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Spin densities studied in momentum space
Journal of Physics and Chemistry of Solids 61 (2000) 345–352 www.elsevier.nl/locate/jpcs
Spin densities studied in momentum space M.J. Cooper*, J.A. Duffy Department of Physics, University of Warwick, Coventry CV4 7AL, UK
Abstract After approximately ten years development, Compton scattering studies of magnetisation density with circularly polarised synchrotron radiation are beginning to provide unique information about the site-dependent distribution of spin density and the limitations of the approximations used in the electronic structure models, as will be illustrated by a selection of recent results which expose the interpretative problems as well as the limitations of the local density approximation to describe exchange and correlation. Technical developments have been restricted to improvements of degree rather than of kind, with limited resolution and the inability to study “hard” ferromagnets, the most damaging. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: Spin density
1. Introduction This review of the current status of magnetic Compton scattering updates a much more comprehensive review by Sakai [1] published in 1996, to which the reader is referred for further detail. The existence of “magnetic effects” in the incoherent scattering cross-section predates the discovery, let alone the exploitation, of synchrotron radiation. Explicit formulations of the Klein–Nishina cross-section for every conceivable combination of photon polarisation (incident and detected) and electron spin were set out by Lipps and Tolhoek [2] in 1954. The essential requirements for an observable effect being an incident beam that is circularly polarised and a net spin moment (i.e. the sample must be ferro- or ferri-magnetic). The effect was the basis for a Compton polarimeter used to measure the degree of circular polarisation of gamma emissions from beta decaying nuclei in the same decade [3]. The use of the technique to study the spin density of the scatterer followed the suggestion of Platzman and Tzoar [4] and also sparked off interest in magnetic effects in diffraction [5]. The famous “proof of principle” experiment by Sakai and Ono [6] in 1974 relied on cooled radioisotope sources with prohibitively low flux; even the use of dilution refrigerators to achieve the milli * Corresponding author. Tel.: 144-01203-523379; fax: 14401203-524654. E-mail address: [email protected] (M.J. Cooper).
Kelvin source temperatures that lead to circularly polarised gamma emissions failed to make the technique practical. The first synchrotron measurements had to await the construction of an asymmetric insertion device that permitted circularly polarised, “hard”, synchrotron radiation to be extracted: that was the superconducting wiggler magnet at the Daresbury Storage Ring Source [7]. The nomenclature adopted is indicated in Fig. 1, which shows a schematic Compton scattering event. The Compton profile (i.e. the spectral broadening of the Compton-shifted line) can be usefully considered as a Doppler broadening of the due to the motion of the electrons along the direction of the X-ray scattering vector, thus if that direction is chosen as the z-axis of a set of Cartesian coordinates, we can define the Compton profile, J pz ; and its spin-dependent variant, J mag pz ; as: ZZ J pz n p dpx dpy ; Jmag pz
ZZ
1 n " p 2 n # p dpx dpy
where n p is the probability distribution of the electron momenta, i.e. it is the electron momentum density distribution. This quantity contains contributions from all the electrons in the target: n " p 2 n # p relates to just the spindependent momentum distribution. The relationship between electron momentum pz and the experimental
0022-3697/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S0022-369 7(99)00314-5
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Fig. 1. Schematic diagram of the scattering geometry and nomenclature adopted for magnetic Compton scattering. E denotes photon energy, k wavevector and e polarisation. The subscripts i and s refer to the incident and scattered beams, respectively. An elliptically polarised incident beam is used and the polarisation of the scattered beam is not recorded. The electron, from which the photon scatters has a momentum p and a spin polarisation s which is normally aligned alternately parallel and antiparallel to the scattering vector ki 2 ks ; by an external magnetic field.
parameters is given by the following equation: pz E 2 E 1 E E =mc2 1 2 cos f s 2i 2 i s mc Ei 1 Es 2 2Ei Es cos f1=2
2
and pz is normally quoted in atomic units (c 137; e " m 1 of momentum where 1 a:u: 1:99 × 10224 kg m s21 : J mag pz is accessible in experiments in which the incoming radiation is circularly polarised. The relevant cross-section [8], developed from the expressions of Lipps and Tolhoek [2] is: 2 2 d2 s e Es m 2"K dV dEs Ei mc2 k 2 ks 1 1 cos2 f 1 Pl sin2 f 1 i 1 2 cos f mc k cos f 1 ks J pz 1 cos f 2 1Pc s^ i Jmag pz mc 3 Inspection of Eq. (3) shows that the final “magnetic term” can be isolated by changing its sign. This in turn can be achieved either by reversing the hand of polarisation of the incident photon beam (i.e. flipping Pc), or by changing the direction of the unit spin vector s , which is done by reversing the direction of the spin vector with an external magnetic field. 2. Problems of magnetic Compton scattering 2.1. The cross-section In a “classical” approach it is easy to see that the intensity of the spin dependent scattering scales as ki =mc when the electron is excited by the incident electric field of an
electromagnetic wave and the scattered radiation is due to the motion of the magnetic dipole (electric dipole re-radiation gives the dominant charge scattering term). In the conventional X-ray region where this approach might be valid, this is a pitifully small factor. On the contrary the Klein–Nishina cross-section, which is valid at all photon energies, together with the expressions developed from it by Lipps and Tolhoek refer to free, stationary electrons, which are of course a fiction. The approach usually adopted is to take the non-relativistic Hamiltonian and add relativistic correction terms, which can then be treated by perturbation theory. The potential in which the electron is bound only features in so much that it determines the electron’s momentum because of the invocation of the impulse approximation. This is fine for low energies, typical of magnetic diffraction, but it does not necessarily approach the Klein–Nishina expression at high energies, as it must (see Ref. [8] for experimental work up to 1 MeV). Expansions in ki =mc Ei =mc2 do not converge and there has to be a judicious neglect of some terms. The relativistic cross-section has been calculated for a free moving electron semi-relativistically [9], then relativistically by Blatt et al. [10]. Their latter result contains terms in p=mc which are not present in the semi-relativistic approach. They show that whereas the scattering from unpolarised electrons can be predicted by combining the cross-section for a stationary electron with the scattering factor S(K,E) for the moving electron, the same generalisation does not (quite) hold for an electron with spin. The invocation of the impulse approximation allows their result to be applied to bound electrons since in the limit of Ei 2 Es @ EB the electron can be treated as freely moving. As the authors point out, their result is analogous to that for the charge scattering cross-section, in which the ground state Compton profile can be disentangled from other momentum dependent factors in the cross-section by iterative methods [11]. To a first approximation the momentum dependence of the cross-section is linear across the energy range of interest and simply folding the (magnetic) Compton profile about pz 0 effects the correction. A similar, as yet unpublished, derivation had been made by Honkimaki [12]. The difference in approach between magnetic Compton scattering, which has to be dealt with relativistically and magnetic diffraction, which can be approached semi-classically, is well illustrated by the question of whether orbital magnetisation contributes to the Compton cross-section as it does in diffraction. After some confusing results [13], convincing experimental evidence provided the answer “no” [14] in situations where the impulse approximation is valid. This accords with common-sense intuition that orbital momentum can hardly be defined in an instantaneous interaction. A quantitative theoretical argument was put forward by Carra and collaborators [15] and a recent model calculation of hydrogen atom in the 2p state [16] indicates just how
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through their reconstruction is shown in Fig. 2. Experiments of this kind rarely take place because they are prohibitively long. Recently the Maximum Entropy Method has been applied to the reconstruction problem [18,19] and results confirm that certain questions such as the wholly negative or positive nature of the unpaired spin density discussed below, can be addressed without recourse to so many directional views, but detailed reconstruction of structure in n p cannot be conjured out of thin air! 2.3. Interpretation of individual magnetic Compton profiles
Fig. 2. A section through the unpaired spin density distribution n " p 2 n # p in the (001) plane in ferromagnetic iron reconstructed from 14 directional magnetic Compton profiles, measured at a resolution of approximately Dpz , 0:7 a:u: [17].The central dip is a real negative contribution from the 4s-p electrons which are spin polarised antiparallel to the d-like bands.
orbital effects can creep into the cross-section as the impulse approximation fails at low energy transfers. 2.2. Reconstructing the spin density The basic difficulties faced by spin-resolved Compton scattering experiments are not experimental. As Eqs. (1) and (2) show the measured quantity is the projection of the momentum density distribution along a line. It is not a 2D projection, as would be the case in a radiograph or a positron 2D ACAR measurement, which is the most similar momentum density probe. The reconstruction problem is tedious because many views are needed. The tomographic analogue is a problem in which only one ray sum can be measured at a time or in crystallography the determination of the electron density distribution from a series of axial (hoo) Bragg reflections. It is simplest in the case of isotropic distributions (e.g. measurements of polycrystals) where, using various algorithms, one may obtain: n " p 2 n # p 2
1 dJmag p dp 2pp
4
and the inherent noise amplification is evident by the presence of the differential. In anisotropic materials (single crystal measurements) the reconstruction proceeds through the B z functions which are the Fourier transforms of the MCPs. If enough different directional transforms are obtained then the quantity B(r) can be reconstructed and Fourier inverted to give the spin dependent density distribution, n " p 2 n # p: The definition of “enough” is critical and depends upon the symmetry (usually cubic) and the statistical accuracy. The best example so far is the work of Tanaka et al. [17] who reconstructed the momentum density in Fe from 14 directional Compton profiles. A section
There are two specific difficulties in interpreting Compton profiles, spin-resolved or otherwise. The first follows from its nature as a one-dimensional projection. The collapse of all the information makes it impossible to interpret any feature in the profile unambiguously a priori, since its location in z, not x or y, is the only definite piece of information. What this means in practice is that Compton profiles are only interpretable in comparison with a model. The second problem is that in order to predict the Compton profile at any momentum it needs to be known at all momenta. In other words the calculation of J p is more demanding than the that of n p: If the momentum range up to, say 5 a.u., is deemed interesting the calculation of n p need only extend to p 5 but for J p it apparently needs to go to infinity! The actual situation is not so dire for one very important reason: the high momentum behaviour in the condensed matter state must approach that of the (calculable) free atom. The reasoning goes as follows: cohesive energy, which is the difference between the total energy of the condensed state and the isolated free atoms, is very small compared with the total energy and the difference is associated with the “outer” slow moving electrons. The tightly bound core electrons are well described by free atom wave functions. Total energy is related (by a change of sign only) to the kinetic energy in any system governed by Coulomb forces. This is the virial theorem and here it allows us to monitor the total energy of a material by measuring the second moment of the momentum distribution 1=2mkp2 I pl and indeed the second moment of the Compton profile, Skp2 J pl: The strong weighting of the high momentum components in this sum, compared with the small size of the cohesive energy precludes any significant differences between the actual and free atom profiles at high momenta. Thus, the free atom model can be grafted onto a band calculation to describe the high momentum behaviour. It has provided a basis for determining site dependent moments in materials such as CeFe2 and UFe2 [20], because of the characteristically different momentum distributions at the two sites. Conversely, the requirement to follow free atom behaviour as p increases is a useful test of the goodness of the data. The problem of interpreting one projection remains and is exemplified by considering the following question. Does the “dip” at pz 0 in the a spin resolved profile, for example the
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Fig. 3. A high resolution Dpz , 0:19 a:u: Magnetic Compton profile of ferromagnetic iron measured at ESRF by McCarthy [26]. The dashed line is a guide to the eye and the solid line represents the FLAPW calculation of Kubo and Asano [28]. “Umklapp” features at high momenta are more prominent in experiment than theory but the reverse is true for the feature they should “echo” in the first Brillouin zone.
one shown in Fig. 3, mean that there is a negative (i.e. antiparallel) spin component? The answer must be yes if the distribution is isotropic since the Compton profile integral then takes the form: Z∞ I p J q dp where I p 4pp2 Dn p 5 p q and Dn p n " p 2 n # p : J q must always increase with decreasing p as long as n p is positive. But the converse in not true if the spin density is aspherical and the problem of determining whether there is a real negative component or simply a decrease in Dn p at low p for that projection cannot be determined. Several sets of directional data are needed to provide a clue and a reconstruction is essential to “prove” the existence of a true negative contribution. Fig. 2 shows that it is indeed the case for ferromagnetic iron but it is certainly dangerous to estimate the size of the negatively polarised spin component from the size of the central dip of any one profile, as was often done in the early analyses of isolated data sets. Even with an extended analysis of data sets the interpretation of magnetic (and charge) Compton profiles is only complete when there is a concerted theoretical and experimental approach that allows the modelled profile to be stripped down into its band structure ingredients. The case study of Ni, presented below illustrates this point well. 2.4. Resolution The hope that the higher and higher flux of synchrotron radiation sources would soon permit a change to higher resolution in magnetic Compton scattering has only been realised in one or two experiments. The first, by Sakurai et al. [21], used two imaging plates which were switched in
register with the magnetic field to record the “up” and “down” spectra over a four day period. It was four years before another high-resolution experiment, which is described below, was attempted. Magnetic Compton experiments continue to be performed with the energy analysis made by a cooled germanium detector whose resolution is a function of energy. With such detectors it is impossible to improve the momentum space resolution significantly below Dp , 0:4 a:u: [22]. Yet, the first Compton spectrometers for studies of the (total) momentum density were based on dispersive crystal optics arrangement and date back to the days of Dumond [23]. Almost half a century elapsed before a synchrotron-based one was installed at LURE by Loupias and collaborators [24]. The resolution of this spectrometer was , 0.1 2 0.2 a.u., a figure reproduced in more recent versions using image plate recording [21,25] and in the latest scanning version installed at the ESRF [26]. The resolution figure is hardly competitive with the norm in positron annihilation “2D ACAR” (,0.10 a.u.) but is it a considerable improvement on the best currently possible with semiconductor detectors. Unfortunately none of the current or recently available sources has enough flux (quite) to make these spin-resolved experiments feasible without an unreasonable sacrifice in statistical accuracy. Consider the situation in which the magnetic effect is 1% of the total scattering. Only when the accumulated counts reaches 2 × 106 does the magnetic signal have a tolerable ratio (10:1) to the statistical noise. Such a measurement takes the best part of a week of continuous counting with the ESRF scanning Compton spectrometer, for example. Unfortunately, the sample with the highest magnetic effect, Fe, is not necessarily the most interesting, the FLAPW calculation of Kubo and Asano [27] predicts far less structure than is found for example in the case of Ni. A high resolution study on Fe by McCarthy [26], shown in Fig. 3, reveals less to confront theory than the lower resolution study on the more structured Ni profile to be discussed later.
3. Some recent results 3.1. 3d systems-ferromagnetic nickel This example emphasises the need to integrate experimental and theoretical work if individual profiles, or sets of profiles are to be interpreted. It is taken from recent work [28] at ESRF where the use of incident beam energies in excess of 200 keV results in momentum resolution figures which are approximately twice as good as those attained earlier at ,50 keV at other synchrotrons (for example the study [17] shown in Fig. 2). The insertion device together with the Laue monochromator produces in excess of 10 11 photons per second with a degree of circular polarisation between 40 and 50% on the target. This means that, with
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Fig. 4. The MCP of nickel in the [110] crystal direction. (a) The upper figure shows the experimental results (open circles), obtained at a momentum resolution of 0.43 a.u., together with a number of the electronic structure calculations as indicated in the box. The models all predict a larger spin moment than the measured 0.56 m B and this difference is evident in the overestimation of the MCP at low momentum. The letters identify features in the model band profiles delineated in the lower figure (b). Here there is no resolution broadening.
a sample like Ni which has a spin moment of 0.56 m B, approximately 8 × 106 counts in the magnetic profile can be accumulated within 36 h (the charge peak contains 1:3 × 108 counts: Fig. 4(a) shows one of the resulting MCPs, the resolved direction was [110] (three other directions were measured). Ignoring for a moment the superimposed model curves the data set shows a very pronounced central dip which is present but to differing degrees in the other orientations. Does this correspond to a negative spin density in whole, or in part, or not at all? In this case, a LMTO calculation was done by the same authors within both the LSDA and GGA approximations. The result of the latter is shown in Fig. 4(b) from which it is clear that the main contribution to the “volcano” is the 5th d-like band which is entirely positive, i.e. the spins are aligned with the field. There is a genuine negative spin polarisation contribution,
which comes from the first and second 4sp-like bands but it is not the major cause of the observed feature. In Fig. 4(a), the two LMTO calculations are superimposed on the observed profile together with a FLAPW prediction of Kubo and Asano [27]. All the models fit the data at high momenta as was argued for earlier, this is unsurprising, but they all predict spin moments (given by the area under the MCP) which are too large. This is a significant shortcoming in the theory; it is not clear that the MCPs can simply be rescaled to equalise them since the size of the moment derives from the size of the exchange splitting and if that changes the lineshape may also change. Nonetheless re-scaling does not remove the discrepancy. There is a difference in the spin-resolved profiles. Returning to the main figure it is clear that the predicted feature, A, in the spin-resolved profile is absent in the
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Fig. 5. The upper diagram shows the MCP of gadolinium measured along the hexagonal axis, together with three model calculations. The full potential calculation appears to be marginally worse near p 0: The lower figure reproduces the Gd MCP together with that of the insulator EuO. The lines are 4f free atom [32] free atom profiles fitted between 3 and 10 a.u.
data set, yet features at C, E, and F which might be thought of as high momentum components relating to the “A” feature in the first Brillouin zone are present in the data. Clearly, there is a common failing in the potential used in all three calculations. In fact, all directional profiles show deeper fissures than theory predicts. The d-like fifth band governs the higher momenta shoulders and predicts the correct Fermi surface. It is thought that the discrepancy between experiment and theory is most likely due to the failure of the description of the exchange-correlation effects. Interestingly McCarthy’s results for Fe, shown in Fig. 3 show a similar
effect; although the magnetic profile is less structured the major discrepancy between experiment and theory is again in the first Brillouin zone. 3.2. 4f systems Gd and EuO Gadolinium has been measured at varying levels of accuracy from 1987 onwards [29]. It is a soft magnet and has a large spin moment. Many of these early measurements were made at liquid nitrogen temperatures for convenience but a higher temperature is needed to avoid the phase where the caxis moments are canted. The main problem with Gd is not
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experimental, it is the simple fact that the structure is hexagonal and calculations n p for non-cubic systems are not well advanced. We performed LMTO calculations [30] within the LSDA and the GGA approximations and also had access to the FLAPW calculations of Kubo and Asano [31]. Experiment and theory are shown together in Fig. 5(a) where it is evident that: (a) the FLAPW theory does not produce any improvement, in fact it is worse than the LMTO method, in contrast to the situation in Ni; (b) the GGA does not improve on the LSDA approximation, this may be due to the more localised (4f) nature of the magnetisation. The adequacy of the LMTO to describe this hexagonal system is significant because it can be used in Gdbased alloys such as Gd-Y where the magnetic properties are driven by the electronic structure. Finally Fig. 5(b) compares the MCPs of Gd with that of EuO, the real interest in the latter is to establish whether there is an induced moment on the oxygen site, which might be expected as a consequence of the super-exchange model of magnetism in this material. That cannot be deduced from the raw data until current band modelling has been completed but it is interesting to see the comparison between the conductor, Gd, with a positive spin polarised conduction electron contribution. The origin of the small dip in the EuO profile is under investigation.
4. Conclusions The extraction of information from MCPs requires a concerted experimental and theoretical approach; this in turn requires that the model is good enough for the differences between it and the data to be meaningfully interpreted: reconstruction will have to figure more prominently in the interpretation, so that n p rather than J mag p can be considered. Yet the information is there because the photon is a “clean” probe of an equilibrium ground state property, and as such it provides one of the most promising methods of testing LSDA and other assumptions underlying electronic structure calculations in ferromagnets where other synchrotron based techniques either involve excited state (e.g. dichroism, photo emission) or are relatively undeveloped (e.g. diffraction). It can also be used to search for induced moments on “non-magnetic” sites (e.g. Y in Gd-Y, O in EuO) but in all applications empirical interpretation is all but impossible: experiment needs to go hand-in-hand with theory if spin-resolved densities are to be understood by this technique.
Acknowledgements All synchrotron studies are the result of team efforts and we have relied heavily on the work of numerous colleagues in this brief summary. Especial thanks are due to J.E. McCarthy (ESRF and Warwick) for her collaboration in
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the ESRF experiments and permission to reproduce Fig. 3 from her thesis and to S. Dugdale (Geneva and Bristol) for performing the LMTO calculations cited here. The support of the EPSRC in the UK and the ESRF for beamtime and experimental support are particularly acknowledged. References [1] N. Sakai, J Appl Cryst. 29 (1996) 81. [2] F.W. Lipps, H.A. Tolhoek, Physica 20 (1954) p. 85 and p. 395. [3] M. Goldhaber, L. Grodzins, A.W. Sunyar, Phys Rev 109 (1958) 1015. [4] P.M. Platzman, N. Tzoar, Phys Rev B 2 (1970) 3556. [5] F. deBergevin, M. Brunel, Acta Cryst. A 37 (1981) p. 314 and p. 324. [6] N. Sakai, K. Ono, Phys. Rev. Lett. 37 (1976) 351. [7] M.J. Cooper, S.P. Collins, D.N. Timms, A. Brahmia, P.P. Kane, R.S. Holt, D.N. Laundy, Nature 333 (1988) 151. [8] J.E. McCarthy, M.J. Cooper, V. Honkima¨ki, T. Tschetscher, P. Suortti, S. Gardelis, K. Ha¨ma¨la¨inen, D.N. Timms, Nucl. Instrum. Meth. A 401 (1997) 463. [9] H. Grotch, E. Kazes, G. Bhatt, D.A. Owen, Phys. Rev. A 27 (1983) 243. [10] G. Bhatt, H. Grotch, E. Kazes, D. Owen, Phys. Rev. A 28 (1983) 2195. [11] R. Ribberfors, Phys. Rev. B 12 (1975) p. 2067 and p. 3136. [12] V. Honkimaki, private communication. [13] S.P. Collins, M.J. Cooper, S.W. Lovesey, D. Laundy, J. Phys.: Condens. Matter 2 (1990) 6439. [14] M.J. Cooper, E. Zukowski, S.P. Collins, D.N. Timms, F. Itoh, Y. Sakurai, J. Phys.: Condens. Matter 4 (1992) 399. [15] P. Carra, M. Fabrizio, G. Santoro, B.T. Thole, Phys. Rev. B 53 (1996) R5994. [16] D. Kekchrakos, K.N. Trohidou, S. Taddei, Phys. Rev. B 56 (1997) 10812. [17] Y. Tanaka, N. Sakai, Y. Kubo, H. Kawata, Phys. Rev. Lett. 70 (1993) 1537. [18] L. Dobrzynski, A. Holas, Nucl. Instrum. Meth. A 383 (1996) 589. [19] L. Dobrzynski, A. Holas, Phys. Rev. B, 1998, submitted for publication. [20] P.K. Lawson, M.J. Cooper, M.A.G. Dixon, D.N. Timms, E. Zukowski, F. Itoh, H. Sakurai, Phys. Rev. B 56 (1997) 56. [21] Y. Sakurai, Y. Tanaka, T. Ohata, Y. Watanabe, S. Nanao, Y. Ushigama, T. Iwazumi, H. Kawata, N. Shiotani, J. Phys.: Condens. Matter 6 (1994) 9469. [22] J.E. McCarthy, M.J. Cooper, P.K. Lawson, D.N. Timms, S.O. Manninen, K. Hamalainen, P. Suortti, J. Synchrotron Rad. 4 (1997) 102. [23] J.W.M. DuMond, Rev. Mod. Phys. 5 (1933) 1. [24] G. Loupias, J. Petiau, J. de Physique 41 (1980) 265. [25] Y. Sakurai, M. Ito, T. Urai, Y. Tanaka, N. Sakai, T. Iwazumi, H. Kawata, M. Ando, N. Shiotani, Rev. Sci. Instrum. 63 (1992) 1190. [26] J.E. McCarthy, Dissertation for the degree of PhD, University of Warwick, 1997. [27] Y. Kubo, S. Asano, Phys. Rev. B 42 (1990) 4431. [28] M.A.G. Dixon, J.A. Duffy, S. Gardelis, J.E. McCarthy, M.J. Cooper, S.B. Dugdale, T. Jarlborg, D.N. Timms, J. Phys.: Condens. Matter 10 (1998) 2759.
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[29] D.M. Mills, Phys. Rev. B 36 (1987) 6178. [30] J.A. Duffy, J.E. McCarthy, S.B. Dugdale, V. Honkimaki, M.J. Cooper, M.A. Alam, T. Jarlborg, S.B. Palmer, J. Phys.: Condens. Matter 10 (1998) 10391.
[31] Y. Kubo, S. Asano, J. Magn. Magn. Mat. 115 (1992) 177. [32] F. Biggs, L.B. Mendelsohn, J.B. Mann, At. Nucl. Data Tables 16 (1975) 201.
Spin densities studied in momentum space M.J. Cooper*, J.A. Duffy Department of Physics, University of Warwick, Coventry CV4 7AL, UK
Abstract After approximately ten years development, Compton scattering studies of magnetisation density with circularly polarised synchrotron radiation are beginning to provide unique information about the site-dependent distribution of spin density and the limitations of the approximations used in the electronic structure models, as will be illustrated by a selection of recent results which expose the interpretative problems as well as the limitations of the local density approximation to describe exchange and correlation. Technical developments have been restricted to improvements of degree rather than of kind, with limited resolution and the inability to study “hard” ferromagnets, the most damaging. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: Spin density
1. Introduction This review of the current status of magnetic Compton scattering updates a much more comprehensive review by Sakai [1] published in 1996, to which the reader is referred for further detail. The existence of “magnetic effects” in the incoherent scattering cross-section predates the discovery, let alone the exploitation, of synchrotron radiation. Explicit formulations of the Klein–Nishina cross-section for every conceivable combination of photon polarisation (incident and detected) and electron spin were set out by Lipps and Tolhoek [2] in 1954. The essential requirements for an observable effect being an incident beam that is circularly polarised and a net spin moment (i.e. the sample must be ferro- or ferri-magnetic). The effect was the basis for a Compton polarimeter used to measure the degree of circular polarisation of gamma emissions from beta decaying nuclei in the same decade [3]. The use of the technique to study the spin density of the scatterer followed the suggestion of Platzman and Tzoar [4] and also sparked off interest in magnetic effects in diffraction [5]. The famous “proof of principle” experiment by Sakai and Ono [6] in 1974 relied on cooled radioisotope sources with prohibitively low flux; even the use of dilution refrigerators to achieve the milli * Corresponding author. Tel.: 144-01203-523379; fax: 14401203-524654. E-mail address: [email protected] (M.J. Cooper).
Kelvin source temperatures that lead to circularly polarised gamma emissions failed to make the technique practical. The first synchrotron measurements had to await the construction of an asymmetric insertion device that permitted circularly polarised, “hard”, synchrotron radiation to be extracted: that was the superconducting wiggler magnet at the Daresbury Storage Ring Source [7]. The nomenclature adopted is indicated in Fig. 1, which shows a schematic Compton scattering event. The Compton profile (i.e. the spectral broadening of the Compton-shifted line) can be usefully considered as a Doppler broadening of the due to the motion of the electrons along the direction of the X-ray scattering vector, thus if that direction is chosen as the z-axis of a set of Cartesian coordinates, we can define the Compton profile, J pz ; and its spin-dependent variant, J mag pz ; as: ZZ J pz n p dpx dpy ; Jmag pz
ZZ
1 n " p 2 n # p dpx dpy
where n p is the probability distribution of the electron momenta, i.e. it is the electron momentum density distribution. This quantity contains contributions from all the electrons in the target: n " p 2 n # p relates to just the spindependent momentum distribution. The relationship between electron momentum pz and the experimental
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Fig. 1. Schematic diagram of the scattering geometry and nomenclature adopted for magnetic Compton scattering. E denotes photon energy, k wavevector and e polarisation. The subscripts i and s refer to the incident and scattered beams, respectively. An elliptically polarised incident beam is used and the polarisation of the scattered beam is not recorded. The electron, from which the photon scatters has a momentum p and a spin polarisation s which is normally aligned alternately parallel and antiparallel to the scattering vector ki 2 ks ; by an external magnetic field.
parameters is given by the following equation: pz E 2 E 1 E E =mc2 1 2 cos f s 2i 2 i s mc Ei 1 Es 2 2Ei Es cos f1=2
2
and pz is normally quoted in atomic units (c 137; e " m 1 of momentum where 1 a:u: 1:99 × 10224 kg m s21 : J mag pz is accessible in experiments in which the incoming radiation is circularly polarised. The relevant cross-section [8], developed from the expressions of Lipps and Tolhoek [2] is: 2 2 d2 s e Es m 2"K dV dEs Ei mc2 k 2 ks 1 1 cos2 f 1 Pl sin2 f 1 i 1 2 cos f mc k cos f 1 ks J pz 1 cos f 2 1Pc s^ i Jmag pz mc 3 Inspection of Eq. (3) shows that the final “magnetic term” can be isolated by changing its sign. This in turn can be achieved either by reversing the hand of polarisation of the incident photon beam (i.e. flipping Pc), or by changing the direction of the unit spin vector s , which is done by reversing the direction of the spin vector with an external magnetic field. 2. Problems of magnetic Compton scattering 2.1. The cross-section In a “classical” approach it is easy to see that the intensity of the spin dependent scattering scales as ki =mc when the electron is excited by the incident electric field of an
electromagnetic wave and the scattered radiation is due to the motion of the magnetic dipole (electric dipole re-radiation gives the dominant charge scattering term). In the conventional X-ray region where this approach might be valid, this is a pitifully small factor. On the contrary the Klein–Nishina cross-section, which is valid at all photon energies, together with the expressions developed from it by Lipps and Tolhoek refer to free, stationary electrons, which are of course a fiction. The approach usually adopted is to take the non-relativistic Hamiltonian and add relativistic correction terms, which can then be treated by perturbation theory. The potential in which the electron is bound only features in so much that it determines the electron’s momentum because of the invocation of the impulse approximation. This is fine for low energies, typical of magnetic diffraction, but it does not necessarily approach the Klein–Nishina expression at high energies, as it must (see Ref. [8] for experimental work up to 1 MeV). Expansions in ki =mc Ei =mc2 do not converge and there has to be a judicious neglect of some terms. The relativistic cross-section has been calculated for a free moving electron semi-relativistically [9], then relativistically by Blatt et al. [10]. Their latter result contains terms in p=mc which are not present in the semi-relativistic approach. They show that whereas the scattering from unpolarised electrons can be predicted by combining the cross-section for a stationary electron with the scattering factor S(K,E) for the moving electron, the same generalisation does not (quite) hold for an electron with spin. The invocation of the impulse approximation allows their result to be applied to bound electrons since in the limit of Ei 2 Es @ EB the electron can be treated as freely moving. As the authors point out, their result is analogous to that for the charge scattering cross-section, in which the ground state Compton profile can be disentangled from other momentum dependent factors in the cross-section by iterative methods [11]. To a first approximation the momentum dependence of the cross-section is linear across the energy range of interest and simply folding the (magnetic) Compton profile about pz 0 effects the correction. A similar, as yet unpublished, derivation had been made by Honkimaki [12]. The difference in approach between magnetic Compton scattering, which has to be dealt with relativistically and magnetic diffraction, which can be approached semi-classically, is well illustrated by the question of whether orbital magnetisation contributes to the Compton cross-section as it does in diffraction. After some confusing results [13], convincing experimental evidence provided the answer “no” [14] in situations where the impulse approximation is valid. This accords with common-sense intuition that orbital momentum can hardly be defined in an instantaneous interaction. A quantitative theoretical argument was put forward by Carra and collaborators [15] and a recent model calculation of hydrogen atom in the 2p state [16] indicates just how
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through their reconstruction is shown in Fig. 2. Experiments of this kind rarely take place because they are prohibitively long. Recently the Maximum Entropy Method has been applied to the reconstruction problem [18,19] and results confirm that certain questions such as the wholly negative or positive nature of the unpaired spin density discussed below, can be addressed without recourse to so many directional views, but detailed reconstruction of structure in n p cannot be conjured out of thin air! 2.3. Interpretation of individual magnetic Compton profiles
Fig. 2. A section through the unpaired spin density distribution n " p 2 n # p in the (001) plane in ferromagnetic iron reconstructed from 14 directional magnetic Compton profiles, measured at a resolution of approximately Dpz , 0:7 a:u: [17].The central dip is a real negative contribution from the 4s-p electrons which are spin polarised antiparallel to the d-like bands.
orbital effects can creep into the cross-section as the impulse approximation fails at low energy transfers. 2.2. Reconstructing the spin density The basic difficulties faced by spin-resolved Compton scattering experiments are not experimental. As Eqs. (1) and (2) show the measured quantity is the projection of the momentum density distribution along a line. It is not a 2D projection, as would be the case in a radiograph or a positron 2D ACAR measurement, which is the most similar momentum density probe. The reconstruction problem is tedious because many views are needed. The tomographic analogue is a problem in which only one ray sum can be measured at a time or in crystallography the determination of the electron density distribution from a series of axial (hoo) Bragg reflections. It is simplest in the case of isotropic distributions (e.g. measurements of polycrystals) where, using various algorithms, one may obtain: n " p 2 n # p 2
1 dJmag p dp 2pp
4
and the inherent noise amplification is evident by the presence of the differential. In anisotropic materials (single crystal measurements) the reconstruction proceeds through the B z functions which are the Fourier transforms of the MCPs. If enough different directional transforms are obtained then the quantity B(r) can be reconstructed and Fourier inverted to give the spin dependent density distribution, n " p 2 n # p: The definition of “enough” is critical and depends upon the symmetry (usually cubic) and the statistical accuracy. The best example so far is the work of Tanaka et al. [17] who reconstructed the momentum density in Fe from 14 directional Compton profiles. A section
There are two specific difficulties in interpreting Compton profiles, spin-resolved or otherwise. The first follows from its nature as a one-dimensional projection. The collapse of all the information makes it impossible to interpret any feature in the profile unambiguously a priori, since its location in z, not x or y, is the only definite piece of information. What this means in practice is that Compton profiles are only interpretable in comparison with a model. The second problem is that in order to predict the Compton profile at any momentum it needs to be known at all momenta. In other words the calculation of J p is more demanding than the that of n p: If the momentum range up to, say 5 a.u., is deemed interesting the calculation of n p need only extend to p 5 but for J p it apparently needs to go to infinity! The actual situation is not so dire for one very important reason: the high momentum behaviour in the condensed matter state must approach that of the (calculable) free atom. The reasoning goes as follows: cohesive energy, which is the difference between the total energy of the condensed state and the isolated free atoms, is very small compared with the total energy and the difference is associated with the “outer” slow moving electrons. The tightly bound core electrons are well described by free atom wave functions. Total energy is related (by a change of sign only) to the kinetic energy in any system governed by Coulomb forces. This is the virial theorem and here it allows us to monitor the total energy of a material by measuring the second moment of the momentum distribution 1=2mkp2 I pl and indeed the second moment of the Compton profile, Skp2 J pl: The strong weighting of the high momentum components in this sum, compared with the small size of the cohesive energy precludes any significant differences between the actual and free atom profiles at high momenta. Thus, the free atom model can be grafted onto a band calculation to describe the high momentum behaviour. It has provided a basis for determining site dependent moments in materials such as CeFe2 and UFe2 [20], because of the characteristically different momentum distributions at the two sites. Conversely, the requirement to follow free atom behaviour as p increases is a useful test of the goodness of the data. The problem of interpreting one projection remains and is exemplified by considering the following question. Does the “dip” at pz 0 in the a spin resolved profile, for example the
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Fig. 3. A high resolution Dpz , 0:19 a:u: Magnetic Compton profile of ferromagnetic iron measured at ESRF by McCarthy [26]. The dashed line is a guide to the eye and the solid line represents the FLAPW calculation of Kubo and Asano [28]. “Umklapp” features at high momenta are more prominent in experiment than theory but the reverse is true for the feature they should “echo” in the first Brillouin zone.
one shown in Fig. 3, mean that there is a negative (i.e. antiparallel) spin component? The answer must be yes if the distribution is isotropic since the Compton profile integral then takes the form: Z∞ I p J q dp where I p 4pp2 Dn p 5 p q and Dn p n " p 2 n # p : J q must always increase with decreasing p as long as n p is positive. But the converse in not true if the spin density is aspherical and the problem of determining whether there is a real negative component or simply a decrease in Dn p at low p for that projection cannot be determined. Several sets of directional data are needed to provide a clue and a reconstruction is essential to “prove” the existence of a true negative contribution. Fig. 2 shows that it is indeed the case for ferromagnetic iron but it is certainly dangerous to estimate the size of the negatively polarised spin component from the size of the central dip of any one profile, as was often done in the early analyses of isolated data sets. Even with an extended analysis of data sets the interpretation of magnetic (and charge) Compton profiles is only complete when there is a concerted theoretical and experimental approach that allows the modelled profile to be stripped down into its band structure ingredients. The case study of Ni, presented below illustrates this point well. 2.4. Resolution The hope that the higher and higher flux of synchrotron radiation sources would soon permit a change to higher resolution in magnetic Compton scattering has only been realised in one or two experiments. The first, by Sakurai et al. [21], used two imaging plates which were switched in
register with the magnetic field to record the “up” and “down” spectra over a four day period. It was four years before another high-resolution experiment, which is described below, was attempted. Magnetic Compton experiments continue to be performed with the energy analysis made by a cooled germanium detector whose resolution is a function of energy. With such detectors it is impossible to improve the momentum space resolution significantly below Dp , 0:4 a:u: [22]. Yet, the first Compton spectrometers for studies of the (total) momentum density were based on dispersive crystal optics arrangement and date back to the days of Dumond [23]. Almost half a century elapsed before a synchrotron-based one was installed at LURE by Loupias and collaborators [24]. The resolution of this spectrometer was , 0.1 2 0.2 a.u., a figure reproduced in more recent versions using image plate recording [21,25] and in the latest scanning version installed at the ESRF [26]. The resolution figure is hardly competitive with the norm in positron annihilation “2D ACAR” (,0.10 a.u.) but is it a considerable improvement on the best currently possible with semiconductor detectors. Unfortunately none of the current or recently available sources has enough flux (quite) to make these spin-resolved experiments feasible without an unreasonable sacrifice in statistical accuracy. Consider the situation in which the magnetic effect is 1% of the total scattering. Only when the accumulated counts reaches 2 × 106 does the magnetic signal have a tolerable ratio (10:1) to the statistical noise. Such a measurement takes the best part of a week of continuous counting with the ESRF scanning Compton spectrometer, for example. Unfortunately, the sample with the highest magnetic effect, Fe, is not necessarily the most interesting, the FLAPW calculation of Kubo and Asano [27] predicts far less structure than is found for example in the case of Ni. A high resolution study on Fe by McCarthy [26], shown in Fig. 3, reveals less to confront theory than the lower resolution study on the more structured Ni profile to be discussed later.
3. Some recent results 3.1. 3d systems-ferromagnetic nickel This example emphasises the need to integrate experimental and theoretical work if individual profiles, or sets of profiles are to be interpreted. It is taken from recent work [28] at ESRF where the use of incident beam energies in excess of 200 keV results in momentum resolution figures which are approximately twice as good as those attained earlier at ,50 keV at other synchrotrons (for example the study [17] shown in Fig. 2). The insertion device together with the Laue monochromator produces in excess of 10 11 photons per second with a degree of circular polarisation between 40 and 50% on the target. This means that, with
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Fig. 4. The MCP of nickel in the [110] crystal direction. (a) The upper figure shows the experimental results (open circles), obtained at a momentum resolution of 0.43 a.u., together with a number of the electronic structure calculations as indicated in the box. The models all predict a larger spin moment than the measured 0.56 m B and this difference is evident in the overestimation of the MCP at low momentum. The letters identify features in the model band profiles delineated in the lower figure (b). Here there is no resolution broadening.
a sample like Ni which has a spin moment of 0.56 m B, approximately 8 × 106 counts in the magnetic profile can be accumulated within 36 h (the charge peak contains 1:3 × 108 counts: Fig. 4(a) shows one of the resulting MCPs, the resolved direction was [110] (three other directions were measured). Ignoring for a moment the superimposed model curves the data set shows a very pronounced central dip which is present but to differing degrees in the other orientations. Does this correspond to a negative spin density in whole, or in part, or not at all? In this case, a LMTO calculation was done by the same authors within both the LSDA and GGA approximations. The result of the latter is shown in Fig. 4(b) from which it is clear that the main contribution to the “volcano” is the 5th d-like band which is entirely positive, i.e. the spins are aligned with the field. There is a genuine negative spin polarisation contribution,
which comes from the first and second 4sp-like bands but it is not the major cause of the observed feature. In Fig. 4(a), the two LMTO calculations are superimposed on the observed profile together with a FLAPW prediction of Kubo and Asano [27]. All the models fit the data at high momenta as was argued for earlier, this is unsurprising, but they all predict spin moments (given by the area under the MCP) which are too large. This is a significant shortcoming in the theory; it is not clear that the MCPs can simply be rescaled to equalise them since the size of the moment derives from the size of the exchange splitting and if that changes the lineshape may also change. Nonetheless re-scaling does not remove the discrepancy. There is a difference in the spin-resolved profiles. Returning to the main figure it is clear that the predicted feature, A, in the spin-resolved profile is absent in the
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Fig. 5. The upper diagram shows the MCP of gadolinium measured along the hexagonal axis, together with three model calculations. The full potential calculation appears to be marginally worse near p 0: The lower figure reproduces the Gd MCP together with that of the insulator EuO. The lines are 4f free atom [32] free atom profiles fitted between 3 and 10 a.u.
data set, yet features at C, E, and F which might be thought of as high momentum components relating to the “A” feature in the first Brillouin zone are present in the data. Clearly, there is a common failing in the potential used in all three calculations. In fact, all directional profiles show deeper fissures than theory predicts. The d-like fifth band governs the higher momenta shoulders and predicts the correct Fermi surface. It is thought that the discrepancy between experiment and theory is most likely due to the failure of the description of the exchange-correlation effects. Interestingly McCarthy’s results for Fe, shown in Fig. 3 show a similar
effect; although the magnetic profile is less structured the major discrepancy between experiment and theory is again in the first Brillouin zone. 3.2. 4f systems Gd and EuO Gadolinium has been measured at varying levels of accuracy from 1987 onwards [29]. It is a soft magnet and has a large spin moment. Many of these early measurements were made at liquid nitrogen temperatures for convenience but a higher temperature is needed to avoid the phase where the caxis moments are canted. The main problem with Gd is not
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experimental, it is the simple fact that the structure is hexagonal and calculations n p for non-cubic systems are not well advanced. We performed LMTO calculations [30] within the LSDA and the GGA approximations and also had access to the FLAPW calculations of Kubo and Asano [31]. Experiment and theory are shown together in Fig. 5(a) where it is evident that: (a) the FLAPW theory does not produce any improvement, in fact it is worse than the LMTO method, in contrast to the situation in Ni; (b) the GGA does not improve on the LSDA approximation, this may be due to the more localised (4f) nature of the magnetisation. The adequacy of the LMTO to describe this hexagonal system is significant because it can be used in Gdbased alloys such as Gd-Y where the magnetic properties are driven by the electronic structure. Finally Fig. 5(b) compares the MCPs of Gd with that of EuO, the real interest in the latter is to establish whether there is an induced moment on the oxygen site, which might be expected as a consequence of the super-exchange model of magnetism in this material. That cannot be deduced from the raw data until current band modelling has been completed but it is interesting to see the comparison between the conductor, Gd, with a positive spin polarised conduction electron contribution. The origin of the small dip in the EuO profile is under investigation.
4. Conclusions The extraction of information from MCPs requires a concerted experimental and theoretical approach; this in turn requires that the model is good enough for the differences between it and the data to be meaningfully interpreted: reconstruction will have to figure more prominently in the interpretation, so that n p rather than J mag p can be considered. Yet the information is there because the photon is a “clean” probe of an equilibrium ground state property, and as such it provides one of the most promising methods of testing LSDA and other assumptions underlying electronic structure calculations in ferromagnets where other synchrotron based techniques either involve excited state (e.g. dichroism, photo emission) or are relatively undeveloped (e.g. diffraction). It can also be used to search for induced moments on “non-magnetic” sites (e.g. Y in Gd-Y, O in EuO) but in all applications empirical interpretation is all but impossible: experiment needs to go hand-in-hand with theory if spin-resolved densities are to be understood by this technique.
Acknowledgements All synchrotron studies are the result of team efforts and we have relied heavily on the work of numerous colleagues in this brief summary. Especial thanks are due to J.E. McCarthy (ESRF and Warwick) for her collaboration in
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the ESRF experiments and permission to reproduce Fig. 3 from her thesis and to S. Dugdale (Geneva and Bristol) for performing the LMTO calculations cited here. The support of the EPSRC in the UK and the ESRF for beamtime and experimental support are particularly acknowledged. References [1] N. Sakai, J Appl Cryst. 29 (1996) 81. [2] F.W. Lipps, H.A. Tolhoek, Physica 20 (1954) p. 85 and p. 395. [3] M. Goldhaber, L. Grodzins, A.W. Sunyar, Phys Rev 109 (1958) 1015. [4] P.M. Platzman, N. Tzoar, Phys Rev B 2 (1970) 3556. [5] F. deBergevin, M. Brunel, Acta Cryst. A 37 (1981) p. 314 and p. 324. [6] N. Sakai, K. Ono, Phys. Rev. Lett. 37 (1976) 351. [7] M.J. Cooper, S.P. Collins, D.N. Timms, A. Brahmia, P.P. Kane, R.S. Holt, D.N. Laundy, Nature 333 (1988) 151. [8] J.E. McCarthy, M.J. Cooper, V. Honkima¨ki, T. Tschetscher, P. Suortti, S. Gardelis, K. Ha¨ma¨la¨inen, D.N. Timms, Nucl. Instrum. Meth. A 401 (1997) 463. [9] H. Grotch, E. Kazes, G. Bhatt, D.A. Owen, Phys. Rev. A 27 (1983) 243. [10] G. Bhatt, H. Grotch, E. Kazes, D. Owen, Phys. Rev. A 28 (1983) 2195. [11] R. Ribberfors, Phys. Rev. B 12 (1975) p. 2067 and p. 3136. [12] V. Honkimaki, private communication. [13] S.P. Collins, M.J. Cooper, S.W. Lovesey, D. Laundy, J. Phys.: Condens. Matter 2 (1990) 6439. [14] M.J. Cooper, E. Zukowski, S.P. Collins, D.N. Timms, F. Itoh, Y. Sakurai, J. Phys.: Condens. Matter 4 (1992) 399. [15] P. Carra, M. Fabrizio, G. Santoro, B.T. Thole, Phys. Rev. B 53 (1996) R5994. [16] D. Kekchrakos, K.N. Trohidou, S. Taddei, Phys. Rev. B 56 (1997) 10812. [17] Y. Tanaka, N. Sakai, Y. Kubo, H. Kawata, Phys. Rev. Lett. 70 (1993) 1537. [18] L. Dobrzynski, A. Holas, Nucl. Instrum. Meth. A 383 (1996) 589. [19] L. Dobrzynski, A. Holas, Phys. Rev. B, 1998, submitted for publication. [20] P.K. Lawson, M.J. Cooper, M.A.G. Dixon, D.N. Timms, E. Zukowski, F. Itoh, H. Sakurai, Phys. Rev. B 56 (1997) 56. [21] Y. Sakurai, Y. Tanaka, T. Ohata, Y. Watanabe, S. Nanao, Y. Ushigama, T. Iwazumi, H. Kawata, N. Shiotani, J. Phys.: Condens. Matter 6 (1994) 9469. [22] J.E. McCarthy, M.J. Cooper, P.K. Lawson, D.N. Timms, S.O. Manninen, K. Hamalainen, P. Suortti, J. Synchrotron Rad. 4 (1997) 102. [23] J.W.M. DuMond, Rev. Mod. Phys. 5 (1933) 1. [24] G. Loupias, J. Petiau, J. de Physique 41 (1980) 265. [25] Y. Sakurai, M. Ito, T. Urai, Y. Tanaka, N. Sakai, T. Iwazumi, H. Kawata, M. Ando, N. Shiotani, Rev. Sci. Instrum. 63 (1992) 1190. [26] J.E. McCarthy, Dissertation for the degree of PhD, University of Warwick, 1997. [27] Y. Kubo, S. Asano, Phys. Rev. B 42 (1990) 4431. [28] M.A.G. Dixon, J.A. Duffy, S. Gardelis, J.E. McCarthy, M.J. Cooper, S.B. Dugdale, T. Jarlborg, D.N. Timms, J. Phys.: Condens. Matter 10 (1998) 2759.
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