Neutron Scattering from Magnetic Materials part 14 pot

If strong enough, such terms mightproduce a rotation of the polarization vector around the interaction vector different fromπ which can be only determined by spherical neutron polarimetr

Inelastic neutron polarization analysis 389

Fig 16 Energy dependence of I (ω) = I+(ω) − I−(ω) and I (ω) = I+(ω) + I−(ω) in CsMnBr3close to the

antiferromagnetic point (1/3, 1/3, 1), at a temperature of 9.2 K and a field of 4 T.

The difference I (ω) = I+(ω) − I−(ω) represents the polarization dependent (chiral)

part of the scattering ( I (ω) ∝ Mch) As previously discussed (see Section 2.2), the chiral

contribution should be an odd function of ω at ω/ kBT 1, corresponding to the relation

ex-implying γC≈ 0.84 These values are found in good quantitative agreement with the Monte

Carlo calculations of Kawamura [62], thus confirming the chiral universality class

Chiral fluctuations in the itinerant helimagnet MnSi. MnSi is a very interesting

itinerant-electron weak ferromagnet with a Curie temperature TC ≈ 29 K [68,69] The

mag-netic structure in zero field is a long-period spiral described by the propagation vector

k= (τ, τ, τ) with τ ≈ 0.017 and equivalents deduced by symmetry [70] Theoretical work

by Bak and Jensen [71] have shown that the spiral order in MnSi was due to the existence

of a sizable Dzyaloshinskii–Moriya vectorial interaction term, resulting from the trosymmetric arrangement of Mn atoms in the cubic unit cell Consequently, the spiralarrangement in MnSi must be very different from that arising in more conventional heli-magnets with competing exchange interactions because in MnSi it should be single handed,

noncen-right or left depending on the sign of the Dzyaloshinskii–Moriya vector DDM= DDMˆk In

MnSi, LPA measurements [70] have clearly demonstrated that the spiral is right handed,

even in zero field The chiral nature of magnetic fluctuations above TChas been recentlyinvestigated by Roessli et al [72], by measuring the difference between inelastic spectra

taken with P0parallel and antiparallel to the scattering vector Q (that is to say associated

with xx longitudinal components) Typical results are given in Figure 17 for temperatures

390 L.P Regnault

Fig 17 Difference neutron counts for incident polarization parallel and perpendicular to Q in MnSi at T= 31 K

and T = 40 K, showing the strength of chiral fluctuations close to TC.

T = 31 K and T = 40 K which, among others, show the growing of chiral critical

fluc-tuations as T approaches TC Contrary to CsMnBr3, in MnSi the chiral flucfluc-tuations can

be directly observed without disturbing the system by the application of a magnetic field(intrinsic chirality versus field-induced chirality) In most cases, the chiral contribution isdeduced from LPA measurements of longitudinal components of the polarization along the

scattering vector, namely xx or ¯xx However, according to Section 2.3, the chiral terms

can as well be obtained directly from SNP measurements of the transverse components

P yx or P zx , which both depend on Mch Such a determination has been successfully taken on the elastic contributions in MnSi below TC[73]

In this section we will show how the SNP may help to solve the full problem, namelythe determination of the complete set of correlation functions involving the nuclear andmagnetic degrees of freedom The spin-Peierls compound CuGeO3appeared to be a verysuitable candidate to perform such a determination owing to the strong spin–lattice inter-actions existing in this material In Section 3, it was shown that the nuclear structure factor

associated with the hypothetic hybrid mode was very small, amounting to at best 0.3% of the main magnetic structure factor at q= π However, if the hybrid correlation function be-

tween the magnetic and structural degrees of freedom was strong enough, it might give rise

to a sizable NMI term and, thus, a small rotation of the polarization away from P0should

be detected Maleyev [9] and Cepas et al [76] have considered theoretically an S = 1/2

spin-Peierls system described by dynamical Heisenberg (H) and Dzyaloshinskii–Moriya

(DM) interactions, both given by the generic term

i,j



α,β V αβ (R j− Ri)S i α S j β, where

V αβ (R j− Ri) = Jβ (R j− Ri)δ αβfor the former and

γ D γ (R j− Ri)ε γ αβfor the latter,

where the D γ coefficients are the components of the Dzyaloshinskii–Moriya vector whichdepend on the position of all ions involved in the superexchange paths Maleyev [9–11],

by using the standard perturbation theory, has found that the INMI terms are related to the

Inelastic neutron polarization analysis 391three-spin susceptibilities (or, equivalently, to the three-spin correlation functions), follow-ing the expression

In (57), the magnitude of NMI terms appears controlled by the first angle bracket which

may be nonzero if the lattice excitations modulate both Rn and the V components For a

system presenting no long-range order, as it is the case for CuGeO3, the scalar berg exchange terms give no contribution to the NMI and only the Dzyaloshinskii–Moriyaterm can contribute According to the general theory [10,11,76], the expression giving the

Heisen-INMI terms contain factors 1/(ω2− ω2

λ ), where ω λ is the phonon energy and λ labels

the different phonon (acoustic or optic) branches Thus, the INMI terms must be a priorisearched near scattering vectors for which the magnetic and nuclear contributions are nottoo far in energy Obviously, the amplitude of INMI terms should also depend drastically

on the polarization of the phonon involved in the interference process (remember that in aninelastic-neutron-scattering experiment one cannot measure simultaneously the spin com-ponents and the atomic displacements along the same direction) In order to verify thesepredictions, a spherical polarization analysis of the 2-meV mode on CuGeO3must be per-formed by using the CRYOPAD device As previously explained, the procedure consists

of determining, for each couple (Q, ω), the 18 components P αβ and P ¯αβ (α, β = x, y, z)

of the polarization matrix after the appropriate background corrections Table 2 gives the

values of P αβ and P ¯αβ (α, β = x, y, z) and the corresponding error bars for the spin-Peierls

mode at Q= (0, 1, 1/2) [74,75] From Table 2, one can deduce the longitudinal

Spherical polarization analysis on the 2-meV mode at Q= (0, 1, 1/2) in CuGeO3.

Longitudinal and transverse components of the polarization matrix

agree-is very close to 1, which again means that the nuclear contribution associated with the

putative hybrid mode at q = π should be very small: NN+ω/σ0 = (0.003 ± 0.003), as

previously found from LPA Applying the set of relations given in Section 2.2, it has beenpossible to get relatively reliable estimates of the chiral term and the real and imaginaryparts of INMI terms:

Thus, within the error bars, the chiral term and the real parts of INMI terms appear to be

very small, while the imaginary parts (especially I z) appear finite, although at the limit

of the accuracy of the present CRYOPAD version (roughly ±0.01) To summarize the

results of the inelastic SNP on the spin-Peierls mode in CuGeO3, the magnitude of INMI

terms is found to be at best 1% of the main magnetic signal At q ≈ π and ¯hω ≈ ∆SP,the magnetic and lattice excitations can be considered as almost completely decoupled.The main reason for this negative result comes from the fact that in CuGeO3, ∆SP ωph

Hybrid modes may exist at other q or energy values, in particular those corresponding to

the continuum [77] Unfortunately, despite several tentatives, they have not been detected

so far

4 Conclusion

In this chapter, we have shown how powerful the neutron polarimetry is for the tion of inelastic spectra in condensed-matter physics The most frequently used method,namely the longitudinal polarization analysis (LPA), is best suited for all studies requiringthe accurate determination of dynamical spin–spin correlation functions involving only thelongitudinal components of the generalized susceptibility tensor Among others things, the

investiga-Inelastic neutron polarization analysis 393LPA allows us to obtain separately the structural and various longitudinal magnetic struc-

ture factors (e.g., the P0-dependent parts, chiral terms, anisotropy of various spin–spincorrelation functions) Basically, the LPA can be used to solve problems for which thenuclear–magnetic interference terms can be neglected If strong enough, such terms mightproduce a rotation of the polarization vector around the interaction vector different fromπ

which can be only determined by spherical neutron polarimetry (SNP) In principle, themeasurement of the nine coefficients of the polarization matrix should allow us to unam-biguously solve the full problem and, finally, should allow us to determine the variousnuclear–nuclear, magnetic–magnetic and nuclear–magnetic correlation functions

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CHAPTER 9

Polarized Neutron Reflectometry

C.F Majkrzak

Center for Neutron Research, National Institute of Standards and Technology,

Gaithersburg, MD 20899, USA E-mail: chuck@rrdjazz.nist.gov

K.V O’Donovan

Center for Neutron Research, National Institute of Standards and Technology,

Gaithersburg, MD 20899, USA, University of Maryland, College Park, MD 20742, USA

and University of California, Irvine, CA 92697, USA

N.F Berk

Center for Neutron Research, National Institute of Standards and Technology,

Gaithersburg, MD 20899, USA

Contents

1 Introduction 399

2 Fundamental theory of neutron reflectivity 401

2.1 Wave equation in three dimensions 402

2.2 Refractive index 404

2.3 Specular reflection from a perfectly flat slab: The wave equation in one dimension 404

2.4 Specular reflection from a film with a nonuniform SLD profile 408

2.5 Born approximation 410

2.6 Nonspecular reflection 411

3 Spin-dependent neutron wave function 412

3.1 Neutron magnetic moment and spin angular momentum 412

3.2 Explicit form of the spin-dependent neutron wave function 413

3.3 Polarization 415

3.4 Selecting a neutron polarization state 417

3.5 Changing a neutron’s polarization 417

NEUTRON SCATTERING FROM MAGNETIC MATERIALS

Edited by Tapan Chatterji

Published by Elsevier B.V.

397

398 C.F Majkrzak et al.

4 Spin-dependent neutron reflectivity 424

4.1 Spin-dependent reflection from a magnetic film in vacuum referred to reference frame of film 424

4.2 Magnetic media surrounding film 438

4.3 Coordinate system transformation 439

4.4 Selection rules “of thumb” 442

4.5 Three-dimensional polarization analysis 444

4.6 Elementary spin-dependent reflectivity examples 445

5 Experimental methods 448

6 An illustrative application of PNR 450

6.1 Symmetries of reflectance matrices 451

6.2 Basis-independent representation 453

6.3 Front–back reflectivity of idealized twists 455

6.4 PNR of actual systems 457

Appendix 462

References 470

Polarized neutron reflectometry 399

1 Introduction

Advances in our understanding of the structure and properties of matter have so oftendepended upon finding the right probe for studying a given problem This was appreciated

long ago, at the beginning of the evolution of modern science In his masterpiece Faust,

Johann Wolfgang von Goethe wrote about the legend of Doctor Faust, who bargained hisimmortal soul to the Devil, Mephistopheles, in exchange for unlimited knowledge Early

in the story, Faust ponders the relationship between humankind and the universe [1, Part I,Scene I]:

Mysterious even in open day,

Nature retains her veil, despite our clamors:

That which she doth not willingly display

Cannot be wrenched from her with levers, screws, and hammers.

Fortunately, for anyone in the 21st century interested in the structure of condensed matter

on the atomic and nanometer length scales, the considerable efforts of our predecessorshave led to the development of a remarkable collection of sophisticated and exquisitelysensitive probes (far surpassing the capabilities of levers, screws and hammers) Thesenewfound tools are so powerful, in fact, that making a pact with Mephistopheles may nolonger be necessary!

Polarized neutron reflectometry (PNR) is one such probe that is particularly well suitedfor determining the nanostructures of magnetic thin films and multilayers Different types

of magnetometers ordinarily yield only average magnetization values, integrated over theentire volume of the specimen, whereas other probes which do possess a higher degree

of spatial resolution, such as scanning electron microscopy with polarization analysis(SEMPA) [2], are specifically surface sensitive because their relatively strong interactionwith matter limits penetration Together with magnetic X-ray scattering, PNR provides aunique means of “seeing” the vector magnetization with extraordinary spatial detail wellbeneath the surface For neutrons this sensitivity to atomic magnetic moments comes aboutbecause the neutron itself possesses a magnetic moment and neutrons can be obtainedwith a wavelength comparable to interatomic distances More specifically, the specular re-flection of polarized neutrons, namely, coherent elastic scattering for which the angles ofincidence and reflection of the neutron wave vector relative to a flat surface are of equalmagnitude, can be analyzed to yield the in-plane average of the vector magnetization depthprofile along the surface normal By measuring the reflectivity (the ratio of reflected toincident intensities) over a sufficiently broad range of wave vector transfer Q, subnanome-

ter spatial resolution can be achieved The specular geometry is depicted schematically inFigure 1 Furthermore, by tilting Q away from the surface normal, the resulting projection

of Q parallel to the surface of the film allows in-plane fluctuations of the magnetization,

which give rise to nonspecular scattering, to be sensed Unlike optical or electron croscopy, neutron and X-ray reflectometry do not directly provide real-space images ofthe objects of interest Because neutron and X-ray wavelengths are of the order of the di-mensions of the objects being viewed, the information about shape and composition that

mi-is contained in the reflected radiation pattern, including the vector magnetization depthprofile, must be extracted by mathematical analysis

400 C.F Majkrzak et al.

Fig 1 Schematic illustration of the structural information which can be deduced from a polarized neutron flectivity measurement, performed under specular conditions, on a typical model magnetic film The film is flat and composed of alternating layers of ferromagnetic (the magnetization direction in a given layer is indicated

re-by an arrow) and nonmagnetic material The reflected intensity, measured as a function of the glancing angle of

incidence θ or wave vector transfer Q, can be analyzed to obtain the in-plane average of the chemical

compo-sition and vector magnetization as a function of depth along the surface normal, with subnanometer resolution under certain conditions, as is described in detail in the text Reflectivities for a polarized incident beam can be differentiated according to whether the spin state of the neutron changes (“flips”) upon reflection from a magnetic

PIand layer tions M are such that the non-spin-flip reflectivity contains information only about the chemical compositional

magnetiza-depth profile whereas the spin-flip reflectivity reveals the vector magnetization magnetiza-depth profile.

Given the ability to obtain the vector magnetization profile by PNR, a multilayered ture composed of ferromagnetic films separated by intervening layers of another materialcan be systematically studied to reveal various fundamental magnetic behaviors For in-stance, the strength and range of the interlayer magnetic interaction between ferromagneticlayers can be examined as a function of layer thickness, crystallographic orientation, thestrain associated with lattice mismatch (in single-crystalline films), the electronic states(e.g., super-, normal-, semiconducting or insulating) and magnetic configurations (co- andnoncollinear) of the intervening layers, and on chemical interdiffusion (e.g., of hydrogen).Investigations of magnetic domain size and orientation and the effects of finite layer thick-ness can also be performed by measuring both specular and nonspecular spin-dependentscattering

struc-Polarized neutron reflectometry 401

A broad range of related problems of fundamental scientific and technological interestinvolving magnetic thin films can be addressed using PNR, especially when employed inconjunction with other techniques such as magnetic X-ray reflectometry and complemen-tary real-space probes Nonetheless, the realization of this research potential depends to acertain extent on the capability of growing nanostructures with atomic scale precision by avariety of thin-film vapor deposition techniques, e.g., molecular beam epitaxy

It is not the purpose of this chapter to review the multitude of magnetic thin-film systemswhich have been studied with PNR There exists a substantial literature on this subject,including a number of review articles [3,4] The primary goal here, rather, is to describethe fundamental concepts of the theory and experimental methodology of PNR As it turnsout, a significant fraction of the description of the basic reflection process for polarizedneutrons can be applied to magnetic X-ray reflection and to more general reflectometrystudies involving nonmagnetic materials

2 Fundamental theory of neutron reflectivity

For our purposes, we can (fortunately) ignore the intricate internal workings of the quarkscomprising a neutron having an energy in the millielectron volt range and concentrate ononly a few resultant properties that are relevant to its interaction with condensed matter

As it happens, we can accurately represent the neutron, according to quantum mechanics,

in the simplest of terms as a plane wave of wave vector k propagating undistorted through

free space, as represented in Figure 2 Because we are concerned with elastic scatteringprocesses in which the neutron in interacting with matter or magnetic field neither gains norloses energy (i.e., the total energy of the neutron is conserved), the neutron wave functionand corresponding time-independent Schrödinger wave equation of motion do not exhibitany explicit dependence on time (This does not imply, however, that the neutron has zero

Fig 2 Idealized representation of a neutron as a plane wave propagating in space (the planar wave fronts of

constant phase are assumed to extend laterally to infinity).

2.1 Wave equation in three dimensions

The mathematical representation of a neutron plane wave in three dimensions is given by

Ψk, r

where the neutron wave vector k = kx ˆx + ky ˆy + kz ˆz and its position in space r = x ˆx +

y ˆy + zˆz The square of the modulus of the neutron wave function, |Ψ |2= Ψ∗Ψ , is

inter-preted as the probability that a given neutron can be found at a specific location in space

with a particular momentum mv = h/λ = ¯hk (where m is the neutron mass, v is its

ve-locity, λ is its wavelength, h is Planck’s constant and ¯h = h/(2π)) The description of the

neutron as a single plane wave implies that it extends infinitely in all directions A morerealistic representation would be a wave packet consisting of a coherent superposition ofplane waves, having a distribution of different wave vectors, which results in a localization

in space consistent with a corresponding uncertainty in momentum – an inescapable tum phenomenon (see, for example, the text by Merzbacher [5]) Nonetheless, in regard tothe wave equation of interest, for the problem at hand (equation (2)), it turns out that thesingle, plane-wave representation of the neutron is remarkably accurate in practice Thus,unless necessary to do otherwise, we will treat the neutron so

quan-Now the neutron interacts with matter primarily through a nuclear potential and a netic potential which affect the magnitude of k The strengths of these potentials are ef-

mag-fectively characterized by scalar “coherent” scattering lengths, although for the magneticcoupling a scattering-angle-dependent “form factor” is also necessary at sufficiently highwave vector transfers (due to the relatively extended spatial distribution of unpaired elec-tron spin density which gives rise to the magnetic potential) We will ignore absorptive aswell as incoherent interactions for the time being and also make the assumption throughoutthat any nuclear spins in the materials considered are completely disordered (see, e.g., thebook by Bacon [9] for a discussion) To properly account for the neutron magnetic mo-ment and its interaction with a magnetic potential requires that the neutron be represented

by a more complicated wave function consisting of two components, each of which hasthe form of a plane wave Discussion of the magnetic interaction will be postponed untilSection 3

At present there exist no coherent neutron sources analogous to a photon laser: the trons in a beam can be taken to be independent of and effectively noninteracting with oneanother Therefore, we can avoid the concept of beams altogether in describing the re-flection process and focus on one neutron at a time, as represented by a plane wave, andcompute probabilities that a neutron is, say, reflected or transmitted by a particular filmstructure

neu-Polarized neutron reflectometry 403Since we are concerned here with structural (instead of vibrational) information, we canlimit our considerations to elastic scattering processes, as mentioned earlier Thus, a time-independent Schrödinger wave equation predicts the evolution of a single neutron planewave in its interaction with material

can be found, for example, in the text by Merzbacher [5].) Now in vacuum, V ( r) = 0 so

that the total energy E of the neutron is equal to the kinetic energy alone Consequently,

of density N (number of scattering centers, e.g., atoms, per unit volume), the potential

energy is given by [10,11]

V =2π¯h2

m N b=2π¯h2

where it is assumed that the material consists of only a single isotope of a given element

possessing a coherent scattering length b and ρ is defined as the scattering length density

(SLD) If absorption were present, then the scattering length would include an imaginarycomponent, but, as stated above, such a possibility will be ignored for the discussion heresince it only complicates the derivations without adding any essential insight in most cases;for neutron reflection, absorption is rarely an appreciable effect For multicomponent ma-terials the SLD can be generalized to

404 C.F Majkrzak et al.

where M is the number of distinct types of isotopes present.

Now because the total energy of the neutron is conserved in an elastic process, we can

equate the kinetic energy of the neutron in vacuum with the constant total energy E within

any material medium Then, substituting the RHSs of (4) and (5) into (2) and simplifying,

we obtain the three-dimensional wave equation



∇2+ k2

2.2 Refractive index

We can impose conservation of energy once again to define a neutron refractive index

analogous to that employed in ordinary light optics In vacuum, E is given by (4), whereas

2.3 Specular reflection from a perfectly flat slab: The wave equation in one dimension

Consider next the reflection of a neutron plane wave from an idealized slab of material

of thickness L and infinite lateral extent in the plane of the film, as depicted in Figure 3.

This slab is perfectly flat, smooth and homogeneous In Figure 3, the reflection of thewave is depicted to be specular in nature, i.e., as mentioned earlier, the angles of incidenceand reflection are equal in magnitude It will be shown in the following discussion thatthis specular condition is indeed the only possibility for reflection if there are no material

density fluctuations along x or y in plane and the density is a function of z alone, along the surface normal: i.e., ρ(x, y, z) = ρ(z) (Here, since |ki| = |kf|, Q = kf− ki impliesthat | Q | = 2k0sin(θ ).) We will have to solve the general equation of motion (11) for a

nonuniform SLD, specifically for a region of space where abrupt changes in the potential

occur along z.