The understanding of rare-earth metal nanostructures has advanced significantly over the decade since the first materials were grown. There can be no doubt that the present state of development remains rudimentary, however, and that a number of the most fundamental questions still remain unanswered. In writing this review we have nevertheless been aware that the general context and relative interest of the field for future research has become more clearly defined over the intervening years. It is therefore fair to ask whether the research field remains vital relative to contemporary alternatives. The following concluding comments attempt to address these questions.
It is undoubtedly true that the present capability for synthesizing new magnetic nanostructures from the rare-earth metals greatly exceeds the limited range of materials explored to date. In assessing the future opportunities we note that entire areas of compelling interest are eliminated by the constrained selection of materials in current use. Almost all completed studies have employed c-axis materials grown from a few selected metals, specifically Gd, Dy, Er and Ho, mainly with Y or Lu, and occasionally each other. Of these restrictions in scope, the limitation to (0001) growth perhaps constrains conceptual advances the most. Only a single investigation has explored magnetic properties of nanostructures grown along the a or b axes, and this with a view to fundamental matters of the induced exchange between spins. This remains the case despite the accessibility of growth procedures starting from sapphire for the controlled growth of (1100), (1120) and (1102) materials, the latter at least with tunable vicinal
72 C.E FLYNN and M.B. SALAMON
tilts (see sect. 3.4). As an example note that for any crystal orientation other than c- axis growth, magnetostriction in the helical phases is no longer fully constrained
locally.
Instead the magnetic state must be dressed in frozen phonons that describe the oscillatory state of strain, as indeed must also be the case for ferromagnetic states of superlattices grown in these orientations, and to relative degrees that offer a probe of magnetostrictive behavior on the atomic length scale. A perceptive application of such ideas surely offers a broad latitude in the design and synthesis of novel ordered states that have previously been inaccessible.
Rare-earth nanostructures also serve as tools to study fundamental problems in the magnetism of materials in various restricted geometries. There remains the question as to the possibility of creating 2D magnetic nanostructures that conform to an idealized Hamiltonian for which the theory is tractable and interesting. Some distance has been covered in the study of thin layers that approach the 2D limit, and a number of investigations relate to the way magnetic coherence is established in rare-earth superlattices at their antiferromagnetic and ferromagnetic transitions, but the connection with the expectations of idealized theoretical models remains to be established. Given the robust and predictable character of their magnetic moments and their interactions, these lanthanide metal systems have much to offer in the area of 2D magnetic behavior. Much the same can be said for the interlayer coupling; the way in which antiferromagnetic coherence is established at the Nrel temperature remains, as noted above, quite poorly understood at the time of writing.
These nanostructures also offer an attractive approach to several fundamental problems of epitaxy. Of particular interest is the interaction of epitaxial strain with magnetic behavior, including stress relaxation during growth, and the relaxation at chosen temperatures of magnetoelastic strains introduced and selected by means of a magnetic field. The study of fluctuation, domain and surface processes in clamped magnetostrictive phases, like the analogous processes in ferroelectrics, remain badly understood in the face of considerable technical and scientific interest in connection with potential applications to switching, transduction and manipulation.
It may be unnecessary to note in addition that attention to date has largely been confined just to the magnetic properties of a few heavy lanthanides in simple nanostructures.
The remaining metals, including the light lanthanides, are at least of equal interest, and promise still less conventional properties owing, for example, to the thermal excitation of modified core configurations that may be expected with confidence when a single 4f subshell is only partly occupied. Striking behavior has begun to be reported also in the electronic properties of rare-earth nanostructures where magnetotransport, for example, has exhibited anomalies that point both to strong modifications of bandstructure and important geometrical effects on interfacial scattering processes. As yet only few, isolated facts are available to prompt further activity in these new directions.
With the investigation of so many phenomena scarcely begun it seems appropriate to conclude this review by reminding the reader briefly of those features of the field that are now well established.
SINGLE-CRYSTAL NANOSTRUCTURES 73 First, the means are available for synthesis of rare-earth nanostructures in a variety of crystal orientations, and for a variety of materials choices, all with interfacial mixing restricted to very few monolayers. The available methods are adapted to the preparation of structures of order 1 ~tm thick and lateral dimensions of inches, and the crystal quality may be regarded as quite satisfactory for metals, at least in the sense that growth defects play a limited role in the phenomena of most interest.
Second, the results available so far serve to confirm the role of induced exchange through the conduction electrons as the primary interaction that determines the ordering characteristics o f the lanthanide moments. This has been most clearly established for the Dy/Y system, for which the helical magnetic waves propagate through many layers of nanostructures and so afford a precise determination of the periodicity and range of the interaction along the c-axis. In this same system alone, the reduced interaction range in, rather than normal to, the basal plane, has been established by experiments on b- and c-axis samples.
Third, and finally, it has been established that the lanthanide magnetism is in general remarkably robust, and in particular is insensitive to the interfaces, even in crystals only a few atomic layers thick. A reservation of critical importance in this regard is the central role o f the state of strain in the description of the magnetic behavior. Specifically it has been established for Dy that epitaxial strains ~-t-2% are sufficient to double the Curie temperature or completely suppress the ferromagnetic phase. The twin assets of robustness and strain sensitivity make these materials at one time both ideal systems with which to explore epitaxial effects, and attractive models with which new states of magnetic order may be designed and synthesized.
Acknowledgements
We are greatly indebted to numerous colleagues who have contributed to our understand- ing of the rare-earth metals and rare-earth nanostructures. These include Drs. R.W. Erwin, M.V. Klein, and J.J. Rhyne, and our past close co-researchers J. Borchers, R. Beach, R.-R. Du, B.A. Everitt, B. Park and E Tsui. To the latter two we owe a special debt for permission to include researches which remain, as yet, published only in their theses at the University of Illinois at Urbana-Champaign. The bulk of our research in this area, which forms a substantial part of this review, was supported by grants from the National Science Foundation. Most of the synthesis detailed in sect. 3 was funded by the Department of Energy, as was a portion of the synchrotron radiation work from our laboratories. Thanks are owed to the Office of Naval Research for funding part of the recent work on thin-film magnetostrictive behavior, which is a central theme of the review.
Appendix: Self-energy and pair interactions in linear response
A common problem has two bodies simultaneously perturbing a linear medium. Examples are two neighboring charges in a dielectric continuum, two bowling balls on a mattress,
74 C.P. FLYNN and M.B. SALAMON
or two spins interacting through an electron liquid - the problem of interest here. The term linear has a specific meaning. If bodies 1 and 2 at rl and r2 separately cause linear perturbations xl ( r - rl) and x 2 ( r - r2), then acting together they cause a perturbation
x ( r ) = x l ( r - r l ) + x 2 ( r - r 2 ) (A1)
for all rl and r2. Here we are concerned with the self-energies El and E2 of interaction between the separate bodies and the linear medium, and the total energy Et when the two bodies interact with the medium simultaneously. The interaction energy is then defined
a s
El2 = Et - E1 - E2. (A2)
Thus defined, the interaction energy is negative when the energy is lowered by the interaction.
The calculations that follow employ a model in which an electronic medium is disturbed by point charges ql and q2. Suppose that in response to a charge q at r = 0 the disturbed medium produces a potential qv(r) proportional to q. Then the net electrostatic potential is
V' = q + q v ( r ) + Vo,
(A3)
F
with/10 the interior potential in the absence of perturbing charges. Because the energy of the medium itself must depend on the square of the disturbance we may write it ½fly2(0), and the net energy E associated with the interaction between the charge and the material is
[
E = qv(O) + ½/302(0) = ½fl v(0) + 2fl' (A4)
This shows that the system minimizes its energy with v ( 0 ) = - q / f l with a net energy relaxation of
q2
E = 2fl - ½flY(O). (A5)
By differentiation the energy change when q ~ q + 6q is
r E - qr~q - v(O)rq. (A6)
The result is just the energy of the added charge 6q in the existing potential v(0), because this relaxed state is an energy minimum.
SINGLE-CRYSTAL NANOSTRUCTURES 75 We need to calculate the relaxation energy when ql and q2 are placed at rl and r2. For convenience suppose first that fractions x of the charges are in place, and the system is relaxed with a net potential
V(r) = xqlv(r - rl) + xq2v(r - r2). (A7)
Because the relaxed system is at an energy minimum, the energy change dE, when added charges dxql and dxq2 are placed at rl and r2, is Vldxql + V2dxq2, with /I1, /I2 t h e potentials at rl and r2 due to external charges. Hence
d E = (ql [ ~ + qlu(0) + q2u(a)]+ q2 [ ~ + q2u(0) + qlu(a)])x dx. (A8) with a = I r l - r 2 [ . By integrating from x = 0 to x = 1, the total relaxation energy for the full charges is now found as
El2 = g(ql + q22) v(0) + qlq2 + u(a) . (A9)
Here the first term is the sum of the two self-energies (eq. A5), and the second term is the desired interaction energy:
E12 = qlq2 ( l + u(a)) .
(A10)
Note that the result is just the energy needed to place the second charge in the medium with the material frozen in its relaxed configuration about the first charge (or vice versa).
It includes both the direct interaction and the material-mediated interaction.
This prescription is quite general in linear response and is readily adapted to other cases. In the case of a medium of dielectric constant e, for example, the induced polarization gives ql q2 [e -1 - 1 ]/a and the interaction energy becomes qlq2/ea. When this is augmented by the response of charged diffusing species, the induced term becomes qlq2[e-le -rDa- 1]/a and the interaction energy is qlq2e-rDa/ca with t¢~ l the Debye- H/ickel screening radius. For charges in a simple electron liquid the required dielectric response is the Lindhart function (see below).
These considerations are readily extended to obtain the interaction when two dis- joint distributed charges Pl (r) and p2(r) interact. For linear response the energy (A10) is
merely summed over all 6-function pairs to find
f f darldar2p(rOp(ra)[Ira-r~l-l+o(Ira-r~l)].
( A l l ) In a system with many disjoint perturbations the excess of the energy over the sum of the separate self-energies is just the sum ~ E/j of pair terms like eq. (A11) over all pairs/j.This is how the sum over pairs enter expressions for the RKKY energy.
76 C.R FLYNN and M.B. SALAMON
Our present interest is in the case of spins interacting through an electron liquid, for which the direct term is just the dipolar interaction. The material-mediated term depends on the interaction between the core spin S and the conduction electrons. It is usually assigned the model form
.(1) . S a ( r -
rl)p(rl)
d3rl, (A12)E = Jsf $
with s the conduction electron spin and
p(r)
the electron density. Suppose that p is the uniform initial density of conduction states with s parallel to a core spin S1. From eq. (A12) the density disturbance is6pup(r)
= ~Jsf S l p u(r), (A13)in which
u(r)
is the disturbance for a perturbation ofunit
energy and the 1 isIsl.
The spin-down disturbance is the same but negative. From eqs. (A10), (A12) and (A13) the induced interaction with a second spin $2 having interaction j~) at a, and including the contribution of both spin senses, is= 1 S x .(1).(z)P2U(r) (A14)
E l 2 2( S l " 2)Jsf Jsf P
This is the RKKY result. It depends only on the interaction strengths ~ .(1) ~iJsf p at the two sites and
the fractional
density disturbanceu(r)/p
per unit interaction energy, which is the same for the full electron liquid as for either subband. As mentioned above, this energy is supplemented by the direct term, which is just the dipolar interaction.References
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