Magnetoelastic stresses in rare-earth thin films and superlattices

We report on the study of the magnetoelastic behavior of some rare-earth based thin films and superlattices (SL`s). Magnetoelastic stress (MS) measurements (by using a cantilever capacitive technique) within a wide range of temperatures (10-300 K) and magnetic fields (up to 12 T) have been performed...

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Datum:	2001
Hauptverfasser:	Arnaudas, J.I., Ciria, M., de la Fuente, C., Benito, L., del Moral, A., Ward, R.C.C., Wells, M.R., Dufour, C., Dumesnil, K., Mougin, A.
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Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2001
Schriftenreihe:	Физика низких температур
Schlagworte:	Низкотемпеpатуpная магнитостpикция магнетиков и свеpхпpоводников
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spelling	irk-123456789-1300112018-02-05T03:02:34Z Magnetoelastic stresses in rare-earth thin films and superlattices Arnaudas, J.I. Ciria, M. de la Fuente, C. Benito, L. del Moral, A. Ward, R.C.C. Wells, M.R. Dufour, C. Dumesnil, K. Mougin, A. Низкотемпеpатуpная магнитостpикция магнетиков и свеpхпpоводников We report on the study of the magnetoelastic behavior of some rare-earth based thin films and superlattices (SL`s). Magnetoelastic stress (MS) measurements (by using a cantilever capacitive technique) within a wide range of temperatures (10-300 K) and magnetic fields (up to 12 T) have been performed. We derive expressions relating the cantilever curvatures and the magnetoelastic stresses in anisotropic thin films and SL`s (for cubic symmetry) deposited onto crystalline substrates. The magnetoelastic energy associated to the interfaces and the epitaxial stress dependence of the volume MS has been investigated by studying the basal plane MS in Hon/Lu₁₅ and in Ho₁₀/Ym SL`s: we obtain interface MS even higher than the volume ones and the effect in bulk`s MS of the epitaxial strain is large. In Dy/Y and Er/Lu SL's we also deduce the MS contributions but, for Er/Lu, incomplete saturation leads to inconclusive results. Although the latter case also happens in Ho/Tm SL`s, MS clearly shows anisotropy competition. In TbFe₂ (t) / YFe₂(1000 Å) (300 Å < t < 1300 Å) epitaxial bilayers, we determine all the MS allowed by the symmetry and show that epitaxial stress strongly modifies the tetragonal MS. The thermal dependence of MS parameters is also analysed. 2001 Article Magnetoelastic stresses in rare-earth thin films and superlattices / J.I. Arnaudas, M. Ciria, C. de la Fuente, L. Benito, A. del Moral, R.C.C. Ward, M.R. Wells, C. Dufour, K. Dumesnil, A. Mougin // Физика низких температур. — 2001. — Т. 27, № 4. — С. 342-363. — Бібліогр.: 54 назв. — англ. 0132-6414 PACS: 75.70., 75.80.+q http://dspace.nbuv.gov.ua/handle/123456789/130011 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Низкотемпеpатуpная магнитостpикция магнетиков и свеpхпpоводников Низкотемпеpатуpная магнитостpикция магнетиков и свеpхпpоводников
spellingShingle	Низкотемпеpатуpная магнитостpикция магнетиков и свеpхпpоводников Низкотемпеpатуpная магнитостpикция магнетиков и свеpхпpоводников Arnaudas, J.I. Ciria, M. de la Fuente, C. Benito, L. del Moral, A. Ward, R.C.C. Wells, M.R. Dufour, C. Dumesnil, K. Mougin, A. Magnetoelastic stresses in rare-earth thin films and superlattices Физика низких температур
description	We report on the study of the magnetoelastic behavior of some rare-earth based thin films and superlattices (SL`s). Magnetoelastic stress (MS) measurements (by using a cantilever capacitive technique) within a wide range of temperatures (10-300 K) and magnetic fields (up to 12 T) have been performed. We derive expressions relating the cantilever curvatures and the magnetoelastic stresses in anisotropic thin films and SL`s (for cubic symmetry) deposited onto crystalline substrates. The magnetoelastic energy associated to the interfaces and the epitaxial stress dependence of the volume MS has been investigated by studying the basal plane MS in Hon/Lu₁₅ and in Ho₁₀/Ym SL`s: we obtain interface MS even higher than the volume ones and the effect in bulk`s MS of the epitaxial strain is large. In Dy/Y and Er/Lu SL's we also deduce the MS contributions but, for Er/Lu, incomplete saturation leads to inconclusive results. Although the latter case also happens in Ho/Tm SL`s, MS clearly shows anisotropy competition. In TbFe₂ (t) / YFe₂(1000 Å) (300 Å < t < 1300 Å) epitaxial bilayers, we determine all the MS allowed by the symmetry and show that epitaxial stress strongly modifies the tetragonal MS. The thermal dependence of MS parameters is also analysed.
format	Article
author	Arnaudas, J.I. Ciria, M. de la Fuente, C. Benito, L. del Moral, A. Ward, R.C.C. Wells, M.R. Dufour, C. Dumesnil, K. Mougin, A.
author_facet	Arnaudas, J.I. Ciria, M. de la Fuente, C. Benito, L. del Moral, A. Ward, R.C.C. Wells, M.R. Dufour, C. Dumesnil, K. Mougin, A.
author_sort	Arnaudas, J.I.
title	Magnetoelastic stresses in rare-earth thin films and superlattices
title_short	Magnetoelastic stresses in rare-earth thin films and superlattices
title_full	Magnetoelastic stresses in rare-earth thin films and superlattices
title_fullStr	Magnetoelastic stresses in rare-earth thin films and superlattices
title_full_unstemmed	Magnetoelastic stresses in rare-earth thin films and superlattices
title_sort	magnetoelastic stresses in rare-earth thin films and superlattices
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2001
topic_facet	Низкотемпеpатуpная магнитостpикция магнетиков и свеpхпpоводников
url	http://dspace.nbuv.gov.ua/handle/123456789/130011
citation_txt	Magnetoelastic stresses in rare-earth thin films and superlattices / J.I. Arnaudas, M. Ciria, C. de la Fuente, L. Benito, A. del Moral, R.C.C. Ward, M.R. Wells, C. Dufour, K. Dumesnil, A. Mougin // Физика низких температур. — 2001. — Т. 27, № 4. — С. 342-363. — Бібліогр.: 54 назв. — англ.
series	Физика низких температур
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fulltext	Fizika Nizkikh Temperatur, 2001, v. 27, No. 3, p. 342–363 A rn auda s J. I. , Cir ia M., de la Fu ente C., Ben it o L., del Mo ral A. , War d R. C. C., Wells M. R., Dufou r C., Dume snil K., an d Mou gin A. Mag net oelastic str esses in ra re-ea rth t hin film s and sup erlat ticesA rna udas J. I., Ciria M., de la Fue nte C., Benito L. , del Mor al A., War d R. C. C., Wells M. R. , Du four C. , Du mesn il K., and Moug in A.Mag neto elastic str esses in rar e-ear th th in f ilm s a nd supe rlatt ic es Magnetoelastic stresses in rare-earth thin films and superlattices J. I. Arnaudas, M. Ciria, C. de la Fuente, L. Benito, and A. del Moral Departamento de Magnetismo de So′lidos, Departamento de Fisica de la Materia Condensada and ICMA, Universidad de Zaragoza and CSIC, 50071 Zaragoza, Spain E-mail: arnaudas@posta.unizar.es R. C. C. Ward and M. R. Wells Dept. of Physics, Clarendon Laboratory, Oxford OX1 3PU, U.K. C. Dufour, K. Dumesnil, and A. Mougin Laboratoire de Me′tallurgie Physique et de Science des Mate′riaux, Universite′ Henri Poincare′, Nancy, Franc \ e Received October 26, 2000 We report on the study of the magnetoelastic behavior of some rare-earth based thin films and superlattices (SL’s). Magnetoelastic stress (MS) measurements (by using a cantilever capacitive technique) within a wide range of temperatures (10–300 K) and magnetic fields (up to 12 T) have been performed. We derive expressions relating the cantilever curvatures and the magnetoelastic stresses in anisotropic thin films and SL’s (for cubic symmetry) deposited onto crystalline substrates. The magnetoelastic energy associated to the interfaces and the epitaxial stress dependence of the volume MS has been investigated by studying the basal plane MS in Ho n /Lu 15 and in Ho 10 /Y m SL’s: we obtain interface MS even higher than the volume ones and the effect in bulk’s MS of the epitaxial strain is large. In Dy/Y and Er/Lu SL’s we also deduce the MS contributions but, for Er/Lu, incomplete saturation leads to inconclusive results. Although the latter case also happens in Ho/Tm SL’s, MS clearly shows anisotropy competition. In TbFe 2 (t)/YFe 2 (1000 A° ) (300 A° < t < 1300 A° ) epitaxial bi- layers, we determine all the MS allowed by the symmetry and show that epitaxial stress strongly modifies the tetragonal MS. The thermal dependence of MS parameters is also analysed. PACS: 75.70.–i, 75.80.+q 1. Introduction The study of the magnetoelastic (ME) properties of magnetic thin films and layered nanostructures, where magnetic layers are interleaved by non-mag- netic ones or by other having different magnetic properties, presents considerable interest, not only because of fundamental reasons, but also due to the technological applications of these artificial sys- tems. For example, the success of molecular beam epitaxy (MBE) of rare-earth (RE) superlattices (SL’s) has allowed the investigation of the novel magnetic properties exhibited by the well known RE metals when thin layers of a rare earth are interleaved with non-magnetic layers (e.g., Lu or Y) or with layers made from a different rare earth. Although much research has been performed to date in relation with different magnetic properties [1–3] of these nanostructures, it was only recently that direct magnetoelastic stress measurements were per- formed in RE-based superlattices [4–6]. The know- ledge of the magnetoelastic behavior of RE SL’s is important because of the influence of the ME en- ergy in the spin configuration and magnetic proper- ties of such systems. Several RE/Y and RE/Lu SL’s have been studied and some of the differences in the magnetic behavior observed on having Y or Lu as interleaving layers are currently attributed to the different epitaxial strain, tensile or compressive, respectively, occuring in both cases. For instance, in bulk Ho the basal plane ME energy is one of the © J. I. Arnaudas, M. Ciria, C. de la Fuente, L. Benito, A. del Moral, R. C. C. Ward, M. R. Wells, C. Dufour, K. Dumesnil, and A. Mougin, 2001 agents driving the magnetic structure from helical to a ferro-cone phase [7]; interestingly, in Ho/Y SL’s this ferro-cone phase is suppressed, unlike in Ho/Lu systems, where the effect of the com- pressive stress stabilizes the ferromagnetic phase [3,8], the FM transition temperature increasing for decreasing Ho fraction. Regarding the RE SL’s, in this paper we will review the magnetoelastic stress studies we carried out in two series of Ho/Lu and Ho/Y superlattices, as well as in Dy/Y and Er/Lu SL’s and in the more complex SL system, Ho/Tm. On the other hand, and concerning nanostruc- tured systems suitable for applications, some RE alloys are, in principle, the logic choice. So, new devices, such as microsystems actuators working at room temperature, should be based on alloys with very large magnetostriction at moderately low fields. The well-known REFe2 (RE: rare earth or rare-earth alloy) Laves phases appear as good candi- dates to fulfil the necessary requirements for appli- cations. The magnetostriction of bulk single-crystal and polycrystalline REFe2 was thoroughly studied in the 1970’s and a noticeable effort has been made in the last years to produce amorphous and polycry- stalline RE-TM (TM: transition metal) films and spring-magnet type multilayers of with improved magnetoelastic properties [9,15]. However, the in- trinsic magnetoelastic behavior of thin-film samples had not been compared with that of crystalline bulk materials due to the lack of REFe2 single-crystal films. The recent success in growing by MBE high quality epitaxial films of these Laves phase magnets [16] has opened the possibility of perform such kind of studies [17,18]. In this article we present magne- toelastic stress measurements performed in a series of TbFe2/YFe2 (110) single-crystal bilayers. First of all we will describe in detail the founda- tions of the experimental technique employed, a cantilever method. This is important because the analysis of the cantilever deflections, in the case of crystalline films and substrates, differs markedly from that performed for isotropic plates, from which it is obtained the following expression relat- ing the cantilever deflection and the magnetostric- tion [19,20]: ∆ = 3λshms   L hsubs    2 Ems Esubs (1 + νsubs) (1 + ν ms ) , where λsubs is the magnetostriction; L, the length of the plate; hsubs and hms , the substrate and film thicknesses, respectively; E and ν are the Young moduli and Poisson ratios; the subscripts ms and subs refer to film and substrate, respectively. For films or SL’s having hexagonal symmetry an adequ- ate formula, expressed in terms of the magnetoelastic stresses Bp µl , instead of in terms of the magnetos- triction coefficients λjk µ , has been derived [4,21]. The case of anisotropic substrates and the boundary conditions imposed by the experimental situation, have also been considered [4] and will be outlined below for films having cubic symmetry. The ana- lysis of the hexagonal symmetry case can be found elsewhere [6]. 2. Determination of magnetoelastic stresses in anisotropic thin films and superlattices: case of cubic symmetry In the following we deal with a situation slightly more complicated than the occurring in hexagonal systems with cylindrical symmetry. In the cubic case, to be able to determine all the relevant mag- netoelastic stress parameters from the experiments, we should perform measurements with the sample cut along different crystalline directions, and conse- quently, we need to obtain the corresponding ex- pressions relating the magnetoelastic stresses and the plate curvatures. 2.1. Evaluation of the elastic energy of a thin plate We consider a thin plate where the length l and width w are much smaller than the thickness h (in our samples h is 10−2 times the other dimensions). Since, in our particular case, the samples were grown in the (110) plane, we have cut the samples using two types of configurations, which will be considered to get practical expressions. In the first configuration, our coordinate system axes y′ and z′ are parallel to the sample sides of dimensions l1 and w1 , respectively, and x′ axis is normal to the growing plane, being y′ axis \|\| [1 __ 10], z′ axis \|\| [001] and x′ axis \|\| [110]. In the second one, our coordi- nate system axes y′′ and z′′ are parallel to the new sides of dimensions l2 and w2 , respectively, and x′′ = x′, being y′′ axis \|\| [1 __ 11] and z′′ axis \|\| [11 __ 2]. To save space, we will label the coordinate systems OX’Y’Z’ as S’ and OX"Y"Z" as S"; also the axes and superscripts corresponding to both systems will be denoted with a single label, s, for a more compact notation, being s = ′ or ′′, respectively, unless otherwise specified. The boundary conditions on the plate’s surfaces are σiknk = Pi , where nk are the components of the normal to the plate surface, σik are the stress tensor components and Pi are the components of the external pressure applied to the plate. Because of the strength of the internal co- Magnetoelastic stresses in rare-earth thin films and superlattices Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 343 hesive forces, much higher than the external forces, we can consider that σisxs = 0 (is = xs, ys and zs) on the plate surface. Within a thin plate, these compo- nents of the stress tensor are small compared with the other ones, and can also be neglected [22]. We will now assume a pure bending of the plate, i.e., bending without torsion; in this case, it is easy to show that the longitudinal strains are given by [23] ε y s y s = xs Rys , εzszs = xs Rz s , εy szs = 0 . (1) In Eq. (1), xs is the coordinate of the plane where the strains are evaluated (xs = 0 is the neu- tral plane); Ris (i s = ys, zs) (with s = ′ and ′′) are the radii of the curvature of the neutral plane of the plate in the planes: y′x′ and z′x′ planes in the first case, and y′′x′′ and z′′x′′ planes for the second one. The elastic energy density for the plate in our both situations: eelas s = 1 2 σ i s j s ε i s j s (2) is obtained by writing the stresses in terms of the curvatures and the compliance coefficients sisjsksls , in the way σ y s y s = xs (ŝ1s1s ŝ2s2s − ŝ1s2s 2 )      ŝ2s2s Ry s − ŝ1s2s Rz s      , (3) σ z s z s = xs (ŝ1s1s ŝ2s2s − ŝ1s2s 2 )      ŝ1s1s Rz s − ŝ1s2s Ry s      , (4) where ŝ1s1s = s1s1s − s1s6s 2 s6s6s , ŝ2s2s = s2s2s − s2s6s 2 s6s6s ; ŝ1s2s = s1s2s − s1s6s s2s6s s6s6s , (5) and where the conventional abbreviated subscripts have been used for the compliance coefficients. Thus, applying the boundary conditions and substi- tuting Eqs. (1) and (3), (4) in Eq. (2), we obtain eelas s = 1 2      Cy s y s Ry s 2 + 2Cyszs Rys Rz s + Cz s z s Rz s 2      xs2 , (6) where Cy s y s = ŝ2s2s ŝs , Cz s z s = ŝ1s1s ŝs , Cyszs = − ŝ1s2s ŝs (7) with ŝs = ŝ1s1s ŝ2s2s − ŝ1s2s 2 . 2.2. Determination of the magnetoelastic stresses from the curvature of the plate We should now evaluate the energy density of the plate, i.e., the magnetic film, or SL, plus the substrate. This energy is contributed by: a magne- toelastic one associated to the magnetic sample, which is the origin of the bending of the plate under an applied magnetic field, and by the elastic en- ergies from the substrate and from the magnetic sample. 2.2.1. Magnetoelastic and elastic density en- ergies of the magnetic sample. To first approxima- tion it is assumed that the magnetoelastic hamilto- nian terms are quadratic in the spin components and linear in the strains. The magnetoelastic and elastic energy are then written considering only single-ion contributions and taking into account the point- symmetry group, 4⁄m 3 __ 2⁄m, for the REFe2 interme- tallic compounds, in the form [13] e mel+el s = b0(ε xsx s + ε y s y s + ε z s z s) + b1(α x s 2 ε x s x s + αys 2 ε y s ys + α z s 2 ε z s z s) + b2(α x sα y s εx s y s + α x sα z s εx s z s + α z sα y s εz s y s) + + 1 2 c1s1s (εx s x s 2 + ε y s y s 2 + εzs z s 2 ) + c1s2s (εx s x s εz s z s + ε y s y s εz ss + εxsxs εy s y s) + 1 2 c4s4s (εx s ys 2 + ε x s z s 2 + ε zsy s 2 ) . (8) In Eq. (8) the represented energy densities emel′ and emel′′ are written in the S′ and S′′ reference systems, respectively; αis denote the direction cosines of the macroscopic magnetisation, M; bi (i = 0, 1 and 2) are the magnetoelastic coupling parameters (or magnetoelastic stresses); ε’s and c’s are the carte- sian strain and elastic components, respectively, of REFe2 compounds, for the corresponding reference system. To evaluate the energy density of the system: substrate-magnetic sample we will assume that the strains in the film or SL are uniform because of it small thickness, compared with the substrate one. This means that, in Eq. (1), the variable xs can be J. I. Arnaudas et al. 344 Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 substituted by the constant value βhsubs , the dis- tance between the neutral plane and the bottom of the film or SL, expressed as a fraction β of the substrate thickness hsubs . The boundary conditions, extended to the vol- ume of the sample, imply the minimization of its total density energy: emel+el s with respect to εxsxs , εxsys , εxszs for our two measurement configurations. So, after that, the Eq. (8) can be rewritten, using Eq. (1), and leave Eq. (8) as a function of the two principal curvatures: 1/Ry′ and 1/Rz′ , for S′, and 1/Ry′′ and 1/Rz′′ , for S′′, and of the position of the neutral plane, β (the same for both cases). Thus, the integration of the total energy densities to the corresponding volumes of the substrate and mag- netic film, give rise the total energies: for S′, E′ mel tot = (α′ \|\| [010]′) = A ∫ (β−1)h subs βh subs 1 2      Cy′y′ Ry′ 2 + 2Cy′z′ Ry′ Rz′ + Cz′z′ Rz′ 2      x′2dx′ + + A ∫ βh subs βh subs +h ms      b0    x′ Rz′ c11 − c12 + 2c44 c11 + c12 + 2c44 + x′ Ry′ 4c44 c11 + c12 + 2c44    + b1    x′ Ry′ c12 c11 + c12 + 2c44    + + b2    x′ Ry′ c11 + c12 c11 + c12 + 2c44 + x′ Rz′ c12 c11 + c12 + 2c44         dx′ , (9) and for S′′, E′′ mel tot (α′′ \|\| [010]′′) = A ∫ (β−1)h subs βh subs 1 2      Cy′′y′′ Ry′′ 2 + 2Cy′′z′′ Ry′′Rz′′ + Cz′′z′′ Rz′′ 2      x′′2 dx′′ + + A ∫ βh subs βh subs +h ms      − b0 3    x′′ Ry′′ c12 − c11 − 10c44 c11 + c12 + 2c44 − x′′ Rz′′ c12 − c11 − 4c44 c11 + c12 + 2c44    − b1 9    x′′ Ry′′ c12 − c11 − 10c44 c11 + c12 + 2c44 + + 2 x′′ Rz′′ c12 − c11 − 4c44 c11 + c12 + 2c44      + 2b2 9    x′′ Ry′′ 4c11 + 5c12 + 4c44 c11 + c12 + 2c44 + x′′ Rz′′ c12 − c11 − 4c44 c11 + c12 + 2c44         dx′′ , (10) where A is the area of the plate. 2.2.2. Relation between curvatures and magne- toelastic stresses parameters. For the magnetic sample, we have assumed it is uniformly strained, so: εy′y′ = βhsubs/Ry′ and εz′z′ = βhsubs/Rz′ for S′, and εy′′y′′ = βhsubs/Ry′′ and εz′′z′′ = βhsubs/Rz′′ for S′′ performing the integration operation of Eqs. (9) and (10), we obtain: for S′, E′ mel tot (α′ \|\| [010]′) = Ahsubs 3   β2 − β + 1 3    1 2      Cy′y′ Ry′ 2 + 2Cy′z′ Ry′Rz′ + Cz′z′ Rz′ 2      + Ahms      b0      βhsubs Rz′ c11 − c12 + 2c44 c11 + c12 + 2c44 + Magnetoelastic stresses in rare-earth thin films and superlattices Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 345 + βhsubs Ry′ 4c44 c11 + c12 + 2c44    + b1      βhsubs Ry′ c12 c11 + c12 + 2c44      + b2      βhsubs Ry′ c11 + c12 c11 + c12 + 2c44 + βhsubs Rz′ c12 c11 + c12 + 2c44           , (11) and for S′′, E′′ mel tot (α′′ \|\| [010]′′) = Ahsubs 3   β2 − β + 1 3    1 2      Cy′′y′′ Ry′′ 2 + 2Cy′′z′′ Ry′′Rz′′ + Cz′′z′′ Rz′′ 2      + Ahms      − b0 3      βhsubsν Ry′′ c12 − c11 − 10c44 c11 + c12 + 2c44 − − βhsubs Rz′′ c12 − c11 − 4c44 c11 + c12 + 2c44      − b1 9      βhsubs Ry′′ c12 − c11 − 10c44 c11 + c12 + 2c44 + 2 βhsubs Rz′′ c12 − c11 − 4c44 c11 + c12 + 2c44      + + 2b2 9    βhsubs Ry′′ 4c11 + 5c12 + 4c44 c11 + c12 + 2c44 + βhsubs Rz′′ c12 − c11 − 4c44 c11 + c12 + 2c44         . (12) In this situation, the minimization of the energies given in Eqs. (11) and (12) with respect to the four independent parameters: β, Ry′′ −1 and Rz′′ −1 for S′ and Rz′′ −1 for S′′, leads to β = 2/3 and to the following expressions: ∂E′tot(α \|\| [1 __ 10]) ∂Ry′ −1 = σ~ ([1 __ 10], [1 __ 10]) + + 2 3 b2(c11 + c12) + 2c44(b1 + 2b0) c11 + c12 + 2c44 = 0 , (13) ∂E′tot(α \|\| [1 __ 10]) ∂Rz′ −1 = σ~ ([1 __ 10], [001]) + + (b2 − b1)c12 + b0(c11 − c12 + 2c44) c11 + c12 + 2c44 = 0 , (14) ∂E′′tot(α \|\| [1 __ 11]) ∂Rz′′ −1 = σ~ ([1 __ 11], [11 __ 2]) − − 2 9 (b1 − b2 + 2b0)      − c11 + c12 − 4c44 c11 + c12 + 2c44      = 0 , (15) where we have defined the σ~(α, β) MEL stresses as follows, σ~ ([1 __ 10], [1 __ 10]) ≡ 1 6 hsubs 2 hms      Cy′y′ Ry′ + Cz′y′ Rz′      , (16) σ~ ([1 __ 10], [001]) ≡ 1 6 hsubs 2 hms      Cz′′z′′ Rz′′ + Cy′′z′′ Ry′′      , (17) σ~ ([1 __ 11], 11 __ 2]) ≡ 1 6 hsubs 2 hms      Cz′′z′′ Rz′′ + Cy′′z′′ Ry′′      . (18) Notice that we have minimized the total energies for the three different cases to obtain the minimum number of equations needed to obtain the three relevant MEL stresses parameters, b0 , b1 , and b2 . By solving the system of Eqs. (13), (14), and (15) we obtain these MEL parameters as a function of the MEL stresses which can be determined from the experimental results (see Sec. 3 below). In this way just with an unique magnetostrictive experiment, it has been possible to determine all the second order MEL contributions to the total MEL energy, which, in fact, is the most relevant one. Let us compare the above expressions with the obtained ones for the hexagonal case, when the magnetization of the sample is parallel to the x axis (αx = 1, αy = αz = 0). In this situation, the minimi- zation of the total energy of the system with respect to the three independent parameters, β, 1/Rx and 1/Ry , leads to the expressions 1 9 Ahsubs 3      Cxx Rx + Cxy Ry      − 2 3 Ahsubshms    � + 1 4 Bγ,2   = 0 , (19) J. I. Arnaudas et al. 346 Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 1 9 Ahsubs 3      Cxy Rx + Cyy Ry      − 2 3 Ah subshms    � − 1 4 Bγ,2   = 0 , (20) and β = 2/3, which has been already substituted in Eqs. (19) and (20). In these equations � represents a complicated combination of symmetry elastic con- stants, cjk µ , and magnetoelastic stresses, Bp µl , for the magnetic sample, which reads � = − 1 2         B1 α,0 − 1 3 B1 α,2   √3 c12 α + c22 α c11 α + 2/√3 c12 α + 1/3c22 α − −   B2 α,0 − 1 3 B2 α,2   √3 c11 α + c12 α c11 α + 2/√3 c12 α + 1/3c22 α      . (21) Note that the volume magnetoelastic constants ap- pearing in Eq. (21) include the exchange contribu- tion to the volume strain. We have obtained the expressions (13), (14), and (15) and (19) and (20), which relate the principal curvatures of a free plate with the magne- toelastic stresses originated upon the application of a magnetic field. However, this is not the real situation in our experiments. In the next paragraph we explain the actual situation as well as details about the experimental technique and charac- teristics of the measured samples. 3. Experimental technique and samples 3.1. Experimental arrangement Our samples, which are rectangular (typical plane dimensions of about 10 mm × 10 mm), are clamped along one of their edges (cantilever con- figuration) (Fig. 1). The exact analytical solution of this kind of problem is unfeasible, because the boundary conditions can not be properly imposed. To take into account the effect of the clamping on the deflection of the plate, finite element modeling [24] and approximate analytical solutions [25] have been proposed. However, we cannot apply the re- sults obtained in these approaches to the problem, which are based on the determination of the deflec- tion of the free end of the clamped plate. In our experimental set-up, we use a capacitive technique which allows us to measure the change of capacit- ance related with the overall curvatures of the plate. Since the equation representing the plate’s surface cannot be known, we associate this change of capacitance to a single curvature of the plate. Our hypothesis is that the curvature of every line parallel to the clamping line is much smaller than the curvature which appears perpendicularly to the clamping direction. We will neglect such small curvatures not only close to the clamping line but also for the whole plate. This approximation overes- timates the value of the deflection and, hence, of the magnetoelastic parameters deduced from it, which can have an error of order 5% [26] for our approximately squared samples. Thus, in the case of cylindrical symmetry, for the sample clamped along the y direction (Fig. 1,a) we will only consider Eq. (19), with Ry −1 = 0, i.e., ∂Etot ∂Rx −1 = 1 9 Ah subs 3 Cxx Rx − 2 3 Ahsubshms    � + 1 4 Bγ,2   = 0 , (22) while for experiments with the sample clamped along x, we will use Eq. (20), with Rx −1 = 0, which reads ∂Etot ∂Ry −1 = 1 9 Ah subs 3 Cyy Ry − 2 3 Ahsubshms    � − 1 4 Bγ,2   = 0 . (23) From Eqs. (22) and (23) it is practical to define the magnetoelastic stresses acting along the OX and OY directions, σ~(x, x) = � + 1 4 Bγ,2 , σ~(x, y) = � − 1 4 Bγ,2 , (24) where the first letter within the brackets stands for the direction of the applied magnetic field and, the second one, for the direction along which the plate deforms longitudinally under the effect of the mag- netoelastic stresses. Subtraction of the above stres- ses allows us to obtain the magnetoelastic stress parameter Bγ,2. Thus, we get Fig. 1. Experimental arrangement of the cantilever: bending along the direction of the applied magnetic field (a); bending perpendicular to the direction of the field (b). a b Magnetoelastic stresses in rare-earth thin films and superlattices Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 347 Bγ,2 = 2[σ~(x, x) − σ~(x, y)] = 1 3 hsubs 2 hms      Cxx Rx − Cyy Ry      . (25) Therefore, by performing two curvature measure- ments, according to the two different configurations shown in Fig. 1, we can separate the parameter Bγ,2, associated with the magnetoelastic strain ε1 γ = 1⁄2 (εxx − εyy) within the (a, b) basal plane of the hcp structure of the magnetic sample. The subtraction eliminates, not only the volume strain, ε1 α , and tetragonal strain, ε2 α , terms but also the effect of the differential thermal expansion between magnetic sample and substrate in the zero-field capacitance. Note that both kind of measurements should be performed with the field applied along the magnetic easy direction of the sample, i.e., the a or b axes of the basal plane for rare-earth-based samples, provided that their growing plane is the basal plane. In the case of cubic symmetry we should perform at least three curvature measurements to be able to determine the three relevant MEL stress parame- ters, as we described in the previous section. Ex- pressions similar to Eq. (25), in which the only elastic constants appearing are those of the sub- strate, are not obtained now. Instead of it, by solving the system of Eqs. (13), (14), and (15), we get for b0 , b1 and b2 relationships where the radii of curvature are multiplied by complex combina- tions of all the elastic constants, for the substrate and the magnetic sample. 3.2. Cantilever capacitive technique The determination of the curvature of the plate is performed by means a capacitive cell, in which the metallic magnetic sample deposited on the substrate acts as one of the electrodes. The rest of the cell was made in copper, and annealed at 800 K to improve its behavior under thermal cycling. Measurements where done either in a three terminal capacitor configuration with a AEL Ltd. ratio bridge (sen- sitivity: 10−6 pF), or in a two plates capacitor configuration with an Andeen–Hagerling 2500 A capacitance bridge (sensitivity: 5⋅10−7 pF). No dif- ferences were found between both methods, except for a small difference in the signal to noise ratio in favour of the latter case. In this situation, cantil- ever deflections as small as 10−9 m can be detected for a typical sample length of 10 mm. For small deflections, it is straightforward to show that a curvature R−1 of the cantilever pro- duces a capacitance change given by the expression ∆C = − C0 2L2/6ε0AR , (26) where C0 is the capacitance of the parallel plate capacitor, A and L respectively, are the area and length of the plate; ε0 is the permittivity of vacuum. For instance, for hexagonal systems, expression (25), together with Eq. (26), gives the magnetoe- lastic stress Bγ,2 in terms of the different experimen- tal values for both kind of measurements, i.e., sample clamped along y and x directions, in the form Bγ,2 = − 2ε0hsubs 2 hms × ×      Cxx A(x,x)∆C(x,x) L(x,x) 2 C0(x,x) 2 − Cyy A(x,y)∆C(x,y) L(x,y) 2 C0(x,y) 2      , (27) where Cxx and Cyy are given by expressions similar to those of Eq. (7) (see Ref. 6), with the com- pliance coefficients corresponding to the substrate material, and the labels (x, x) and (x, y) refer to the two measuring configurations. In the experimental setup the capacitive cell is placed within a continuous flow cryostat, allowing us to measure between 1.7 and 300 K. Magnetic fields up to 12 T, produced by a superconducting coil, can be applied parallel to the plane of the sample. 3.3. Samples characteristics The magnetoelastic stress experiments reported in this paper were performed in Ho films and in Ho/Y, Ho/Lu, Dy/Y, Er/Lu, and Ho/Tm super- lattices and in different REFe2 films and bilayers. All the RE superlattices were grown by molecu- lar beam epitaxy using a Balzers UMS 630 facility. The rare-earth metals grow epitaxially onto a Nb metal layer deposited on a sapphire substrate [3]. Both the body-centered-cubic Nb and hexagonal- close-packed rare-earth metals grow with their re- spective close-packed atomic planes parallel to the substrate plane. The epitaxial relationships are {112 __ 0} Al2O3 \|\| {110} Nb \|\| {0001} RE, resulting the a axis of the rare earth at an angle of 5° with [0001] Al2O3 . The crystalline structure of the superlat- tices was investigated using a triple-axis x-ray dif- fractometer, giving an interface width of ± 2 lattice planes [3]. The sapphire substrates, of initial thick- ness of 500 µm, were thinned down to 150 µm to increase the sensitivity of the cantilever method (note the high value of the elastic constants of sapphire, ∼ 400 GPa). After the thinning, the mag- J. I. Arnaudas et al. 348 Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 netic-sample thickness to substrate thickness ratio remains very small (∼ 10−3). This allows us to disregard the terms of type (Ri Rj) −1, appearing in Emel tot , which arise from the elastic energy of the magnetic sample and which are hma/hsubs times smaller than the corresponding ones in the elastic energy of the substrate. From x-ray diffraction measurements performed in our Ho/Y and Ho/Lu superlattices, between RT and 10 K [28], we know that the epitaxy is good: the misfit between basal plane lattice par- ameters of Ho and Lu (or Y) in the superlattice is very small (e.g., for the hexagonal a lattice para- meter: (aY − aHo)/aHo = 2.1⋅10−3, at 45 K in a [Ho40/Y15]50 SL). The REFe2 films studied in this work have been also grown by MBE. Sapphire substrates were co- vered with a niobium buffer (the epitaxial re- lationships being: [11 __ 1] Nb \|\| [0001] Al2O3 and [112 __ ] Nb \|\| [101 __ 0] Al2O3). After that, a very thin iron layer of 15 A° thick was deposited at 820 K onto the (110) niobium plane reacting with it to produce a NbFe-ϕ alloy on the surface [29]. RHEED surface analyses have shown a 2D rectangular lattice, where the lattice parameters were close to those of a C15 cubic Laves phase in (110) plane, 7.0 ± 0.1 A° and 4.8 ± 0.1 A° . The TbFe2 layers were obtained by co-deposition of the rare-earth and iron constitu- ents, keeping the substrate at 820 K for YFe2 layer deposition, and reducing the temperature to 620 K for TbFe2 layer in order to avoid interdiffusion. At this level the epitaxial relationships are: [11 __ 0] REFe2 \|\| [11 __ 0] Nb and [001] REFe2 \|\| [001] Nb. Finally, all the samples were overcoated with a 200 A° thick layer of Y to protect them from oxidation. A small average roughness (≈ 25 A° ) of the layers was confirmed by Atomic Force Microscopy [30] in all the samples. However TbFe2 (300 A° )/YFe2 (1000 A° ) sample presented islands quite close each other, in fact quasi-continuously spread on the plane, and elong- ated along [110] direction. The composition of the bilayers was checked by microprobe analysis. The stoichiometric composition was found to be within ± 2% for REFe2 (RE = Tb and Y). The thickness was estimated by the calibration of the evaporation rates, using quartz balances and optical sensors with a 10% of error; x-ray diffraction scattering con- firmed that all the films grown epitaxially on the NbFe-ϕ buffer. They exhibited an average ≈ 0.6% lattice expansion along [110]. The mismatch lattice parameter between TbFe2 and YFe2 layers induces a shear strain at the interface, εxy 0 ≈ − 0.61, − 0.58, − 0.57, and − 0.54% for TbFe2 samples thickness of 300, 600, 1000, and 1300 A° , respectively [30,31]. The dispersion observed along the [220] orientation corresponds to an average mosaic spread of −∼ 1.5°, while the Bragg peaks broadening along this orien- tation also indicates a mean coherence length ran- ged between 200 and 400 A° . In addition, analyses of Mo..ssbauer spectra carried out on the samples indi- cate that [11 __ 1] is the magnetization easy direction in absence of applied magnetic field at room tem- perature [30]. However, a careful analysis of these results has also concluded that the magnetization easy axis would be strongly influenced by the shear deformation existing in the growing plane, which is temperature dependent [30]. 4. Magnetoelasticity of Ho films We have measured the magnetoelastic stress in two epitaxial Ho films, 5000 and 10000 A° thick [32]. The interest in these measurements is double: first, they are useful for checking the experimental system and the validity of the hypotheses assumed in the analysis explained in Sec. 2, as well as the clamping conditions; second, the results obtained in Ho films can be a reference for magnetoelastic stress measurements in Ho-based superlattices, although the studied samples, being so thick, are expected to Fig. 2. Magnetoelastic stress isotherms σ~(b, a) for the 5000 A° Ho film. Magnetoelastic stresses in rare-earth thin films and superlattices Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 349 exhibit a bulk-like behavior in their magnetic and magnetoelastic properties. In Figs. 2 and 3 we show the magnetoelastic stress isotherms obtained for the 5000 A° Ho film between 10 and 140 K. The measurements were performed applying the magnetic field B along the b-easy direction of holmium and with the sample clamped along the b axis, σ~(b, a) in Fig. 2, and along the a axis, σ~(b, b) in Fig. 3. The magnetoelastic stress isotherms display changes in the slope at certain critical fields that can be ascribed to field induced transitions to dif- ferent magnetic phases. However, the study of the phase diagrams and the analysis of the differences observed in the critical fields when compared with the bulk Ho values (in the films, the critical fields are higher [33], are not aims of the present work. We will only stress that, at the maximum applied field of 12 T, for temperatures below ∼ 90 K, we have reached a ferromagnetic phase and we can safely assure that the sample is saturated along the easy axis. Subtraction of the values of both magnetoelastic stresses (see Eq. (25)), for the different tempera- tures and at the maximum field of 12 T, gives the temperature variation of the magnetoelastic stress parameter Bγ,2, which is plotted in Fig. 4 for the 5000 A° film. We show in the same figure the fit of the experimental results by using the expression given by the standard Callen and Callen theory for the magnetoelastic coupling [27], which reads Bγ,2(T) = Bγ,2(0)Î5/2 [ � −1(m(T))] , (28) where m is the reduced magnetization of the Ho film (m = M(T)/M(0)); Î5/2 is the reduced hyper- bolic Bessel function, Il+1/2/I1/2 , of order l = 2; � −1 is the inverse Langevin function. The 0 K value of the Bγ,2 stress obtained from the fit was Bγ,2(0) = 0.29 ± 0.02 GPa. Similar results were ob- tained for the 10000 A° film, for which Bγ,2(0) = 0.28 ± 0.02 GPa. Both values are in good agreement with the basal-plane-symmetry-breaking magnetoe- lastic stress parameter determined for bulk Ho, in which an extrapolation from B = 3 T and T = 70 K gives Bγ,2(0) = 0.275 GPa [34]. From the above results, i.e., the thermal depend- ence and the values of the magnetoelastic stress Bγ,2, we conclude that the origin of the magnetoe- lastic strain in our Ho films is a single-ion crystal electric field interaction. In addition, as antici- pated, this strain is not affected by any surface effect, likely due to the present high volume to surface ratio. Then, in this case, Bγ,2 ≡ Bν γ,2 . Let us label this volume stress parameter obtained for bulk-like films as Bν0 γ,2 . 5. Magnetoelastic behavior of Ho/Lu superlattices Magnetization and MEL stress measurements have been performed in a series of [HonHo /LunLu ]50 superlattices, with nHo ranging from 8 to 85 atomic planes. Here we only report on the stress measure- ments in the HonHo /Lu15 samples, although the magnetization isotherms and isofields also provide valuable information about the different field-in- duced and zero-field magnetic phase transitions [33]. In this regard, the most remarkable fact is the effect of the compressive strain in the basal plane of Ho layers due to the Lu interleaving layers; this compression, and the corresponding c-axis expan- sion, stabilizes the ferromagnetic phase [3,8]. More- over, the ferromagnetic transition temperature in- creases for decreasing Ho fraction and the helifan phases, intermediary between the helix and fan phases, existing in bulk Ho [35] and also observed by us in the Ho films, do not appear in the Ho/Lu Fig. 3. Magnetoelastic stress isotherms σ~(b, b) for the 5000 A° Ho film. Fig. 4. Thermal variation of the basal plane magnetoelastic stress parameter for the 5000 A° Ho film. The continuous line is the fit to Bγ,2(0)Î5/2[ � −1(m(T))], with Bγ,2(0) = 0.29 GPa. J. I. Arnaudas et al. 350 Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 superlattices. In the Ho/Lu superlattices with low number of atomic planes of Ho, even the fan phase is absent [33] (Fig. 5). In Fig. 6 we display, as an example, the σ~(b, a) and σ~(b, b) magnetoelastic stresses for the thin Ho layers superlattice Ho8/Lu15 . The isotherms were done in the temperature range from 10 to 160 K (the lower limit of 10 K was chosen to avoid the superconductivity of the Nb layer present in the samples, appearing below 9 K, which uncontrol- lably modifies the magnetization and the magneto- elastic stress measurements). The magnetoelastic stress isotherms show anomalies similar to those displayed by the magnetization isotherms. We as- cribe the abrupt jumps at the lowest temperatures to transitions to a ferromagnetic state (e.g., see in Fig. 6 the σ~(b, a) isotherms, at T ≤ 40). Smoother anomalies at higher temperatures, which are better detected in the field derivative of the stress iso- therms, correspond to helix-fan or helix-ferromag- net transitions. In the paramagnetic phase, T > TN , we observe the usual B2 thermodynamic behavior. The basal plane magnetoelastic stress parameter Bγ,2, which appears in the magnetoelastic energy [6], is determined by using Eq. (25). However, there is a difference with the case of the Ho films: for a superlattice [AtA /BtB ]r which has only a volume fraction FA = tA/(tA + tB) of material A undergoing magnetoelastic strain, the value that we actually obtain from the experiment is Bmeas γ,2 = = Bγ,2fA . In our Ho/Lu superlattices we consider that only the Ho layers behave magnetoelastically. Also, because the c-axis parameters of both Ho and Lu are very similar, the ratio between thicknesses fA can be well approximated by the ratio between numbers of atomic layers. Therefore, for A = Ho and B = Lu, we have Bmeas γ,2 = Bγ,2 n Ho nHo + nLu . (29) For convenience, as we will see below, we have plotted in Fig. 7 the values, at 10 K and 12 T, of Bmeas γ,2 (nHo + nLu) (with nLu = 15) as a function of the number of Ho atomic planes in the bilayer repeat, nHo . In the next section we will see that the observed non-linear dependence of the product Bmeas γ,2 (nHo + nLu) with nHo denotes the presence of additional terms in the magnetoelastic stress, i.e., in the present SL’s Bγ,2 is not simply Bν0 γ,2 (the volume magnetoelastic stress obtained for the Ho films). If the latter case would be true we would obtain the dashed line shown in Fig. 7, as it is obvious from Eq. (29). 5.1. Analysis of the basal plane cylindrical symmetry breaking magnetoelastic stress In a previous work [4], to explain the anomalous thermal dependence of the basal plane magnetoelas- tic stress in a [Ho6/Y6]100 superlattice, we showed that it was necessary to consider that the associated magnetoelastic parameter Bγ,2 had a linear depend- Fig. 5. Magnetic phase diagram for the Ho22/Lu15 superlattice. The critical fields Bc were obtained from magnetoelastic stress and magnetization measurements; FM and H correspond to the ferromagnetic and helix phases; AFB is a phase where FM Ho blocks are antiferromagnetically coupled [8]. Fig. 6. Magnetoelastic stress isotherms for the Ho8/Lu15 super- lattice. a b Magnetoelastic stresses in rare-earth thin films and superlattices Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 351 ence on the strain (this implies to have a non-linear dependence on the strain in the magnetoelastic energy). In this paragraph we will outline a model [36] which takes into account this fact, as well as the effect of the interfaces in the basal plane mag- netoelastic stress in superlattices. (a) Let us assume that the volume magnetoelas- tic parameter is, to first order in the strain, Bν γ,2 = Bν0 γ,2 + Dν γ ε (30) where Dν γ = (∂Bv γ,2/∂ε)ε=0 accounts for a possible modification of Bν γ,2 due to the epitaxial strain ε. This strain occurs when the two materials, A and B constituting the superlattice adjust their different bulk lattice parameters to a common one, having a intermediate value. If the lattice mismatch, defined in terms of the bulk’s basal-plane a-lattice par- ameters of both materials as e0 = (aB − aA)/aB, is not too high (for A = Ho and B = Lu we have e0 = − 0.017, at 10 K), it is reasonable to assume that A and B will accommodate their lattice par- ameters to a single value aA = aB = asl (see end of Sec. 3) and that no misfit dislocations will appear [37]. In Eq. (30) we have assumed isotropic basal plane strains, i.e., εxx = εyy = ε. With the above hypotheses it is easy to relate the basal-plane strains of elements A and B through the mismatch e0 , obtaining εA = asl A − aA aA = asl B − aB + aB − aA aA −∼ −∼ asl B − aB aB + aB − aA aB = εB + e0 . (31) The preceding relation allows to express the misfit elastic energy of a bilayer A/B in terms of the epitaxial strain εA. Minimization of that energy, considering free x − y interfaces for the bilayer, with no stress component in the z direction, gives us the equilibrium value of the basal plane epitaxial strain [33]: εA = e0 tB kAkB −1tB + tA , (32) where kA,B are the following combinations of elastic constants for the A and B elements: kA,B = c11 α,A,B      1 − 2c13 A,B c 33 A,B      2 − − 1 √3 c12 α,A,B      1 −      2c13 A,B c33 A,B      2      + 1 12 c12 α,A,B      1 + 2c13 A,B c33 A,B      2 . Now, Eq. (30) can be written in the form Bν γ,2 = Bν0 γ,2 + Dν γ,2 e0 t B kAkB −1t B + tA , (33) where kHo/kLu is close to 1, for A being a rare-earth element and B being Y or Lu. (b) The interfacial contribution to the magnetoe- lastic stress is now treated similarly to the case of the interface anisotropy: the symmetry breaking at the surfaces or interfaces was related with the appearance of an additional term in the magnetoc- rystalline anisotropy, this term being proportional to the surface-to-volume ratio 1/t, where t is the magnetic film (or block, in superlattices) thickness [38]. Then, considering the two surfaces or inter- Fig. 7. Bmeas γ,2 (nHo + nLu) for the series Hon/Lu15 (●), with 8 < n < 85. The line is the function Bγ,2nHo , where Bγ,2 is given by Eq. (34), with the fitting parameters Bν0 γ,2 = 0.27 GPa, B̂surf γ,2 = – 7 GPa and Dν γ,2 = – 135 GPa, taking e0 = (aLu − aHo)/aLu = − 0.017. The dashed line is Bν0 γ,2nHo (a). Same as (a) in a double-logarithmic plot, including the values for the two Ho films (❍) (b). J. I. Arnaudas et al. 352 Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 faces per magnetic block, we add an interfacial term 2Bsurf γ,2 /tA to the volume magnetoelastic stress of Eq. (33), obtaining Bγ,2 = Bν0 γ,2 + 2B̂surf γ,2 nA + Dν γ,2e0 t B kA kB −1tB + tA , (34) where B̂surf γ,2 = Bsurf γ,2 (cA/2)−1 is the value of the inter- facial magnetoelastic stress expressed in units of energy per unit volume of the superlattice (cA/2 is the (a, b)-planes spacing for element A). For the HonHo /LunLu superlattices, substitution of Eq. (34) in Eq. (29) gives Bmeas γ,2 = = nHo nHo + nLu      Bν0 γ,2 + 2B̂surf γ,2 nHo + Dν γ,2e0 nLu kHokLu −1nHo + nLu      with kHo/kLu = 0.95 and e0 = − 0.017. Now, from Eq. (35), the advantage of plotting Bmeas γ,2 (nHo + nLu) vs. nHo becomes clear. In Fig. 7 we show the fitting of the experimental values to the function Bγ,2nHo , where Bγ,2 is given by Eq. (34). The parameters giving the best fit are Bν0 γ,2 = 0.27 GPa, B̂surf γ,2 = = − 7 GPa and Dν γ,2 = − 135 GPa. In a log–log plot (Fig. 7) we have included the values for two Ho films studied, also satisfying Eq. (35). Note that the obtained Bν0 γ,2 agrees with the value of Bγ,2 for bulk holmium [34], which supports the proposed model. Interestingly, we have found a quite high value of the epitaxial stress increase of the un- strained volume stress parameter (Dν γ,2e0nLu/(nLu + + nHo) ≈ 5.4⋅Bν0 γ,2 for nHo = 8). Also, we have de- duced a interfacial contribution to the magnetoelas- tic stress which is larger and of opposite sign to the volume stress (2B̂surf γ,2 ≈ − 25⋅Bν0 γ,2 ; e.g. 2B̂surf γ,2 /nHo ≈ ≈ − 6.4⋅Bν0 γ,2 for nHo = 8). The interfacial stresses found in the present HonHo /Lu15 superlattices are even higher than the highest values previously repor- ted in Cu/Ni/Cu trilayers (B̂surf γ,2 ≈ − 6⋅Bν0 γ,2) [39]. 5.2. Temperature variation of the magnetoelastic stresses The study of the thermal dependence of the magnetoelastic stresses serves as a separate test of the previous analysis and the ensuing fitting pa- rameters. For volume-originated distortions like Bν0 γ,2 and Dν γ,2 , the standard Callen and Callen theory [27] predicts a temperature variation of the type Î5/2[ � −1(m(T))] (see Eq. (28)) . For the interface magnetoelastic parameter Bsurf γ,2 , a depen- dence type m4(T) at low temperatures is predicted [40] if we assume that, at the interfaces, the spin dimensionality is D = 2 (some interface anisotropy could account for this reduction in dimensionality). In the high temperature regime both volume and interface stress parameters should display the same thermal dependence, type m2(T) [27,40]. Therefore, we have analyzed the temperature variation of Bγ,2 by using the expression Bγ,2 = (Bν0 γ,2 + Dν γ,2ε) Î5/2[ � −1(m)] + 2B̂surf γ,2 nHo mα , (36) where α = 4 at low temperatures and α = 2 at temperatures close to TN . In Fig. 8 we show the temperature variation of Bγ,2 at 12 T for two of the studied samples, together with the theoretical fits obtained by using Eq. (36). Fig. 8. Thermal dependence of Bγ,2 for the Ho30/Lu15 and Ho40/Lu15 superlattices. The lines are theoretical fits using Eq. (36), which includes interface Bsurf γ,2 and volume Dν γ,2 terms, the latter associated to the epitaxial strain. Magnetoelastic stresses in rare-earth thin films and superlattices Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 353 Similar satisfactory results were obtained for the other samples, except for the superlattice with nHo = 8, were the fit was poorer. We employed the parameters previously deduced from the analysis of the Ho-thickness dependence of Bmeas γ,2 (nHo + nLu) (see Fig. 7), allowing less than a 9% variation in their values. The interface contribution, varying as m4(T), was included below 50, 80, 55, and 40 K, for the superlattices with nHo = 14, 30, 40, and 45, respectively. These results support the validity of the magnetoelastic parameters obtained and confirm the single-ion crystal electric field origin for the magnetoelastic coupling in the Ho/Lu superlat- tices. 6. Magnetoelastic behavior of Ho/Y superlattices We have carried out magnetoelastic stress meas- urements in the superlattices Ho10/YnY , where nY = 10, 30, and 40. In this case the interface contribution to Bγ,2 remains unchanged, since the Ho layers thickness is kept constant. However, the influence of the epitaxial strain ε on the volume magnetoelastic parameter is expected to be in- creased for increasing values of the Y layers thick- ness (see Eq. (33). In Fig. 9 we show, as an example, the magnetoelastic stress isotherms for the Ho10/Y10 superlattice, where the transitions from helix to ferromagnetic phase at low temperatures are observed in the form of relatively abrupt chan- ges in the stress; at higher temperatures, additional changes in the isotherm slope are associated with intermediary fan phases (the magnetization iso- therms display such kind of changes at the same fields). Above TN the paramagnetic behavior is observed. However, the most striking fact found in the present Ho/Y series is the change of sign of the magnetoelastic stress parameter Bγ,2. In Fig. 10 we show the σ~(b, a) and σ~(b, b) isotherms, at 10 K and 12 T, for the Ho10/Y10 and Ho10/Y40 superlat- tices. For comparison, the values obtained for the 5000 A° film are also shown. Note that, in order to compare the experimental values of σ~(i, j) corre- sponding to samples with different Ho volume frac- tions, we have plotted in Fig. 10 the measured stresses multiplied by (tY + tHo)/tHo , according to Eq. (29) applied to the Ho/Y case (we denote these normalized stresses as σ~mag(i, j). The mag- netoelastic stress Bγ,2 = 2[σ~mag(b, b) − σ~mag(b, a)] changes its sign along the series, as it is shown in Fig. 11, where the 10 K and 12 T values of Bγ,2 are plotted against the volume fraction of Y. In this series, the interface contribution, inversely propor- tional to the Ho thickness, must be constant. There- fore, the plot in Fig. 11 reveals the dependence of the magnetoelastic stress with the epitaxial strain ε on the Ho blocks, which is proportional to the Y volume fraction (see Eq.(32)). The line in Fig. 11 is a fit to the model Eq. (34), where A = Ho and B = Y, nHo = 10, kHo/kY = 1.05 and e0 = 0.023. The parameters obtained from the fit are Bν0 γ,2 = 0.27 GPa, B̂surf γ,2 = 2.16 GPa, and Dν γ,2 = − 56 GPa (the values Fig. 9. Magnetoelastic stress isotherms σ~(b, a) and σ~(b, b) for the Ho10/Y10 superlattice. Fig. 10. Magnetoelastic stress isotherms σ~mag(b, a) ( ), and σ~mag(b, b) (- - -) at 10 [4] K for the Ho10/Y10 and Ho10/Y10 superlattices and for the 5000 A° Ho film. J. I. Arnaudas et al. 354 Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 for bulk Ho and for the 5000 A° Ho film have not been included in the fit, although they are plotted in Fig. 11 to emphasize the change of the sign of Bγ,2 in the Ho10/YnY superlattices, for the film, tY = 1000 A° , the Y seed thickness). For this series the constant interfacial contribution to the magne- toelastic stress is near two times the unstrained volume magnetoelastic stress (2B̂surf γ,2 /nHo ≈ 1.6⋅Bν0 γ,2 , with nHo = 10). The epitaxial stress contribution to the unstrained volume stress parameter depends on the Y thickness but it always has opposite sign to Bν0 γ,2 (e.g., Dν γ,2e0nY/(nY + nHo) ≈ − Bν0 γ,2 , with nHo = 10 and for nY = 40). The explanation of the change of sign of Bν γ,2 is another confirmation of the validity of the model assumptions, mostly the existence of a strain de- pendence of the volume magnetoelastic parameter, represented by the term Dν γ,2ε in Eq. (34). The parameter Dν γ,2 is negative for both the Ho/Lu and Ho/Y studied series; however, the different sign of the lattice mismatch e0 in both cases makes possible to observe the change of sign of Bγ,2 only in the Ho/Y series. In order to understand the origin of the interface contribution to the magnetoelastic stress, a model of crystal electric field including screening due to the conduction band electrons has been proposed [41]. In this model, a Hartree-Fock-type dielectric constant and a Gaussian distribution, of half-width b, for the electronic charge density of the Ho3+ ions are assumed. The calculated values of Bν0 γ,2 and Bsurf γ,2 , for the Ho/Lu superlattices, agree in sign and magnitude with those determined in our experi- ments, indicating that both the volume and, more interestingly, the interface magnetoelastic stresses have their origin in the single-ion interaction with the distorted crystal electric field. The experimental values can be well reproduced if the Fermi energy of the interface, before epitaxy, is smaller than the volume one and if we take a density of states close to the obtained one from the bulk RAPW band- structure calculations for Dy. The difference be- tween the Fermi-energy values is simply, but quan- titatively, explained assuming the formation of Ho3+ interface weakly localized electron states close to the Ho/Lu ideal contact plane [41]. However, two cautions are suggested: small systematic vari- ations with nHo of the volume contribution Bν0 γ,2 (and of Dν γε) will result in variations of Bsurf γ,2 , although such an effect cannot be ascertained with our available experimental data; however, the sign and order of magnitude should probably be pre- served, as both our volume and interface MEL parameters must be single-ion properties. Therefore, a more refined theory of the conduction electron band structure for the whole superlattice could predict changes in the Ho volume MEL stress ex- tended to a certain number of layers within the Ho block. 7. Competing anisotropies in the magnetoelastic behavior of Ho/Tm superlattices In this section we present a study of two Ho/Tm SL’s: Ho8/Tm16 and Ho30/Tm16 , nearly isomor- phous to the corresponding Hon/Lu15 samples ana- lysed in Sec. 5. The differences when substituting Ho for Lu are, however, of fundamental import- ance. Now, the Tm «spacers» between Ho blocks are, not only magnetic, but also have the magneti- zation easy axis along c. In fact, bulk thulium has its magnetic structure longitudinally modulated along the c axis below TN = 58 K [42]. Its strong axial anisotropy hardly leaves that its magnetic moments been tilted out of the c axis when a magnetic field is applied perpendicularly (within the basal-plane). This deviation is lesser than a 2% of the total moment at low temperatures and high- fields, −∼ 10 T [43]. However, a magnetic field ap- plied parallel to the c axis can break the modulated structure along this axis, and then gets a fully aligned magnetization state, −∼ 2700 emu/cm3 at 0 K, for magnetic field higher than 2.8 T [43]. Thus, having the Ho/Lu system as a well- known reference, we have chosen Ho/Tm SL’s to study the presumable competition between the ani- sotropies of holmium and thulium layers in a SL system. By doing magnetization and MEL stress measurements we have analysed the differences which are observed when comparison with the hol- Fig. 11. Dependence of Bmag γ,2 with nY/(nY + nHo). The line is a fit to Eq. (34) (see text for details). The values for bulk Ho (❍) and for the 5000 A° Ho film (∆∆) are included. Magnetoelastic stresses in rare-earth thin films and superlattices Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 355 mium-lutecium results is made [44]. In this Ho/Tm SL’s, the analysis of the magnetization gives some insight about the kind of interaction existing be- tween the Ho and Tm layers. However, the MEL stress measurements prove much better than magne- tization whether the holmium layers or thulium ones dominate or not the magnetic behavior in Ho/Tm SL’s. This is because, the MEL stress is a power of the magnetization [27,40]. In this way, a competition between the anisotropies of Ho and Tm ions can be clearly revealed by determining the MEL stresses in the Ho/Tm SL’s. The different sign of the Ho and Tm anisotropies makes opposite the contribution to the magnetostriction from both ions, in particular, for the basal-plane cylindrical symmetry breaking MEL stress that we determine in these hexagonal-symmetry systems. In Fig. 12 we show the σ~(b, b) and σ~(b, a) MEL stress isotherms, for the Ho8/Tm16 and Ho30/Tm16 superlattices. The MEL stress isotherms for the Ho8/Lu15 SL were shown in Fig. 6, and those for the Ho30/Lu15 are similar in their field dependence (saturation at relatively small magnetic fields, H ≥ 0.5 T, and being more saturated when decreas- ing the temperature), although they have different saturation values. As a first approach, and for comparison with the Ho/Lu SL’s, we can analyse the MEL stress data obtained in the Ho/Tm samples assuming that the Tm moments do not leave their magnetization easy axis, even under the action of the maximum applied field of 12 T. In this way we obtain the Bγ,2 MEL parameter, which is shown as a function of the temperature in Fig. 13, for the two Ho/Tm SL’s studied, together with the Bγ,2 values for the iso- morphous Ho/Lu samples. The values of Bγ,2 for the Ho/Tm SL’s are positive at high temperatures and very close to the Ho/Lu ones, showing a broad peak at around 80–90 K. Below 80 K, Bγ,2 start to decrease, reach- ing a maximum slope on decreasing the temperature at −∼ 60 K, and undergoing a change of sign at a certain temperature, which is 60 K for Ho8/Tm16 and 20 K for Ho30/Tm16 . Considering the single- ion CEF origin of the Bγ,2 MEL stress, this strong deviation towards negative values is interpreted as a result of the competition of the Ho and Tm anisotropies. Moreover, the effect of the Tm layers is much more intense in Ho8/Tm16 SL than in Ho30/Tm16 , where the basal-plane anisotropy of holmium blocks dominates the Bγ,2 MEL stress, making the values more positive for this SL, where the thickness of holmium layers is high enough with respect to the thulium ones. This is in good agre- ement with the magnetization results too [44]. So, our MEL stress experiments clearly indicate that the Tm moments are tilted out from the c axis when a magnetic field is applied within the basal plane of the hcp structure (otherwise the values of Bγ,2 would be nearly the same as for the Ho/Lu SL’s), this effect being larger in the case of the Ho30/Tm16 sample than for the Ho8/Tm16 SL. Nevertheless, experiments in higher magnetic fields are needed to attempt the full saturation of these competing anisotropy SL’s, and, in this way, to be able to obtain separately the Tm and Ho MEL stress parameters [44]. 8. Magnetoelastic behavior of Dy/Y and Er/Lu superlattices In dysprosium based SL’s the determination of the easy basal plane cylindrical symmetry breaking Fig. 13. Thermal dependence of Bγ,2 for the Ho8/Tm16 and Ho8/Lu15 SL’s (a) and the Ho30/Tm16 and Ho30/Lu15 SL’s (b). Fig. 12. Magnetoelastic stress isotherms for the Ho8/Tm16 and Ho30/Tm16 superlattices. J. I. Arnaudas et al. 356 Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 magnetoelastic stress, Bγ,2, is of paramount interest, inasmuch as it decisively helps to drive the sponta- neous helical (H) ferromagnetic (F) transition in bulk Dy, at Tc > 85 K [7]. However, it was observed than in Dy/Y superlattices, grown along the c axis, the Y block tensile stress suppresses the H-F transition [1]. In bulk Er the c axis is easy, and the magnetic structure changes from paramagnetic to sinusoidal at TN1 > 85 K, to the (a, c) plane elliptically cycloidal structures at TN2 > 52 K and to ferro- conical phase of wave vector (5/21) c∗ at Tc > 20 K [45,46]. To our knowledge no magnetic phase diag- rams have been traced for Er/Lu superlattices. We have carried out magnetoelastic stress measu- rements in the superlattices (Dyn/Y15) and (Erm/Lu10), with n = 5, 15, 25, and m = 10, 20, 30 atomic planes, from 10 K and in applied magnetic fields up to B = 12 T; the magnetic field was applied within the basal plane, along the a easy axis, for both series. In Fig. 14 we present, as an example, the σ~(a, b) and σ~(a, a) isotherms for B \|\| a, for the [Dy25/Y15]50 SL. We notice changes in slope for certain fields that we tentatively ascribe to the H-fan (FN) and to the FN-F transitions. These fields correspond well with the observed ones on the magnetization isotherms. As we may observe, sa- turation is practically accomplished at 12 T for T < TN . This indicates that we are measuring a MS of crystal electric field origin, once the sample is F. For the Er SL’s in Fig. 14,b we present, as an example, the same kinds of isotherms for the Er30/Lu10]40 SL, where we observe: three changes in slope for T < Tc , two for Tc < T < 40 K, and no features for 40 K < T < TN1 , if we tentatively ascribe the same phases of the bulk to the SL [45,46]. Similar changes in slope were observed in the magnetization isotherms at about the same fields for T < Tc , although for the range 40 K < T < TN1 , a slope change is observed at about 2 T. According to bulk neutron diffraction measure- ments and mean field calculations [45,46] we tenta- tively ascribe the transitions for T < Tc as follows: The transition at Hc1 ≅ 3 T could be to a conical-fan structure of wave vector (1/4) c∗, which, at Hc2 ≅ ≅ 8 T, transforms to an FN structure around the a axis. Finally, the transition at Hc3 ≅ 12 T could be toward an F one along the a axis. Notice that only the transition at Hc1 is abrupt. For the range Tc < T < 40 K, we observe changes of slope at about 1, 2, 5, and 12 T. If we tentatively translate from the bulk [45] these fields would respectively correspond to transitions to commensurable cycloi- dal structures of wave vectors (in c∗ units) 6/23 and 4/15, a fan 2/7 about the basal plane, and toward an F phase along B. If we analyse the thickness dependence of the magnetoelastic stress of the Dy/Y SL’s as we did for the Ho/Lu SL’s, we obtain the values: Bν0 γ,2 = = 0.85 GPa, Dν γ = 5.2⋅102 and Bsurf γ,2 /(c/2) = = − 22.8 GPa [5] (note that the SL with n = 5 can only be included in the analysis duplicating the value of Dν γ , which indicates that the SL’s with very thin Dy blocks are more sensitive to epitaxial strain variations). The effective interface magnetoe- lastic stress 2Bsurf γ,2 tDy is strong compared to the volume term, up to about one order of magnitude larger and of the opposite sign. Fig. 14. Magnetoelastic stress isotherms for the SL’s [Dy25/Y15]50 (a) and [Er30/Y10]40 (b), σ~a ≡ σ~(a, b) and σ~b ≡ σ~(a, a) correspond to SL clamping along the a and b axes. Magnetoelastic stresses in rare-earth thin films and superlattices Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 357 Another test of the CEF origin for the Bγ,2 is to check if it scales with the reduced saturation (at B = 12 T) magnetization m = M(T)/M(0) as m3, according to the standard theory of magneto- striction [27]. In Fig. 15 we compare for the Dy25/Y15 SL the thermal variation of Bγ,2 (12 T) with the one for m3 (12 T); the agreement is satisfactory. For the Ern/Lu10 SL’s we have done a study fully similar to the one above for the Dy/Y SL’s. However, as it can be seen in the isotherms plotted in Fig. 14,b the magnetoelastic stresses are far away from saturation, as it happens with the magnetization [33]. Thus, the analysis of the ex- perimental results in this case is inconclusive (see Fig. 15 and Ref. 5). Magnetoelastic stress experi- ments in larger fields, up to 30 T, which are needed to saturate this SL’s within the growing plane, are in progress. 9. Magnetoelastic behavior of (110) TbFe 2 /YFe 2 bilayers For the TbFe2/YFe2 bilayers, the MEL stress measurements were performed between 10 K and RT and the magnetic field, up to 12 T, was applied within the growing plane (temperatures below 10 K were avoided because of the diamagnetic behavior of the Nb buffer). In Fig. 16 we show, as an example, isotherms corresponding to the TbFe2 (600 A° )/YFe2 (1000 A° ) bilayer, for the stresses: σ~([1 __ 10], [001]), σ~([1 __ 10], [1 __ 10]) and σ~([1 __ 11], [11 __ 2]) (see Eqs. (16), (17), and in Sec. 2.2.2). For the sake of clarity, and because of the similarity be- tween the different isotherms for a given sample, we have only plotted the isotherms obtained at the lowest and the highest temperatures of the measur- ing range. At the beginning, and before starting each measurement, we have demagnetized the sample. Due to the high Curie temperature of TbFe2 , of about 696 K, it was unfeasible with our experimental set-up to get any virgin state by heat- ing up first the sample above 700 K and, then, cooling it down to get the initial demagnetized state. Moreover, at these high temperatures the bilayer structure would be destroyed due to the diffusion between the layers. Instead of this, we have performed, at the measuring temperature, hys- teresis cycles with decreasing amplitude of the maximum applied magnetic field in each cycle, to get a macroscopically demagnetized sample. From the MEL stress isotherms, we can distinguish two kinds of magnetoelastic behavior, depending on the applied magnetic field direction. For those measure- ments where the magnetic field is parallel to [1 __ 11] direction (see Fig. 16,c), the MEL isotherms show a large hysteresis with coercive field values very close to those observed in the magnetisation meas- urements [18]. On the contrary, when the field is applied along [1 __ 10] direction, the MEL curves hardly show hysteresis, the fields for zero MEL stress crossing being even smaller than the corre- sponding ones obtained from magnetisation hys- teresis measurements. In all the cases the coercive fields increase on reducing the thickness of TbFe2 layer. In spite of these differences, it is important to note that all the loops have approximately the same Fig. 15. Comparison of Bγ,2(T) with m3(T) at 12 T, where m is the reduced magnetization, for Dy25/Y15 and Er30/Lu10 SL’s. Fig. 16. The magnetoelastic stress isotherms for TbFe2 (600 A° )/YFe2 (1000 A° ) bilayer: σ([1 __ 10], [001]) (a); σ([1 __ 10], [1 __ 10]) (b), and σ([1 __ 11], [11 __ 2]) (c). For the sake of simplicity we only represent the RT and 10 K isotherms. J. I. Arnaudas et al. 358 Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 «closing field», which has very large values (≈ 6 T at RT). Moreover, at the maximum applied field of 12 T, the σ~(α, β) MEL stresses do not show satura- tion at all. This lack of saturation even occurs when the magnetic field is applied along the easy [1 __ 11] direction, which, however, seems to show the best approach to the saturation. To some extent, magne- tostriction is more sensible to show the lack of the full saturation than magnetization, a usual fact. Therefore, since the magnetization is indeed satu- rated at 12 T [18], we will consider the MEL stress values at 12 T as corresponding to a saturated regime. 9.1. Magnetoelastic stress parameters We have obtained the b0 , b1 , and b2 MEL stress parameters as explained in Sec. 2.2.2. Their thermal dependences for the different TbFe2(t)/YFe2 bilayers are shown in Fig. 17 (the t = 300 A° sample is the only one of the series which does not show a measurable MEL stress). We can see a smooth but monotonous decrease of all the parameters on rising up the temperature, and also that the values of b0 are about one order of magnitude smaller than those of b1 and b2 . Notice that, because of the selected applied field and MEL stress directions, it has been possible to determine directly all the second order MEL stress parameters in the present bilayers. However, b0 was never determined before in bulk TbFe2 , and therefore, we cannot compare its value with those ones for the TbFe2/YFe2 bilayers. We also can observe that b1 , for TbFe2 (1300 A ° )/YFe2 (1000 A ° ) sample and TbFe2 (1000 A ° )/YFe2 (1000 A ° ) sample are very close in value, −∼ – 0.4 GPa at 10 K and 12 T, but for TbFe2 (600 A ° )/YFe2 (1000 A ° ) it is about 25% smaller than in the others. This could indicate that b1 tends to saturate for increasing TbFe2 block thickness in the bilayer. However, from anisotropic magnetostriction, λt s , measurements in polycrystal- line TbFe2 and from measurements in crystalline TbFe2 , and using the relationship λt s = (3/5)λ[100] + + (9/10)λ[11 __ 1] [47–51], we have estimated a 0 K value of b1 −∼ – 0.11 GPa for bulk TbFe2 , which is less than half of those obtained for the t = 1000 A° and 1300 A° studied bilayers. The origin of this enhancement of MEL stress in the bilayers will be analysed in more detail later. It is interesting to note that b2 is 74% of the bulk’s value for TbFe2 (1300 A ° )/YFe2 (1000 A ° ), at RT and 12 T, whereas it is only a 55% of it in the case of TbFe2 (1300 A ° ) on NbFe–ϕ [4]. This would indicate that the use of NbFe–ϕ (15 A° ) as buffer instead of YFe2 (1000 A ° ) decreases the value of the MEL stress, b2 , in the (110) TbFe2 (1300 A ° ) thin films. The reason why this happens is not very well understood at the present, although the epitaxial strain should be playing an important role. The strain originated by the epitaxial growth of the 1300 A° thick TbFe2 layer onto the 1000 A° thick YFe2 buffer is −∼ – 0.54%, at RT [31] and, when the buffer is a 15 A° thick NbFe–ϕ layer, the epitaxial strain increases up to −∼ – 0.64% [30,52]. Therefore such a small misfit strain reduction changes sub- stantially the rhombohedral, b2 , MEL parameter. This could also happen with the observed increase of \|b2\| (see Fig. 16) when t increases because, as the TbFe2 block thickness increases, the shear strain, εxy , decreases [31]. Now, and since we have infor- mation about the in-plane strains in the bilayers, we can perform the same kind of analysis as we did for the RE SL’s, although modified for the current cubic symmetry, to obtain information about the origin of the MEL coupling in the present systems. So, for b2 we can write b2 = b2 v + b2 s t = (b2 v0 + d2 v εxy 0 ) + b2 s t , (37) where b2 v0 is the main volume contribution, d2 v = = (∂b2/∂εxy 0 ) taking into account the non-linear ef- fects on the MEL energy of Eq. (8). This strain, Fig. 17. The temperature dependence of the MEL stress par- ameters: b0 (isotropic MEL stress) (■); b1 (tetragonal MEL stress) (●), and b2 (rhombohedral MEL stress) (▲), for TbFe2(t)/YFe2 (1000 A° ) bilayers at different t, A° : 600 (a); 1000 (b); 1300 (c). Magnetoelastic stresses in rare-earth thin films and superlattices Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 359 within the elastic regime (see Sec. 5.1), for the cubic symmetry reads εxy 0 = ε0 1000 A° β2t + 1000 A° , (38) where ε0 (−∼ – 0.7%)31, is (a⊥ TbFe 2 − a⊥ YFe 2)/〈a⊥〉, being a⊥ REFe 2 the lattice parameters along [220] and β2 ≡ ≡ c44 A /c44 B , where c44 A,B are the shear elastic con- stants for A ≡ TbFe2 and B ≡ YFe2 . Notice that the value of β2 −∼ 0.2 has been deduced from the εxy 0 experimental results (only at RT [30,31] by using Eq. (38) (remember that εxy 0 depend on t and their values have been given in Sec. 3.3). The small value obtained for β2 (<< 1) could indicate that the elas- tic regime assumption that we are taking for granted would not be valid for large t values (we should note that, if we consider a possible disloca- tion energy, we would arrive to a dependence with the thickness similar to that appearing in Eq. (38) [53,54]; so, both contributions, elastic and disloca- tion energy, should be taken into account and this could change the value obtained for β2). The last term in Eq. (37) is the Ne′el interface contribution to the stress. In Fig. 18,b we plot the RT values of b2 versus TbFe2 thickness, as obtained from Fig. 17. The small value of b2 for the TbFe2 (300 A ° )/YFe2 (1000 A° ) bilayer [17], and the bulk TbFe2 value, b2 v0 = – 0.307 GPa [47,48] are also shown. We represent in the same Figure the fit of these values by the model expression given in Eq. (37). The fit is good regardless the reduced number of experi- mental points available. The parameters giving the best fit are: d2 vε0 = 0.0158 GPa and b2 s = 98 GPa⋅A° , and so d2 v = – 2.3 GPa. For instance, for t = 1000 A° , the contributions to b2 amount: b2 v = – 0.313 GPa and b2 s/t = – 0.1 GPa. We see that the overall volume MEL stress is practically the same as the bulk one and three times the interfacial contribu- tion. Clearly non-linear MEL effects on b2 (rhom- bohedral striction) are small for our bilayers. The same kind of analysis can be applied to b1 , i.e., b1 = b1 v + b1 s t = (b1 v0 + d1 vεxx 0 ) + b1 s t , (39) where d1 v = (∂b1/∂εxx 0 )eq with εxx 0 the misfit strain. This strain is εxx 0 =    + ε0 2 − e0    1000 A° β1t + 1000 A° − ε0 2 � , (40) being e0 the lattice mismatch for the bulk TbFe2 and YFe2 (e0 −∼ – 0.19%) and � ≡ 1000 A° t [c11 A c12 B − − c11 B c12 A ]/{[t(c11 A + 2c12 A ) + 1000 A° (c11 B + 2c12 B )][t(c11 A − − c12 A ) + 1000 A° (c11 B − c12 B )]} and β1 ≡ (c11 A + + 2c12 12 A )/(c11 B + 2c12 B ), where c1,j A,B (j = 1, 2) are the elastic constants for A ≡ TbFe2 and B ≡ YFe2 . From the data available in the literature [47], we obtain that � −∼ 0 and β1 −∼ 0.7. In Fig. 18,a we present the RT values of b1 for our bilayers and for bulk TbFe2 , b1 v0 = – 49.5 MPa [47,48,51]. Notice that, for the TbFe2 (300 A° )/YFe2 (1000 A° ) bi- layer, b1 is negligible compared with the other b1 values. Although the value for t = 300 A° could not be experimentally determined, we should notice that b2 in this bilayer is rather small and that b1 < b2 for bulk TbFe2 and for the rest of the bilayers. Again the fit by model Eq. (39) is good in the available region of thickness. Now the fitting parameters are: d1 ν(+ε0/2 − e0) = – 0.32 GPa and b1 s = 110 GPa⋅ A° . At RT, +ε0/2 − e0 = – 0.16% and then, d1 v = 200 GPa. For instance, for t = 1000 A° , the different contributions to b1 are: b1 ν = – 0.24 GPa and b1 s/t = 0.11 GPa, both being of comparable size, but of opposite sign. This is clearly reflected in the minimum exhibited by b1 at t −∼ 1000 A° , because of the competition of b1 ν and b1 s . For b2 both contributions, b2 ν and b2 s , have the same sign and, so, the competition is absent. It is also clear that for Fig. 18. TbFe2 block thickness dependence of the tetragonal MEL stress parameter, b1 (a), and the rhombohedral MEL stress, b2 (b), at RT. The continuous lines represent the best fits using the Eqs. (39) and (37). The horizontal dotted lines are the values of b1 and b2 for bulk TbFe2 . Notice the asymp- totic approximation to these values of b1 and b2 for the bilayers. J. I. Arnaudas et al. 360 Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 b1 the misfit strain strongly affects the bulk’s value, b1 ν0 , although keeping its sign. The same kind of fitting was done for the other measurement temperatures. The temperature vari- ations of the fitted parameters d1 ν(+ε0/2 − e0) and d2 νε0 are shown in Fig. 19,a. Due to the lack of experimental data about ε0 , we cannot separate the MEL parameters d1 ν and d2 ν for the whole range of temperatures, as we did at RT. The thermal vari- ation of d1 νεxx 0 and d2 νεxy 0 is rather different: \|d1 νεxx 0 \| decrease with increasing temperature and \|d2 νεxy 0 \| remains constant. In Fig. 19,b we show the thermal variation of the interface MEL parameters b1 s and b2 s . Noticeable is that b1 s and b2 s are almost the same, although they are associated to quite different MEL stresses modes, b1 (tetragonal stress) and b2 (rhombohedral stress). This means that ∂b2/∂εxy 0 and ∂b1/∂εxx 0 are quite sensitive to the magnetostriction mode. This is not surprising since both are axial modes. To ascertain the spin dimensionality, D, for the different MEL contributions to the MEL stress parameters we analysed their thermal dependence in terms of Callen and Callen standard theory [27,40] of single-ion crystal electric field magnetostriction. Thus, for volume spins (D = 3) we expect: bi ν(T) = = bi ν,0(T) + (di ν(0 K)εi 0(T, t))Î5/2[ � −1(m(T))] (i = 1, 2 and ε1 0 = εxx 0 , ε2 0 = εxy 0 ), where m(T) is the reduced magnetization of Tb sublattice (m(T) = MTb(T)/MTb(0)); � −1 is the inverse Langevin function; Î5/2 is the Bessel reduced function of first kind. In Fig. 19,a we plot the thermal dependences of d1 νεxx 0 and d2 νεxy 0 and the fit by the above formula, where we have used the values of bi ν,0(T) existing in the literature for bulk TbFe2 [47–51]. The agreement is good, indicating Heisenberg spins. On the other hand, for the interfacial MEL stresses we tried two scalings: bi s(T) = bi s(0 K)mα(T) (i = 1, 2) with α = 4 at low temperatures and α = 2 in the high temperature range. This is the scaling predicted for spin dimension D = 2; a scaling with α = 3 for low temperatures indicates an spin dimen- sion D = 3. The reduced magnetization, m(T), was deter- mined by subtracting from the Ms,TbFe2 (T), the weighted iron sublattice magnetization, which was assumed to be the same as for YFe2 [18]. In Fig. 19,b we plot, as an example, the scaling of the interfacial MEL stress parameter bi s(T) (i = 1, 2) with m4(T) and m3(T) (notice that the temperatures are suffi- ciently low and the m2(T) regime does not appear). The agreement is perhaps better with the m3(T) scaling at higher temperatures, but at the lower ones the scalings are equally satisfactory. Therefore it is difficult to decide about the spin dimension- ality at the interfaces; and whether the spins are confined or not to the growing plane. 10. Conclusions The relevance of the magnetoelastic energy asso- ciated to the interfaces and the epitaxial stress dependence of the volume magnetoelastic stress, which implies a non-linear behavior of the volume magnetoelastic energy, has been investigated by studying the magnetoelastic behavior of two series of Ho-based superlattices. Measurements of the basal plane magnetoelastic stress isotherms in HonHo /Lu15 and in Ho10/YnY SL’s, between 10 K and above the Ne′el temperatures, allowed us the determination of interface magnetoelastic stresses which are of the order, even higher, than the volume ones. The modification of the bulk’s magnetoelastic stress due to the epitaxial strain is also found to be important in both series. Moreover, the thermal dependence of the magnetoelastic stress is in agreement with a single-ion crystal field character for the magneto- elastic interaction in the studied superlattices. In Dy/Y and Er/Lu superlattices we have also per- formed MEL stress measurements, obtaining the Fig. 19. Thermal dependence of (a) the non-linear MEL stresses: d1 νεxx 0 (■) and d2 νεxy 0 (❐), and (b) interface MEL stresses: b1 s (●) and b2 s (❍). All of them were obtained from the best fit to b1 and b2 . The dotted lines are the fits according to Î5/2 Callen and Callen law [27]. The dash-dotted line is the thermal dependence for the 2D-spin model [40]. Magnetoelastic stresses in rare-earth thin films and superlattices Fizika Nizkikh Temperatur, 2001, v. 27, No. 3 361 different MEL stress contributions for the Dy/Y samples, but not for the Er/Lu, where incomplete saturation leads to inconclusive results. The latter situation also takes place for Ho/Tm SL’s; never- theless, MEL stresses clearly show up the competi- tion between the axial and in-plane anisotropies of Tm and Ho layers. We have measured the magne- toelastic stress in TbFe2(t)/YFe2 (1000 A° ) (with t = 300, 600, 1000 and 1300 A° ) epitaxial bilayers. The sample with t = 300 A° is the only one which does not show a measurable MEL stress. For the rest of the bilayers, we have been able to determine all the parameters accounting for the different MEL distortions allowed by the cubic symmetry. We have shown that the epitaxial stress strongly modifies the value of the MEL tetragonal stress, which, due to this, increases more than three times its value as compared with the bulk TbFe2 one for a given range of t values. An interface contribution to the MEL stresses has been also determined. As concerns to the origin of both volume and interface contributions, the thermal dependence of the MEL parameters pointed out to a single-ion crystal-elec- tric-field origin for the volume stress contribution and a not well-defined 2D behavior for the spins at the interface. 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