) is approximately 44- 1 x 10 -3 at low temperatures, while the calculated value is 6 x 10 -3 for w = 30 °. This is not a serious discrepancy considering the assumptions concerning the multilayer structure that are required to extract the anomalous strains from the neutron diffraction data and the neglect of interface effects on the magnetoelastic coupling coefficients. The additional forced magnetostriction in a saturating magnetic field was measured to be 1.0 x 10 -3 (Erwin et al. 1987) for [DY151Y14], and the calculated value is also 1.0 x 10 -3. The case of Dy-Lu superlattices is significantly different since there is an extreme enhancement of the Curie temperature from 85 K to the range 140-160 K. Calculations attempting to model these magnetic structures are difficult because the temperature dependence of the different terms in the Hamiltonian must be included. A reasonable model for bulk Dy can be constructed based on the wave-vector dependence of the exchange energy extracted from inelastic neutron scattering measurements (Nicklow 1971, Nicklow et al. 1971). The temperature dependence of this exchange function can be interpreted as arising from its strain derivatives. Using a simple mean-field theory for the temperature dependence of the magnetic moment, one can solve for the temperature dependence of the magnetic structure and the anomalous strains using appropriate values for the magnetoelastic constants. Unfortunately, when the derived exchange and magnetoelastic constants are applied to the superlattice model, the extreme enhancement of the Curie temperature is not obtained. Typically what is found is that the Dy turn angle is considerably reduced but the Curie temperature may only rise 10-20 K. The reason for this is that the added terms in the Hamiltonian of the superlattice due to epitaxial strain give a direct perturbation of the exchange energy, just as the corresponding terms in bulk Dy can account for the temperature dependence of the turn angle. The bare exchange and its strain derivatives have the same dependence on temperature through the renormalization of the magnetic moment. A strong enhancement of the Curie temperature cannot be obtained from this theory, since it has been assumed that a large fraction of the driving energy arises from the single-ion gamma-mode strains which renormalize so strongly with temperature that their contribution is negligible compared to the exchange at high temperatures. The failure of this theory is interesting since it suggests that some other physics related to the superlattice structure might be responsible for the enhancement of Te [Beach et al. 1993]. 8. Coherent magnetic moments and disorder at interfaces The coherent 4f atomic magnetic moment in magnetic layers of the Dy and Er superlattices can be calculated from the observed magnetic scattering intensity relative

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATTICES

49

12

10

Function ( J = 15/2)

••i•ouin

8

zk

2

00

[DY'6 IY2°]8' I Oy Coherent q~ [DylslY,,]~ tL~yerMoment

3b

60

90

m\ m \

120

180

T(K) Fig. 34. Temperaturedependence of the coherent Dy layer moment in [DYlrlY20189 and [DYlsIY14164 superlattices compared to a Brillouin function. Also shown is the total integrated magnetic intensity. (See text.) to the nuclear intensities. The resulting temperature dependence of the moment is shown in fig. 34 for Dy superlattices compared to a Brillouin curve (J = ~ for Dy) and is given in fig. 35 for Er superlattices compared to the experimental bulk c-axis and basal plane moments. For the Er superlattices the data analysis was based on a simple parameterization of the intermediate temperature spin state. Note that the low-T value of the c-axis moment is 8#B as in bulk Er (total free ion moment = 9/zB), and with a rapid fall off at increasing temperatures. The 5 K basal plane moment (helical) is about 3.5#B, which is reduced from the bulk value of 4#B. For Dy, the calculated atomic moments for both superlattices fall below the Brillouin curve for the 10#B free ion moment of Dy. Figure 34 also shows the total overall magnetic intensity obtained from a difference pattern between scans taken at temperature (T) minus that for T > TN. This difference intensity, which should include all the magnetic scattering (coherent and incoherent), does accurately follow the Brillouin curve. These results imply some degree of disordered moment, presumably in the interface layers, which does not contribute to the coherent moment. Additional evidence for this is found in a temperature-dependent weak broad diffuse scattering that underlies the [0002] nuclear peak. This is shown by the enlarged plot in fig. 36 for a [DY15[Y14164 superlattice at 10 K. The shaded area is a representation of the broad incoherent scattering underlying the (0002) diffraction peak at Qz = 2.24 ,~-1. The broad distribution reflects ferromagnetic moment correlations of a range of order

50

J.J. RHYNE and R.W. ERWIN

l g

C-axis

•

[Er23~{t9] zx

•

[grl3lYz6]

3

basal

[Er321Y21] 0

%

% % ~

_

a

bulk

•

% %

% %

% %

~ 2 3 1 1 9 ]

{D bulk g r C-axis - -

o

A

-

2

0

%

\ k

o .1"4 _

{D

•

\\ \

•

G}

\ \

II 0

¢

\\

t

I

I

I

40

0

30

T(K) Fig. 35. The c-axis and basal plane moment components for [Er32]Y21], [Er23[Y19], and [Eq31Y26 ] superlattices (TN = 78.0, 78.5, and 72.5 4- 1 K, respectively) compared to bulk Er (TN = 84 K) given by the continuous lines. The ordering temperature for the basal plane components is about half the value for bulk Er. The cartoons at the top of the figure represent the ordering progression of the c-axis

moment componentas the temperature is lowered. 10 ~ . The reduction of the coherent moment in the interface layers has also been found in [Gd-Y] superlattices by Majkrzak et al. (1986) and has been calculated directly from magnetic diffraction data on [GdIY] using polarized synchrotron radiation data by Vettier et al. (1986). The best fit model to the diffraction data as shown in fig. 37 assumes the full Gd moment (5.6#B for T = 150 K) at the center of the Gd layer with a reduction in the planes near the interface. Interface disorder effects are presumably responsible for the decreased coherent moments in the Er superlattices also.

9. Residual m o m e n t effects in superlattices The effects of anisotropy, magnetostrictive, and epitaxial strains are also evident in the remanent magnetic state examined in a series of [DyIY] superlattices. Ill diffraction studies fields were applied along a basal plane direction that induced a partial or complete collapse of the helical state in favor of ferromagnetic order. Between each isothermal field application, the field was reduced to zero and the

MAGNETISM IN ARTIFICIAL METALLIC SUPERLAI"FICES

51

12 IDylslY14164 10K 10

:>,

8 6

42 jiJ 1.80

1.95

2.10

2.25

2,40

2.55

2.70

ez(~, 1) Fig. 36. Enlarged plot of the (000/) scattering in the [DY151Y14164superlattice showing the broad magnetic peak underlying the pattern at the (0002) position. This peak, reproduced by the shaded area, reflects short-range ferromagnetic correlation of some of the moments, presumably those in the interface layers. This explains the reduced coherent layer moment (fig. 34).

.:L n-"

Gd21Y21 T=ISOK

6.0

g.. 5°t '1 ~

4.(?

~ s°l ~ 2.0 ~

1.0

W

"

20

i

r

2O O LAYER NUMBER

Fig. 37. Models for the spatial magnetic moment modulation in a [Gd211Y21] superlattice. The dashed curve follows the chemical modulation, and the solid curve, which gives the best fit to the data, represents the full 6.0/~B moment (T = 150 K) at the center of the Gd layer with a smooth decrease in the interface planes (after Vettier et al. 1986).

52

J.J. RHYNE and R.W. ERWIN

after 18 kOe

[DYlsIY9 ]100 I0 K Residual Intensity Nuc,. Peaks 1

2'°1

after IOkOe /'~

after

[ /

/

~

t

).,Je~Ul

H=O

l

/1~/

"l

~

36

J~ l~"

H=O

/'~

30

" II

24

I ~

after 18 kOe after 3kOe

after i IOkOe

.,~

~

LU 18"

~ 120

12

lay,. Iv. ],oo 0 rL~/"°'~ ~ ..,~t,~:

IOK

0 tgo 2.oo 2.1o

2.10 2.20 2.30 2.40 Oz(~," )

Residual Intensity Q- Mog Satellites

o z (~-')

Fig. 38. (Left) Excess ferromagnetic intensity remains on the (0002) nuclear peaks following application of the field, which indicates the strong metastability of the induced ferromagnetic state at 10 K for the superlattice with only 9 Y planes (cf. fig. 39 for the 20 Y plane sample). (Right) Residual intensity for H = 0 at the Q - satellite positions in the [DY161Yg]10o superlattice after applying each of the fields shown.

warmed to 80 K

r

I

after / ' ~ 60kOe ~" I~

[DYI61¥20]S9 Residual Intensity NucLPeake

after

;5 kOe I/" fl

H-O

Y

~- 90

warmed1/q80 K

~1

]

~/~

| ~ 1 /~T

1

[

II

~__~ ~

after ~ / / ' /

IX

after

240

H-O

/

~'

~-,80

1,84 1.94 2.04 Oz(~'-I)

2) ~

~o 0 2.12 2.22 2.32 Oz (~")

otter 60 kOe

.

. 'joy

[DYL61Y20]SS iOK Residual Intensity Q" Satellites

•4

Fig. 39. (Left) Residual intensity for H = 0 at the (0002) peak position in the [DY16]Y20]s9 superlattice that shows that all the ferromagnetic intensity is lost following removal of the applied field. (Right) Except for 3 kOe, the Q - satellites are reformed with significantly broadened peaks, indicating a partial loss of coherence in the remanent state. The coherence is recovered on warming, for example to 80 K.

MAGNETISM IN ARTIFICIAL METALLIC SUPERLA'ITICES i

after

after

~ 40 koe

- after

~ /I

=

I zskoe

,~ | ,~

t

/

~

53

/

. oI ri , zt -

| = ~ 1,90

,

Residual Intensity 0 - Satellites

2.00 2.10 Qz

Fig. 40. Residual intensity at 130 K on the Q - satellites in the [DYlfIYg]89 superlattice following magnetization at the fields shown. The initial state is essentially recovered at high temperatures (e.g., 130 K = 0.79Tc) in all superlattices studied.

residual magnetic intensities at both the Q+ satellite position and at the (0002) ferromagnetic peak were measured. At 10 K in a [Dy 161Y9 ] 100 sample, the remanent state after magnetization showed little tendency to restore the initial zero field helical state (except for magnetizing fields less than about 3 kOe) as confirmed by the weak intensity reappearing at the magnetic sublattice positions in fig. 38 (right), and by the large ferromagnetic intensity remaining on the (0002) peaks in fig. 38 (left). After the field was removed the spins remain metastably locked into the ferromagnetic configuration until the sample temperature was warmed above 100 K in zero field. In a thicker Y sample [Dyl6lY20]89 with lower anisotropy energy density, essentially all the ferromagnetic intensity was lost from the (0002) peak when H was reduced to 0 (see fig. 39 (left)). However, as illustrated in fig. 39 (right), the intrinsic helimagnetic state was only partially reformed, and in particular the remanent state Q peaks are significantly broadened suggestive of a partial loss of interlayer coherence following the magnetization. The total wave vector integrated magnetic intensity in the Q+ remanent peaks is close to that found for the initial H = 0 state. This result is consistent with the disappearance of the ferromagnetic intensity on the (0002) positions in the remanent state. The original coherence was restored by warming the sample as shown by the scan at 80 K. At 130 K where the anisotropy and magnetostrictive energies have become negligible, the original zero field patterns were essentially recovered in all samples for all fields applied up to 40 kOe as illustrated in fig. 40 for the remanent state in a [DY161Y9189 superlattice at 130 K following magnetizations in the fields given. Er superlattices also show the effect of inhibited low temperature reformation of the intrinsic periodic moment state after magnetization. For [Er13.slY251100 in fig. 29b (left) the zero field remanent state following application of a 27 kOe field at 40 K

54

J.J. RHYNE and R.W. ERWIN

(~, 0.5TN) recovers the zero-field-cooled turn angle but the range of coherence is appreciably reduced. 10. Summary Artificial metallic superlattices of the heavy lanthanide elements with yttrium have been shown to exhibit incommensurate (except Gd) periodic magnetic order that is coherent over several bilayer periods. The features of the ordering, including the phase and chirality coherence observed in [Dy[Y] systems, suggest that the order is propagated via a spin density wave in the Y conduction bands stabilized by the 4f spins. Superlattices with the S-state ion Gd behave similarly, except that the interlayer coupling is only collinear ferromagnetic or antiferromagnetic. The existence of the long-range propagating order is found to be independent of the direction of the net moment component (viz., Dy vs. Er) as long as the stacking direction of the superlattice is parallel to the propagation vector of the magnetic structure (e*). In the [Dy[Y] superlattices, the magnetic coherence length is found to be proportional to the reciprocal thickness of the intervening Y layer. However, [DylY] superlattices produced with the growth axis in a basal plane direction are observed to have no long-range interlayer coupling consistent with the large anisotropy in the conduction electron generalized susceptibility, which leads to a large anisotropy in the real space exchange coupling range. Gd-Y and Dy-Lu superlattices show planar ferromagnetic and antiferromagnetic coupling between layers in both c-axis and b-axis growth direction superlattices, where for the latter case the coupling is aided by dipole-dipole energies. The transition from periodic moment states to ferromagnetism observed in bulk samples of Dy and Er is completely suppressed in the Y-based superlattices due to clamping of the magnetoelastic strains. An enhancement in Tc is observed in Lu-based superlattices also arising from stain considerations. In the Dy-Y superlattices, the application of an external field effects a gradual transition to an aligned ferromagnetic state. This transition is preceded at temperatures approaching TN (in samples with relatively thick Y layers) by a loss in interlayer coupling before the helical ordering is broken. The extensive array of interlayer coupling and phase transition phenomena discovered in the superlattices and the disparate nature of many of these phenomena compared to the bulk metals, have provided added insight into some of the fundamental magnetic interactions in the rare earth metals. It is also clear that many facets remain unexplained and are motivation for future theoretical and experimental investigations. For example, the effect of forming a superlattice structure on the electronic band structure and on the detailed forms of x(q) and the exchange interaction, plus the dependence of magnetic coherence range on the non-magnetic interlayer thickness are ripe for further study. A more detailed analysis of the effect of interface strains and dislocation densities on the elastic, magnetoelastic, and exchange energies and their interplay may be expected to lead to a better description of the modification of ferromagnetic transition temperatures and associated transition fields.

MAGNETISM IN ARTIFICIAL METALLIC SUPERLATYICES

55

In the experimental arena, the formation and properties of superlattices with light rare earths are unexplored. These systems can provide information on the effect of epitaxy between hcp metals and the double hexagonal close packed (dhcp) and Sm crystal structures, and the resulting perturbation of the complex multi-Q states in Nd and the exchange-driven ordering of the single ground state in Pr. The growth of superlattices with non-constant layer repeat sequences and of superlattices with multiple magnetic metals (e.g., the Dy-Gd system of Majkrzak 1988) promise to produce new unique complex ordering phenomena. These and other superlattice investigations not yet envisioned can be expected to further enhance our basic understanding of the fascinating realm of rare earth magnetism.

11. Acknowledgements The authors particularly want to thank their long-term collaborators in the study of magnetic superlattices, Myron Salamon, Julie Borchers, and C. Pete Flynn and their students who worked on these problems, Jack Cunningham, Roy Du, Frank Tsui, Shantanu Sinha, Bob Beech, Tony Matheny, and B. Everitt. In addition, the authors have particularly benefited from discussions about magnetic superlattices with Chuck Majkrzak, J. Kwo, Denis McWhan, and Christian Vettier among others, and with Alan Mackintosh, Jens Jensen, and AI Overhauser concerning spin density waves.

References Beach, R.S., J.A. Borchers, R.W. Erwin, C.E Flynn, A. Matheny, J.J. Rhyne, and M.B. Salamon, 1992, J. Magn. Magn. Mater. 104, 1915. Beach, R.S., J.A. Borchers, A. Matheny, R.W. Erwin, M.B. Salamon, B. Everitt, K. Pettit, J.J. Rhyne, and C.P. Flynn, 1993, Phys. Rev. Lett. 70, 3502. Bohr, J., Doon Gibbs, J.D. Axe, D.E. Moncton, K.L. D'Amico, C.E Majkrzak, J. Kwo, M. Hong, C.L. Chien, and J. Jensen, 1989, Physica B 159, 93. Borchers, J., Shantanu Sinha, M.B. Salamon, R. Du, C.P. Flynn, J.J. Rhyne, and R.W. Erwin, 1987, J. Appl. Phys. 61, 4043. Borchers, J.A., G. Nieuwenhuys, M.B. Salamon, C.P. Flynn, R. Du, R.W. Erwin, and J.J. Rhyne, 1989a, J. Phys. (Paris), Colloque C8, 49, 1685. Borchers, J.A., 1989b, PhD Thesis, University of Illinois. Borchers, J.A., M.B. Salamon, R.W. Erwin, J.J. Rhyne, R.R. Du, and C.P. Flynn, 1991a, Phys. Rev. B 43 3123.

Borchers, J.A., M.B. Salamon, R.W. Erwin, J.J. Rhyne, G.J. Nieuwenhuys, R.R. Du, C.E Flynn, and R.S. Beach, 1991b, Phys. Rev. B 44 11814. Cable, J.W., E.O. Wollan, W.C. Koehler, and M.K. Wilkinson, 1965, Phys. Rev. 140 A1896. Callen, E., and H.B. Callen, 1965, Phys. Rev. 139, A455. Callen, E., 1968, J. Appl. Phys. 39, 519. Child, ILR., W.C. Koehler, E.O. Wollan, and J.W. Cable, 1965, Phys. Rev. 138, A1655. Child, H.R., and W.C. Koehler, 1968, Phys. Rev. B 174, 562. Cooper, B.R., 1967, Phys. Rev. Lett. 19, 900. Cooper, B.R., 1968, Phys. Rev. 169, 281. Coqblin, B., 1977, The Electronic Structure of Rare Earth Metals and Alloys: The Magnetic Heavy Rare Earths (Academic Press, London). Durbin, S.M., J.E. Cunningham, and C.P. Flynn, 1982, J. Phys. F 12, L75. Dumesnil, K, C. Dufour, M. Vergnat, G. Marchal, Ph. Mangin, M. Hennion, W.T. Lee,

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