Strain relaxation

OTTO]

4. Magnetism of epitaxial rare-earth crystals

4.2.1. Strain relaxation

The fact that real epitaxial layers of sufficient thickness tend towards bulk behavior demonstrates that the elastic constraint described above is, in practice, relieved. Anelastic processes therefore take place in the growing film, or in any event prior to examination, for example during cooling. That this must necessarily happen is apparent upon energetic grounds. Suppose that an epitaxial strain e originates at the interface between the crystal and its template. The elastic strain energy per unit area is then

cije2d/2,

for a film of thickness d, in which c~ is the appropriate elastic constant. Suppose also that the strain e occurs at an interface for which the strain e0 would correspond to perfect registry, so that e - e0 represents interfacial relaxation. Then the interfacial misfit creates (e-eo)/Ib] interfacial dislocations per unit length, with b the Burgers vector.

This introduces an interfacial energy per unit area of

2wle-eoHbl,

with w the line energy per unit length of dislocation. By minimizing the sum of the two energies with respect to the strain one finds that the epilayer exists at a uniform equilibrium strain e = e0 when d <dc =

2w/[b[cueo,

and at the smaller value e

=2w/Ib]co.d

for d > de.

The critical thickness de, in this simple approach provides the upper bound on the fully pseudomorphic epitaxial crystal of sect. 4.1. Thicker epilayers still remain uniform epitaxial crystals in this approximation, but are less strained.

During the growth of real crystals the strain relief must take place continuously, and some of the anelastic strain most probably remains in the epilayer rather than in the interface. Furthermore, the final state of strain is observed to depend on the growth temperature (Dodson and Tsao 1987, Tsui and Flynn 1995, Tsui 1992) which indicates

SINGLE-CRYSTAL NANOSTRUCTURES 27 that an activated component is present in the kinetics. While the mechanisms of plastic relief themselves are of considerable interest and technical importance, the specific inhomogeneous configurations that are created in this way have no special significance.

Nor are the detailed processes well understood.

This brief background in strain relief can serve to illuminate the resulting effects on thin-film magnetism. The following point is Of particular importance. Suppose that the crystal does, as suggested above, achieve and maintain a specific state of uniform strain e as a result of the growth procedure. Then, to a good approximation, the temperature dependence of its magnetism must just be the section of the epitaxial phase diagram that corresponds to the particular orientation and strain. Indeed the epitaxial phase diagram is obtained precisely from measurements of this type, but for thin, unrelaxed epilayers. A critical question is whether or not thicker, partly relaxed epilayers exhibit identical behavior for the same actual strain e. If this is the case it indicates that the interfacial registry that defines the partly relaxed misfit strain e is independent of any applied magnetic field. Otherwise the field dependence of the misfit would cause the state to pass along a different path across the surface of state as the temperature changes. This discussion has neglected differences of expansion coefficients and anelastic evolution subsequent to growth, but these types of elaborations can be incorporated as needed. The distinction identified here is important in connection with the possibility that mechanical damage accompanies magnetic cycling. There is a further connection with the metastable magnetized state that can be induced in Dy films more than 76 A thick grown on Y substrates.

4.2.2. Effect on the phase diagram

Available results indicate that partial relaxation in fairly thin films does leave them in reasonably uniform states of strain, and therefore must largely be interfacial. The solid points in fig. 7, other than those for films unrelaxed on Lu and Y, actually were taken from partially relaxed thin films (cf. fig. 4). The fact that they conform accurately to the surface defined by unrelaxed systems indicated that their state of strain remains frozen at the part-relaxed value specified by the location of the points along the e-axis. Thus, they approximate epitaxial crystals but with modified interfacial registry.

Most revealing among available data for a wider range of behavior are the critical fields of b-axis Dy grown epitaxiatly on Y (100) with various thicknesses (Tsui 1992). Figure 9 shows how the critical field depends on temperature and film thickness for Dy grown along the b-axis. An approximate continuous surface is drawn through the scatter of the data points. It indicates that the first 50 ]~ has a special temperature-insensitive behavior with a critical field near 10kOe that is relatively large. Films over 100~k thick have critical fields that vary with temperature very much like sections of the c-axis epitaxial surface (fig. 7), with e positive. The results for the thinnest films suggest that relief of the pseudomorphic strain sets in at about 50 A. An alternative explanation that interfacial effects cause strong changes of magnetic state below 50 A can be discounted on the basis of other results described in sect. 4.3. For the thicker b-axis films note that the a-c growth

28 C.P. FLYNN and M.B. SALAMON

<

Fig. 9. Magnetic phase diagram for b-axis Dy/Y superlattices and b-axis films up to 1 ~tm thick. The lower surface (solid lines) separates the helimagnetic and fan phases, the upper surface (dashed lines) marks the saturation fields. The bulk limits are shown in the foreground.

plane is anisotropic and thus quite different from the isotropic in-plane strains of the c-axis case. The existence of close similarities between c-axis and part-relieved b-axis magnetic properties is in this respect remarkable. A straightforward explanation of the data would be available if the b-axis films thicker than 50 A were relaxed to a strained configuration sufficiently similar to that of the partly relaxed basal plane case that the magnetic behaviors also became similar. Detailed exploration of these possibilities has been hindered by the difficulty o f determining the precise strain states of thin epilayers on thick templates.

Similar but less complete results are available for Er films (Tsui 1992). The Nrel temperature changes at most by ~ I K as the film thickness is reduced to 400A.

Results showing the change o f the ferromagnetic transition for Er on Y are presented in fig. 10a. There, the critical fields as functions of temperature for two Er crystals 1750A and 9500A thick are compared with the bulk behavior. Neutron scattering data for the 3950,~ film is shown in fig. 4. Evidently ferromagnetism is never recovered in these films for temperatures above 10K, although the precise behavior

SINGLE-CRYSTAL NANOSTRUCTURES 29

35.0

30.0

25.0 0

"0 20.0

"~ 15.0"

10.0

5.0

0.0 8.0

15.0

12.o I

o 9.0

"o

:2, 6.0 'E rj

3.0

0.o

Er

, i I , i ,

2o.o 40.0 0o.o

T e m p e r a t u r e (K)

80.0

2 0 K

(b)

0,0 4.0 O,O 12.0 16.0 20.0

Film Thickness

(A)

"103

Fig. 10. (a) Critical field vs tempera- ture for bulk Er, 1750]~ and 9500A films; (b) dependence of the critical field at 10K (solid points) and 20K (open circles) on film thickness.

for films l~tm thick may be sensitive to details o f the growth procedure. Figure 10b shows the available results for c-axis films. Both D y and Er exhibit a complete suppression o f ferromagnetism in the thinnest films, as detailed above. However, they differ considerably in the film thickness required to restore bulk properties, as noted in the discussion o f domain effects above. As fig. 10 makes clear, a 3 kOe field is required to restore ferromagnetism at 1 0 K in an Er film as thick as l~tm, while

30 C.P. FLYNN and M.B. SALAMON

a comparable field induces ferromagnetism in a Dy film only 20nm thick at 30K (Kwo et al. 1988).

4.2.3. Effects on the magnetic state

Thus far we have used the shift of phase equilibrium to monitor the influence of epitaxial strain on the competing magnetic phases. Each phase undergoes changes of energy and structure caused by the existing strain. These alter the balance of the free energies and thereby shift the phase boundaries. The quantities of direct physical interest are of course the individual free energy shifts themselves, and the structural changes of the phase from which they arise. In certain cases it is possible to probe structural changes directly, in order to obtain a direct relationship between epitaxy and structure. This is most readily possible for the helimagnetic phases, for which neutron scattering provides a direct probe of both the wavelength and amplitude as a function of tempera~re. Selected results for Dy and Er are discussed in what follows to clarify the present understanding of the strain- induced changes.

In the helimagnetic phase of bulk Dy the turn angle per layer is about 43 ° near the Nrel temperature of 178 K. With decreasing temperature it decreases smoothly to about 26 ° at the Curie point, where it falls immediately to zero in the ferromagnetic phase.

Epitaxy tends to increase the turn angle and to suppress its fall to zero. Even a 200 Dy film remains in the helimagnetic phase at 80 K (5 K below the bulk transition), with a turn angle of 27 °. In a more extensive study of Er films (Borchers et ah 1988a, 1991) the phase angle in films as thick as 9600 A are always larger at a given temperature than in bulk Er. Added evidence of the influence of epitaxial strain is that 600 A of Er grown on c-axis Lu, which gives a basal plane compression, has turn angles systematically smaller than the bulk at each temperature (Beach et al. 1991). Magnetic X-ray scattering confirms these results and shows that Y enhances certain Er lock-in states while Lu suppresses them (see below) (Tanaka et al. 1995).

Efforts have been made to reproduce the observed trends of structure modification by epitaxy using models for the several terms in the free energy of the helimagnetic phase. The main contributions are the exchange energy, the magnetoelastic energy and the Zeeman energy. Of these, the last two can be represented reasonably well by means of measured moments and magnetoelastic data. Unfortunately, the exchange energy is not well known. It is often simulated by the simplest theoretical model that predicts helimagnetic behavior, which employs three exchange constants: J0, coupling nearest neighbors within a plane, J1, nearest-neighbors on adjacent planes, and J2, next-nearest neighbor planes (Cooper 1972). When bulk turn angles are used to fit the model, the exchange constants for the first and second planes turn out to be strongly temperature dependent (Tsui 1992). Further, even within this 3-J model, the turn angles vary from plane to plane due to end effects in thin-film structures (Jensen and Mackintosh 1991, Bohr et al. 1989). In reality the couplings predicted by the RKKY interaction have a long range, and with the awkward feature of near-cancellation among many terms with

SINGLE-CRYSTAL NANOSTRUCTURES 31

opposing signs. A useful basis on which to build an analysis of the thin-film effects remains as yet to be formulated.

Neutron scattering measurements such as shown in fig. 4, reveal similar modifications of wave vector in the CAM phase of Er. As

Q(T)

changes with temperature in bulk Er, as described in sect. 2.2, its magnitude passes through values that are commensurate with the lattice periodicity itself, and lock-in occurs at various commensurate structures.

These are shown as a solid line in fig. 11. Each commensurate structure corresponds to a specific magnetic wave in the lattice. These are longitudinally polarized at higher temperatures, develop a helimagnetic component at intermediate temperatures, and have a cone structure at low temperatures. The behavior of

Q(T)

in thin films and superlattices gains added interest from the opportunity to examine the way epitaxy modifies these well defined magnetic structures and changes their coupling to the lattice. Lock-in states of a similar character can occur for spiral antiferromagnets such as Dy and Ho. They are, however, weak for Dy because its small in-plane anisotropy blurs spin-slip structure, and the stronger lock-in behavior of bulk Ho has not yet been traced into epitaxial films. The present illustrative examples are therefore confined to the case of Er.

Results for the turn angle of two Er (0001) films on Y grown, respectively 860A and 9500 A thick are described here. The strongly perturbed periodicities of (i) the (0002) satellites from the c-axis modulation (CAM: open circles) and (ii) the superposed basal plane spiral (solid circles) at low temperature, are shown for the 860,~ crystal in fig. 1 la.

For comparison the bulk behavior is indicated by the solid line. It is apparent that the wavelength of the CAM is clamped in a narrower range of large q values by the epitaxial constraint. A point of considerable interest is that the specific bulk spin configurations marked 2/7 [4 up-spins and 3 down-spins per magnetic unit cell] and 4/15 [454T354T]

for the bulk are greatly enhanced in the epitaxial film, although others appear to be suppressed. In the c-axis field-dependent magnetizations, shown in fig. 1 lb, these two states (which have net magnetic moments) are marked B 1 and B2, coinciding with sharp features. In thicker epitaxial films the periodicities are less perturbed from bulk values, and the wavelengths of the CAM and basal plane spiral appear more nearly equal. This is illustrated by the example of a 9500 ,~ Er epilayer on Y in fig. 1 lc. The resulting magnetic signature shown in fig. 1 ld consists of sharp features under conditions that correspond to the 2/7 and 6/23 lock-in states.

Recent results by Tanaka et al. (1995) confirm the observations for Y substrates and reveal interesting differences caused by basal plane compression on Lu substrates. The lock-in states are much weaker and the turn angles are uniformly smaller. Lock-in to Q = 0.25 c* takes place just below 30 K.

Without question the results summarized here afford just a first glimpse of a rich field in which the magnetism of epitaxial films responds in an interesting and sensitive manner to the epitaxial constraint. The actual state of strain in this limit depends on both the film thickness and the growth conditions. In tum the magnetic state must depend on the state of strain and other factors that may influence, for example, the magnetic domain structure, in addition to the natural variables of field and temperature. An eventual complete description must include the statistical behavior of the spin-slip system.

32

20.0

¢

8.0

4.0

0.0 0,0

55.0

52,0

49.0

-1

46,0

<¢

43.0

4 0 . 0 0.0

C.P. FLYNN and M.B. SALAMON

0 2 B 2 B1

I I

16.0 20.0 30.0

i

Ca)

f I

40.0 50.0 60.0

Temperature (K)

(b)

F

^{- 612s}

J 114

5 / 2 1

I i I t

2o,o 40.0 60.0 60.0

Temperature (Ix') ^Fig. l lab. (a) Magnetization curves and (b) phase advance per atomic layer for an Er film of thickness 860 ,~.

4.3. The thin-film limit

As the thickness of a film is reduced, the pseudomorphism is improved and the state of strain generally becomes more uniform. At the same time, however, the definition of magnetic phase structure may be complicated by boundary effects, and the sharp symmetry distinctions among alternative phases is blurred. For a recent review of thin- film magnetism see the article by Falicov et al. (1990), which deals mainly with transition

S I N G L E - C R Y S T A L N A N O S T R U C T U R E S 33

12

0

b~

~ 3

0 0 56

~sz

-K

v 48

~ 4 4

I I I I I I I I I

D3 B3 D1 BI

'o 0.5 kG

i I I I I I I I I I I

2 0 4 0 6 0 8 0

T e m p e r a t u r e (K)

I I I I I I I I

~o

======================== 5 / . . . / 1 ~ 4/15 9

I

(d)

40 t I I I I I I I

0 20 40 60 80 100

T e m p e r a t u r e (K) Fig. l l c d . S a m e as fig. 1 la, b, for a 9 5 0 0 / k film.

LO0

metals. The theory of phase transitions and dimensionality is reviewed by Stanley (1971).

Here we are concerned instead with particular effects that are important for rare-earth thin films.

As a specific example to clarify the effect of reduced film thickness consider the helimagnetic phase of Dy, which has a c-axis wavelength of about 30.~. The structure has a net moment, and is no longer clearly an antiferromagnet, when the film thickness

34 C.P. FLYNN and M.B. SALAMON

falls below 30 A. It is reasonable to anticipate, in addition, that the end effects at the two surfaces of the film change the turn angle and make it vary with depth. Still worse, the distinction between a helical phase and a ferromagnet is almost entirely lost as the film thickness is further reduced to about two layers in a typical lanthanide with a magnetic period of 10 atomic layers. Finally, the reduced dimensionality has two profound effects on magnetism by first confining the magnetic interactions to spins located in a thin slice of material, and second by ensuring that all moments exist in an environment of perturbed crystal field. In addition it is likely that continuity in monolayer films is broken by the terrace structure of the original substrate template. These various independent phenomena combine to determine the magnetic behavior in what we term the thin-film limit. The behavior of films has obvious consequences for superlattices which comprise films coupled through spacer layers. A further fundamental motivation is that couplings among successive dilute layers at constant spacing offers the best hope for a future precise probing of the magnetic response function.

Given the severity of the geometrical and physical constraints suffered by the thinnest films, the effect on the magnetism appears remarkably mild down to monolayer film thicknesses. This is the point made in sect. 1 that the magnetism is generally robust. In reviewing these facts we first present relevant results for several metals, together with brief assessments in the light of the available theoretical modeling. The connection between these separate bodies of information is not fully formulated at the time of writing because the development of the subject matter is in its infancy.

4.3.1. Gadolinium

This is an interesting metal for the way the thin film tends towards bulk behavior at large thickness and to the two dimensional limit at dilute coverage. Bulk Gd has its axis of easy magnetization along c (see sect. 2) but the spin moment is canted from c by a large and temperature-dependent angle. There is only short-range order on a 0.1 ~tm scale in the basal plane. No accepted explanation of this behavior is available, but d states that couple to both the 4f moment and the lattice may be responsible.

In thin films the demagnetizing factor, which favors basal plane magnetization, overcomes the c-axis bias O f the bulk, and gives rise to easy alignment in the basal plane rather than perpendicular along c. Neither the film thickness nor the way this crossover takes place has ever been investigated. It may depend on the thickness d of the film relative to the correlation length of the in-plane bulk order.

Further changes caused by interfacial effects and reduced dimensionality may be anticipated as d is decreased to monolayer levels, even when the environment remains nonmagnetic but lanthanide-like. The fact that the thinnest films have spins oriented in the growth plane make them candidates for description by the XY model. Like the Heisenberg model, theory predicts that no true long-range order exists even down to 0 K for interacting systems of XY symmetry, although a Kosterlitz-Thouless transition is expected (Kosterlitz and Thouless 1973). These predictions have been difficult to test in practice because the magnetic dilution of 2D systems causes the signals to fall below

SINGLE-CRYSTAL NANOSTRUCTURES 35

102

I01

,o o

I0 -I

I I

9ML 5 M L ~

T(K)

300

200

100

"1~

0

~ e Tp

• Tg

+°t

+ ~0, ~.5, ,1.0 1.5 2.0 4.0 6.0 8.0 Bulk

Thickness (monolayers)

(a) (b)

Fig. 12. (a) Temperature dependence of the basal-plane magnetization of Gd films of various thickness. The applied field is 50 Oe in each case; (b) the thickness dependences of the Curie-Weiss temperature Tp and spin-glass temperature Tg. At low coverage, both temperatures increase as powers of the film thickness.

the detection level of steady state measurements (Falicov et al. 1990). The lanthanides have localized moments and interactions that can be parameterized simply in a way that resembles the simplified theoretical models. Also, MBE offers a means for preparing materials with many high quality layers of tailored structure spaced by a distance that exceeds the interaction range. Therefore the changes of magnetic behavior that occur as the Gd in rare-earth multilayers is reduced to monolayer thickness is of fundamental interest. From measurements of ac susceptibility Fahrle and Lewis (1994) have reported that Gd grown on W ( l l 0 ) has lost its ferromagnetism at a thickness of 7ML, while Li et al. (1993) find even a single Gd monolayer to be ordered.

The introduction by Tsui, Han, and Flynn of methods that allow the magnetism of rare earths grown on artificial mica to be probed down to the submonolayer level provides new opportunities for the study of 2D effects (Tsui et al. 1993b). Figure 12a from Tsui, Park and Flynn (1995) shows the magnetization produced by a 50Oe field as a function of Gd film thickness and temperature. Even in the 2ML thick film the Gd magnetization approaches saturation at temperatures close to the bulk Tc of 290K.

This displays the robustness in a remarkable degree. Films 1ML and less in thickness exhibit markedly reduced fractional magnetizations which pass through a maximum at a lower temperature Tg. It will become evident below that this is associated with the spin glass behavior. Near 1ML coverage there is a well defined change from saturation magnetization below a specific Curie temperature to much smaller and temperature sensitive magnetizations.