UNIVERSITATIS ACTA UPSALIENSIS
UPPSALA
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1440
Theoretical and Computational Studies on the Physics of Applied Magnetism
Magnetocrystalline Anisotropy of Transition Metal Magnets and Magnetic Effects in Elastic Electron Scattering
ALEXANDER EDSTRÖM
ISSN 1651-6214
ISBN 978-91-554-9753-8
Dissertation presented at Uppsala University to be publicly examined in Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 25 November 2016 at 13:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Patrick Bruno (European Synchrotron Radiation Facility).
Abstract
Edström, A. 2016. Theoretical and Computational Studies on the Physics of Applied Magnetism. Magnetocrystalline Anisotropy of Transition Metal Magnets and Magnetic Effects in Elastic Electron Scattering. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1440. 109 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9753-8.
In this thesis, two selected topics in magnetism are studied using theoretical modelling and computational methods. The first of these is the magnetocrystalline anisotropy energy (MAE) of transition metal based magnets. In particular, ways of finding 3d transition metal based materials with large MAE are considered. This is motivated by the need for new permanent magnet materials, not containing rare-earth elements, but is also of interest for other technological applications, where the MAE is a key quantity. The mechanisms of the MAE in the relevant materials are reviewed and approaches to increasing this quantity are discussed. Computational methods, largely based on density functional theory (DFT), are applied to guide the search for relevant materials. The computational work suggests that the MAE of Fe
1-xCo
xalloys can be significantly enhanced by introducing a tetragonality with interstitial B or C impurities.
This is also experimentally corroborated. Alloying is considered as a method of tuning the electronic structure around the Fermi energy and thus also the MAE, for example in the tetragonal compound (Fe
1-xCo
x)
2B. Additionally, it is shown that small amounts (2.5-5 at.%) of various 5d dopants on the Fe/Co-site can enhance the MAE of this material with as much as 70%. The magnetic properties of several technologically interesting, chemically ordered, L1
0structured binary compounds, tetragonal Fe
5Si
1-xP
xB
2and Hexagonal Laves phase Fe
2Ta
1-xW
xare also investigated. The second topic studied is that of magnetic effects on the elastic scattering of fast electrons, in the context of transmission electron microscopy (TEM). A multislice solution is implemented for a paraxial version of the Pauli equation. Simulations require the magnetic fields in the sample as input. A realistic description of magnetism in a solid, for this purpose, is derived in a scheme starting from a DFT calculation of the spin density or density matrix.
Calculations are performed for electron vortex beams passing through magnetic solids and a magnetic signal, defined as a difference in intensity for opposite orbital angular momentum beams, integrated over a disk in the diffraction plane, is observed. For nanometer sized electron vortex beams carrying orbital angular momentum of a few tens of ħ, a relative magnetic signal of order 10
-3is found. This is considered realistic to be observed in experiments. In addition to electron vortex beams, spin polarised and phase aberrated electron beams are considered and also for these a magnetic signal, albeit weaker than that of the vortex beams, can be obtained.
Keywords: Magnetism, Magnetic anisotropy, DFT, Permanent magnets, Electron vortex beams, Electron microscopy, Electron scattering, Multislice methods
Alexander Edström, Department of Physics and Astronomy, Materials Theory, Box 516, Uppsala University, SE-751 20 Uppsala, Sweden.
© Alexander Edström 2016 ISSN 1651-6214
ISBN 978-91-554-9753-8
urn:nbn:se:uu:diva-304666 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-304666)
List of papers
This thesis is based on the following papers, which are referred to in the text by their Roman numerals.
I Stabilization of the tetragonal distortion of Fe x Co 1 −x alloys by C impurities - a potential new permanent magnet
E. K. Delczeg-Czirjak, A. Edström, M. Werwi´nski, J. Rusz, N. V.
Skorodumova, L. Vitos, O. Eriksson Physical Review B 89, 144403 (2014)
II Electronic structure and magnetic properties of L1 0 binary alloys A. Edström, J. Chico, A. Jakobsson, A. Bergman, J. Rusz
Physical Review B 90, 014402 (2014)
III Increased magnetocrystalline anisotropy in epitaxial Fe-Co-C thin films with spontaneous strain
L. Reichel, G. Giannopoulos, S. Kauffman-Weiss, M. Hoffmann, D.
Pohl, A. Edström, S. Oswald, D. Niarchos, J. Rusz, L. Schultz, S.
Fähler
Journal of Applied Physics 116, 213901 (2014)
IV Toward Rare-Earth-Free Permanent Magnets: A Combinatorial Approach Exploiting the Possibilities of Modeling, Shape
Anisotropy in Elongated Nanoparticles, and Combinatorial Thin-Film Approach
D. Niarchos, G. Giannopoulos, M. Gjoka, C. Sarafidis, V. Psycharis, J.
Rusz, A. Edström, O. Eriksson, P. Toson, J. Fidler, E.
Anagnostopoulou, U. Sanyal, F. Ott, L.-M. Lacroix, G. Viau, C. Bran, M. Vazquez, L. Reichel, L. Schultz, S. Fähler
JOM 67, 1318-1328 (2015)
V From soft to hard magnetic Fe-Co-B by spontaneous strain: a combined first principles and thin film study
L. Reichel, L. Schultz, D. Pohl, S. Oswald, S. Fähler, M. Werwi´nski, A. Edström, E. K. Delczeg-Czirjak, J. Rusz
Journal of Physics: Condensed Matter 27, 476002 (2015)
VI Magnetic properties of (Fe 1 −x Co x ) 2 B alloys and the effect of doping by 5d elements
A. Edström, M. Werwi´nski, D. Iu¸san, J. Rusz, O. Eriksson, K. P.
Skokov, I. A. Radulov, S. Ener, M. D. Kuzmin, J. Hong, M. Fries, D.
Yu. Karpenkov, O. Gutfleisch, P. Toson, J. Fidler Physical Review B 92, 174413 (2015)
Erratum: Magnetic properties of (Fe 1 −x Co x ) 2 B alloys and the effect of doping by 5d elements
A. Edström, M. Werwinski, D. Iu¸san, J. Rusz, O. Eriksson, K. P.
Skokov, I. A. Radulov, S. Ener, M. D. Kuzmin, J. Hong, M. Fries, D.
Yu. Karpenkov, O. Gutfleisch, P. Toson, J. Fidler Physical Review B 93, 139901(E) (2016)
VII Magnetic properties of Fe 5 SiB 2 and its alloys with P, S and Co M. Werwi´nski, S. Kontos, K. Gunnarsson, P. Svedlindh, J. Cedervall, V. Höglin, M. Sahlberg, A. Edström, J. Rusz, O. Eriksson
Physical Review B 93, 174412 (2016)
VIII Enhanced and Tunable Spin-Orbit Coupling in Tetragonally Strained Fe-Co-B Films
R. Salikhov, L. Reichel, B. Zingsem, R. Abrudan, A. Edström, D.
Thonig, J. Rusz, O. Eriksson, L. Schultz, S. Fähler, M. Farle, U.
Wiedwald
Submitted to Physical Review B
IX On the origin of perpendicular magnetic anisotropy in strained Fe-Co(-X) films
L. Reichel, A. Edström, D. Pohl, J. Rusz, O. Eriksson, L. Schultz, S.
Fähler
Submitted to Journal of Physics D: Applied Physics
X Towards a magnetic phase diagram of the Fe 5 SiB 2 -Fe 5 PB 2 alloy system
D. Hedlund, J. Cedervall, A. Edström, S. Kontos, O. Eriksson, J. Rusz, P. Svedlindh, M. Sahlberg, K. Gunnarsson
Manuscript
XI Magnetocrystalline anisotropy of Laves phase Fe 2 Ta 1 −x W x from first principles - the effect of 3d-5d hybridisation
A. Edström
Manuscript
XII New permanent magnets; what to look for, and where L. Nordström, A. Edström, D. Carvalho de Melo Rodrigues, A.
Burlamaqui-Klautau, J. Rusz, O. Eriksson Submitted to Nature Materials
XIII Prediction of a Larger Local Magnetic Anisotropy in Permalloy D. Carvalho de Melo Rodrigues, A. Burlamaqui-Klautau, A. Edström, J. Rusz, L. Nordström, M. Pereiro, B. Hjörvarsson, O. Eriksson Manuscript
XIV Elastic Scattering of Electron Vortex Beams in Magnetic Matter A. Edström, A. Lubk, J. Rusz
Physical Review Letters 116, 127203 (2016)
XV Magnetic effects in the Paraxial Regime of Elastic Electron Scattering
A. Edström, A. Lubk, J. Rusz
Physical Review B (accepted for publication) Reprints were made with permission from the publishers.
Comments on the contributions of the author
In all the papers listed above, I participated in discussions and contributed
to the writing process. In Paper I, I performed VCA and CPA calculations
of magnetic properties. In Paper II, I performed most calculations and had
the main responsibility of writing the paper. In Paper III, Paper VIII, Paper IX
and Paper X, I performed the computational work. In Paper V, I performed the
SPR-KKR calculations. In Paper VI, I performed the SPR-KKR calculations
and had the main responsibility of writing the paper. In Paper XI, I performed
all the work involved. In Papers XIV-XV, I had the main responsibility in
performing analytical work, code implementation and computational work, as
well as writing the papers.
Contents
1 Introduction
. . . .9
1.1 Magnetocrystalline Anisotropy and Permanent Magnets
. . . .10
1.2 Magnetism in the Transmission Electron Microscope
. . .13
2 Elements of the Theory of Magnetism
. . .16
2.1 Relativistic Electrons
. . .16
2.1.1 Non-Relativistic Limit and the Scalar Relativistic Approximation
. . . .19
2.1.2 Spin-Orbit Coupling and the Magnetocrystalline Anisotropy
. . . .22
2.2 Exchange Interactions and the Heisenberg Hamiltonian
. . . .35
2.3 Microscopic Magnetic Fields in a Solid
. . .38
3 Computational Methods
. . .42
3.1 Density Functional Theory
. . .42
3.1.1 FP-LAPW
. . .46
3.1.2 SPR-KKR
. . . .47
3.1.3 Models to Treat Disorder
. . .49
3.1.4 Computing the MAE
. . .52
3.1.5 Exchange Coupling Parameters
. . .60
3.2 Monte Carlo Simulations
. . . .60
3.3 TEM Simulations - The Multislice Approach
. . . .63
3.3.1 Multislice Solution to Paraxial Pauli Equation
. . . .67
4 Results
. . . .69
4.1 Magnetocrystalline Anisotropy and Permanent Magnet Materials
. . .69
4.1.1 Fe 1 −x Co x Alloys
. . .69
4.1.2 (Fe 1 −x Co x ) 2 B
. . . .73
4.1.3 L1 0 Binary Compounds
. . .76
4.1.4 Fe 5 Si 1 −x P x B 2
. . . .79
4.1.5 Fe 2 Ta 1 −x W x
. . .81
4.1.6 Other Interesting Hard Magnetic Materials
. . .82
4.2 Magnetic Effects in Elastic Electron Scattering
. . . .83
4.2.1 Electron Vortex Beams
. . .84
4.2.2 Spin Polarised Scattering
. . .88
4.2.3 Aberrated Probes
. . .88
5 Summary and Conclusions
. . .91
6 Sammanfattning
. . .94
7 Acknowledgements
. . .97
References
. . . .99
1. Introduction
Magnetism has repeatedly revolutionised human society; first about a mil- lennium ago [1], by allowing navigation across the planet with the aid of the compass. Later by allowing the conversion between electrical and me- chanical energy with generators and motors made from permanent magnets, as well as efficient energy transfer using transformers with magnetic cores.
Also more recently, by allowing the enormous and exponentially increasing amounts of digital information, that has become readily available to us, to be stored on magnetic hard drives 1 . However, it was not until the emergence of quantum mechanics and its relativistic extension, in the early 20th century, that the fundamental, microscopic origins of magnetism could even begin to be understood [3]. Due to its theoretical complexity and far reaching tech- nological importance, the field remains under extensive research activity and promises to continue improving our living standard with the development of novel magnetic technology, such as spintronics [4], magnonics [5], skyrmion- ics [6], magnetic cooling and more efficient clean energy production [7], or yet unimagined technological advances.
In recent decades, the understanding and development of the field of mag- netism has benefited greatly from advanced computer simulations [8–10], made possible by the exponential increase in available computer power 2 . It is the purpose of this thesis to use such methods to contribute to two selected top- ics in magnetism. With increasing need for clean energy and transport, recent years have seen a growing production of wind power and hybrid or electric vehicles [7, 12]. Correspondingly, there has been an increase in demand for high energy density permanent magnet materials, made from cheap and read- ily available, constituent elements. Finding such materials is the first topic addressed, as introduced further in Sec. 1.1. With the desire for a continuing size reduction of magnetic technology, e.g. in going beyond 1 Tb/in 2 mag- netic storage capabilities 3 with bit patterned media [13, 14] and related tech- nologies, and because solid state magnetism is of atomic origin, experimental
1
It was estimated that the world’s total information storage exceeded 10
20bytes, i.e. one hun- dred billion gigabytes, already a decade ago, with a doubling rate of slightly more than one year [2]. In recent years, the largest fraction of this information has been stored on magnetic storage devices.
2
The total computing capacity of the 500 most powerful computers in the world has been dou- bling in less than two years during recent decades and reached 593 Pflop/s in 2016 [11].
3
Current magnetic storage technologies, approaching 1 Tb/in
2, represent a 500 million fold in-
crease in storage density compared to the first IBM hard disk drive with a capability of 2 kB/in
2in 1956 [13]. Further increases call for novel magnetic technology in the nano regime.
characterisation techniques capable of reaching high, preferably atomic, spa- tial resolution are highly desirable. This leads us to the second topic, which considers magnetic effects in electron scattering theory, in order to explore new routes aimed at this purpose, as briefly introduced in Sec. 1.2.
1.1 Magnetocrystalline Anisotropy and Permanent Magnets
Permanent magnets are often characterised in terms of the energy product (BH) max , which describes twice the maximum energy density which can be stored in the magnet [15]. As illustrated in Fig. 1.1a), there was a tremendous development in the energy products of available permanent magnets through- out the 20th century, ending with the high performance Nd 2 Fe 14 B magnets discovered in the early 1980’s [16]. As will be discussed further below, this large energy product is the result of a combination of a large saturation mag- netisation and a large enough magnetocrystalline anisotropy energy (MAE), which provides the magnet with coercivity. Nd 2 Fe 14 B has now remained the most high performing permanent magnet for more than three decades and to find an alternative which outperforms it still appears challenging, although it has been theoretically suggested that nanostructured magnets should be able to achieve a (BH) max of 1 MJ /m 3 [17]. From the discovery and over the coming twenty years, Nd 2 Fe 14 B magnets emerged into a billion dollar industry, while still growing, as the magnets became crucial in a wide range of technological applications [7]. The best performing SmCo 5 and Nd 2 Fe 14 B magnets get their unique properties by combining magnetic transition metal elements with rare earth (RE) elements. As illustrated in Fig. 1.1b) there was a huge increase in RE prices between 2010 and 2012. This happened because of economical and political reasons which have been discussed by various authors [7, 18–20] and most RE elements are now considered as critical elements 4 . This has led to significant research efforts towards finding new permanent magnet materials, with the properties of the rare earth transition metal compounds, but without the rare earth elements. One such effort, to which this thesis is a part, is de- scribed in Paper IV.
An upper limit for the energy product can be expressed in terms of the intrinsic material parameter saturation magnetisation M s as [20]
(BH) max < 1
4 μ 0 M s 2 , (1.1)
which shows that a large saturation magnetisation is desirable for a strong permanent magnet. However, a large coercive field H c , which describes how
4
Although there is no general definition of critical elements, it usually takes into account esti-
mates of supply risk and economical importance. Both the EU and USA consider RE elements
as critical [21, 22].
1900 1920 1940 1960 1980 2000 (BH) max (kJ/m 3 )
0 100 200 300 400 500
Steel Alnico Ferrite Sm-Co Nd-Fe-B
(a) Development of energy products of permanent magnets throughout the 20th century.
Reproduced from Ref. [7, 23].
2004 2006 2008 2010 2012 2014
Price in USD/kg
0 200 400 600
Dy-oxide price/5 Ce-oxide price La-oxide price Eu-oxide price/10 Nd-oxide price
(b) Development in RE oxide prices with time from 2004 to 2014. Reproduced from Ref. [24].
Figure 1.1.
difficult it is to rotate the magnetisation of the magnet, is also needed. An additional upper limit for the energy product is μ 0 M s H c , while the coercivity is bounded by the anisotropy field H c < H a . For the important case of a uniaxial crystal, the anisotropy field is
H a = 2K
μ 0 M s , (1.2)
where K is the uniaxial magnetic anisotropy. This implies that a large satu- ration magnetisation is detrimental to the hard 5 magnet properties, unless the anisotropy is correspondingly large. These arguments have been discussed in
5
Hard magnets have large anisotropy and coercivity while the opposite is true for soft magnets.
terms a hardness parameter
κ =
K
μ 0 M s 2 , (1.3)
with the minimum requirement of κ > 1/2 for a material to possibly be useful as a permanent magnet [20]. However, a more realistic requirement has been suggested to be κ > 1 [20, 25], implying K > μ 0 M s 2 , i.e. that the anisotropic energy density is greater than the magnetisation energy density. Together with the requirement that the Curie temperature T C should be well above to oper- ating temperature (usually room temperature or higher), this provides specifi- cations for a good permanent magnet in terms of the three important intrinsic magnetic material properties M S , MAE and T C . The saturation magnetisation and Curie temperature should be as large as possible, with the constraint that K > μ 0 M s 2 . This is useful because these three material properties are available from first principles electronic structure calculations, as will be discussed in Sec. 3. This kind of computational methods are, therefore, expected to be very useful in finding new permanent magnet materials with the desired properties, which is an important part of this thesis.
Table 1.1 contains a summary of some intrinsic (T C , M s , MAE and κ) and extrinsic properties (H c and (BH) max ) of RE and ferrite permanent magnets compared to transition metals bcc Fe and hcp Co. From this table we see that the simple and abundant material bcc Fe has a higher Curie temperature and saturation magnetisation than the best performing RE-transition metal based magnets. However, the MAE is two orders of magnitude smaller, whereby a sizeable coercivity and energy product can never be obtained. It is enough to go to hcp Co to gain one order of magnitude in the MAE, which is because of the hexagonal (uniaxial), rather than cubic, crystal structure, as will be ex- plained in more detail in Sec. 2.1.2. However, hcp Co still has an MAE one order of magnitude smaller than the RE-transition metal magnets, together with a quite high saturation magnetisation. This results in a small value for κ, which makes the material useless as a permanent magnet. The physics of magnetism in RE elements is quite different from that in transition metal ele- ments [9]. In particular, RE elements typically have localised f-electrons with significant orbital magnetic moments and strong spin-orbit coupling (SOC), which, as discussed in Sec. 2.1.2, is the main source of MAE in the relevant materials. In contrast, transition metal magnets usually have a weak SOC and largely quenched orbital magnetic moments. This leads to the grand challenge in finding good permanent magnet materials without RE or other heavy ele- ments with strong SOC. One of the main purposes of this thesis is to find so- lutions to this problem. The theory of MAE in transition metal based magnets is therefore reviewed in Sec. 2.1.2 and possible paths towards large MAE in transition metal magnets are discussed. The conclusions from that discussion are largely also the main message of Paper XII.
After gaining an understanding of the problem at hand and potential paths
towards a solution, computational methods based on first principles electronic
Table 1.1. A summary of the properties of some high performing rare-earth magnets, a ferrite alternative and the transition metals bcc Fe and hcp Co. The relevant per- manent magnet properties provided are Curie temperature T C , coercivity H c , energy product (BH) max , saturation magnetisation M s , magnetocrystalline anisotropy energy (MAE), as well as the hardness parameter κ. Data were taken from Ref. [26–28].
The extrinsic properties depend on the microstructure of the material and should be considered as an estimate of realistic values. An upper limit for the energy product of bcc Fe and hcp Co was estimated from μ 0 M s H c .
Nd 2 Fe 14 B SmCo 5 BaFe 12 O 19 bcc Fe hcp Co
T C (K) 588 1020 740 1043 1388
μ 0 H c (T) 1.21 0.90 0.15 7 · 10 −5 5 · 10 −3
(BH) max (kJ/m 3 ) 512 231 45 < 0.1 < 7
μ 0 M s (T) 1.61 1.22 0.48 2.21 1.81
MAE (MJ/m 3 ) 4.9 17.2 0.33 0.048 0.53
κ 1.54 3.81 1.34 0.11 0.45
structure theory are applied to explore these directions. Mainly, two different implementations of density functional theory (DFT) within the generalised gradient approximation, namely the full potential linearised augmented plane waves and spin polarised relativistic KKR methods, are used, as described in Sec. 3.1. These methods allow for the calculations of saturation magnetisation and MAE, but also other ground state properties, including the Heisenberg exchange coupling parameters J i j (see Sec. 2.2 and Sec. 3.1.5). With the J i j as input, the Curie temperatures can be evaluated, for example using Monte Carlo simulations, as described in Sec. 3.2. Using these different computational methods, the magnetic properties of various transition metal based materials are studied and ways of enhancing the MAE are considered. In particular the effects of dopants or alloying are considered in several different systems, whereby the theory of alloys is briefly reviewed in Sec. 3.1.3.
1.2 Magnetism in the Transmission Electron Microscope
A wide variety of experimental techniques are available to characterise mag-
netic materials; for example, elastic neutron scattering can be used to exam-
ine magnetic structures, while inelastic neutron scattering allows one to study
magnetic excitations, such as magnons. A small scattering cross section, how-
ever, limits neutrons to probe bulk samples and they are not useful to study
magnetism at the nanoscale [9]. Magneto-optical effects (e.g. Kerr or Faraday)
can also be used to probe magnetic materials but, in this case, the diffraction
limit of light limits the spatial resolution and again nanomagnetism is out of
reach. This issue can be solved by the use of x-rays and a powerful technique is
found in x-ray magnetic circular dichroism (XMCD), where one measures the absorption spectra for oppositely circularly polarised x-rays. The x-rays ex- cite core electrons into the conduction band and in magnetic materials, which have different electronic structure for opposite spin channels, a difference in the x-ray absorption is observed for opposite polarisations. By application of the so called sum rules [29, 30] element specific spin and orbital magnetic moments can be probed. Furthermore, by making use of the relation between magnetocrystalline anisotropy and orbital moment anisotropy, pointed out by Bruno [31] and discussed in Sec. 2.1.2, an estimate for the magnetic anisotropy can be found by measuring the orbital magnetic moment for different magneti- sation axes [32].
With continuous reduction in size of magnetic technology, and to allow ex- ploration of novel phenomena in the atomic regime, magnetic characterisation techniques with atomic or subatomic resolution, which is beyond the capa- bility of XMCD (L-edge excitations in Fe are in the order of 700 eV, corre- sponding to a photon wavelength around 1.8 nm), would be of immense value.
Furthermore, XMCD experiments are performed at large scale synchrotron facilities, while it is clearly highly advantageous with more readily available techniques that can be routinely performed in small scale laboratories. A new technique carrying the advantages of XMCD while improving some of the problems was potentially made available with the proposition [33] and exper- imental realisation [34] of an electron analogue of XMCD, namely electron magnetic circular dichroism (EMCD). In EMCD, electron energy loss spec- troscopy (EELS) is used to observe the inelastic scattering of electrons in a transmission electron microscope (TEM). A dichroic signal carrying the same information as XMCD (although separate determination of spin and orbital moments is more challenging than their ratio) [35] is obtained by comparing electron energy loss spectra acquired at points in the diffraction plane cor- responding to momentum transfer obeying certain conditions. In scanning transmission electron microscopy spatial resolution well beyond atomic reso- lution is possible [36] and for the case of EMCD nanometer scale measure- ments have been reported [37]. From this it is clear that EMCD is potentially a very valuable experimental technique for observing magnetic phenomena at the atomic scale. Nevertheless, challenges in obtaining high signal-to-noise ratio and other drawbacks, such as the need for crystalline samples, has thus far hindered it from becoming a routinely used method.
The EMCD technique gained new attention with the discovery of electron vortex beams [38–40] (EVB), i.e. electron beams with a phase winding cor- responding to a well defined, quantised orbital angular momentum (OAM).
Such beams should allow for dichroic signals to be observed in EELS experi-
ments in the TEM [39]. However, computational work has indicated that using
inelastic scattering of EVBs to measure a dichroic signal is only useful in the
atomic resolution regime [41] and so far this is technologically challenging to
achieve, although there have been experimental efforts in that direction [42].
The experimental realisation of EVBs with very large OAM, in the order of hundreds of ¯h [40, 43, 44], potentially allows for another type of mag- netic interaction of the electron beams with magnetic matter that might be experimentally detectable in the TEM. The Pauli equation for an electron in a homogeneous magnetic field B hom can be written (in Coulomb gauge)
ˆp 2 2m + e
m ( ˆL + 2ˆS) · B hom − eV(r)
Ψ(r) = EΨ(r), (1.4)
and there is an interaction between the field and the OAM of the beam which increases linearly with the magnitude of the OAM. Clearly, there is a similar interaction also for the spin angular momentum of the beam but the advantage of the OAM is that it can potentially be increased by two orders of magnitude or more. The effect of homogeneous magnetic fields on EVBs was recently discussed [45] and found to result in a small shift in energy and OAM of the beams, which could potentially be measurable for large fields. To understand the effect of magnetism in a magnetic solid on an EVB, however, requires a more advanced analysis which can be done with the computational meth- ods discussed in Sec. 3.3.1 and input obtained according to the discussion in Sec. 2.3. Sec. 4.2.1 provides a brief introduction to EVBs and a summary of results for the elastic scattering of such beams through magnetic materials presented in Papers XIV-XV.
The possibility to have a large OAM in vortex beams and the difficulty in
obtaining electron beams with a high degree of spin polarisation, both result in
advantages in using EVBs to observe magnetism in the TEM, in comparison
to spin polarised beams. Nevertheless, recent developments in spin polarisa-
tion technology [46] make it interesting to also consider the scattering of spin
polarised electron beams in magnetic materials. In addition to this, it has re-
cently been shown in theory [47, 48] and later experimentally confirmed [49],
that EVBs is only one of several possible phase distributions that can yield
a magnetic signal in EELS experiments, while the alternatives correspond to
phase aberrations. Such aberrations can be controlled in modern aberration
corrected electron microscopes [50], thus providing an interesting alternative
path to observing magnetism in the TEM. Elastic scattering of spin polarised
electron beams is studied in Papers XIV-XV and aberrated electron beams in
Paper XV. These topics are also discussed further in Sec. 4.2.2-4.2.3.
2. Elements of the Theory of Magnetism
This chapter gives an introduction and overview of those areas of the theory of magnetism which are most important to understand the work behind this thesis. It begins, in Sec. 2.1, by discussing the relativistic nature of magnetism and in Sec. 2.1.2, the spin-orbit coupling and its relation to the magnetocrys- talline anisotropy. It continues in Sec. 2.2 by discussing the exchange inter- actions which lead to magnetic ordering and how it can be described in terms of exchange coupling parameters and the Heisenberg Hamiltonian. Sec. 2.3 discusses how the microscopic magnetic fields in a solid can be described starting from a calculation of the spin density, which is important as input for the calculations discussed in Sec. 3.3 and used in Papers XIV-XV.
2.1 Relativistic Electrons
Magnetism arises due to the quantum mechanical spin or orbital angular mo- mentum of electrons. The spin angular momentum was initially rather arti- ficially introduced into the theory of quantum mechanics to explain the fine structure of the hydrogen atom [51]. It was not until Dirac introduced a rel- ativistic wave equation [52, 53] for the electron that the spin became well understood as an intrinsic angular momentum necessary for a Lorentz invari- ant version of quantum mechanics. Moreover, relativistic effects neglected in the Schrödinger equation are of importance in describing electrons in atomic core states and the relativistic spin-orbit coupling (SOC) which, as will be dis- cussed later on, brings in a rich new array of physical phenomena, including the magnetocrystalline anisotropy which is essential for permanent magnets.
Also in the context of transmission electron microscopy a relativistic descrip- tion of electrons is important because of the large kinetic energies involved.
The Dirac equation may, including electromagnetic interactions, be written in the following way [54]
γ μ
i ∂ μ + eA μ
c − mc 2
ψ = 0, (2.1)
where −e is the electron charge, c the speed of light, γ μ are the Dirac matrices, A μ is the electromagnetic potential and m is the electron mass. Alternatively it might, after separating out the time dependence, be written
ααα · (−i¯h∇ + eA)c − eV + βmc 2
ψ = Eψ, (2.2)
where A is the magnetic vector potential, V is the scalar potential, β = γ 0 =
I 2 ×2 0 0 −I 2 ×2
and ααα =
0 σσσ σσσ 0
, (2.3)
where σσσ = (σ x ,σ y ,σ z ) contains the Pauli matrices. For a complete relativis- tic description with many particles and creation and annihilation of these, the Dirac equation should be considered in second quantised form with a solution in terms of operators on a Fock space, rather than elements of a Hilbert space.
However, the necessary insights we require can be obtained already by con- sidering the equation in the first quantised form whereby we restrict ourself to this situation. Furthermore, the many-body Dirac equation is reduced to single particle Kohn-Sham-Dirac equations, of the same form as Eq. 2.2, within rela- tivistic density functional theory [55]. Hence, it is the equation which is solved for all electrons in the SPR-KKR method and for the core electrons only in the FP-LAPW method, as will be further discussed in Sections 3.1.1-3.1.2. Often however, solving Eq. 2.2 is more complicated than what is necessary to de- scribe the phenomena of interest with good accuracy, so that simplifications and approximations can beneficially be applied. One such simplification is to expand the equation in the non-relativistic limit v /c 1 as discussed in Sec. 2.1.1. This naturally introduces a term describing the spin-orbit coupling, which is essential for magnetocrystalline anisotropy, and allows for applying the so called scalar relativistic approximation.
Perhaps the simplest case for which Eq. 2.2 can be solved is the free elec- tron case (A μ = 0), where the solution appears in the form of the usual plane waves [54, 55]. For later purposes, it is more relevant to consider this solution in cylindrical coordinates (r =
x 2 + y 2 , ϕ = tan −1 y x , z = z), where it reads
ψ l (r) =e −iEt/¯h e ik
zz
⎡
⎢ ⎢
⎣
⎛
⎜ ⎝
1 + mc E
2χ
1 − mc E
2cos θσ z χ
⎞
⎟ ⎠e il ϕ J l (k ⊥ r)+
i
1 − mc 2 E
⎛
⎜ ⎜
⎝ 0 0
−β sinθ 0
⎞
⎟ ⎟
⎠e i (l−1)ϕ J l −1 (k ⊥ r)+
i
1 − mc 2 E
⎛
⎜ ⎜
⎝ 0 0 0 α sinθ
⎞
⎟ ⎟
⎠e i (l+1)φ J l+1 (k ⊥ r )
⎤
⎥ ⎥
⎥ ⎦ , (2.4)
where J l (x) are Bessel functions, χ =
α β
(2.5)
and θ is the angle defined by
cos θ = k ⊥
k 2 ⊥ + k 2 z , (2.6)
with k z being the component of the wave vector parallel to the z-direction and k ⊥ the component projected in the plane perpendicular to the z-direction. The energy and wave vector are related via E 2 − m 2 c 4 = c 2 ¯h 2 (k 2 ⊥ + k 2 ). Eq. 2.4 is essentially the relativistic description of the electron vortex beams which are studied in Paper XIV and Paper XV and discussed further in Section 4.2.1. The relativistic electron vortex beam in Eq. 2.4 has been discussed in some detail by Bliokh et al. [56]. It is interesting to compare the cylindrical solution to the free electron Dirac equation to that of the free particle Schrödinger equation, which reads
Ψ l (r) = e −iEt/¯h e ik
zz e il ϕ J l (k ⊥ r ). (2.7) The non-relativistic case in Eq. 2.7 is proportional to the upper two compo- nents of Eq. 2.4, as one would expect since these are the non-zero compo- nents in the non-relativistic limit where E → mc 2 . The lower two components are, however, different from the non-relativistic case as they carry additional contributions proportional to Ψ l±1 . The state in Eq. 2.7 is an eigenstate of the z-projected orbital angular momentum operator and has an orbital angular momentum of l ¯h since L z Ψ l = −i¯h ∂ϕ ∂ Ψ l = l¯hΨ l . The relativistic state ψ l , on the other hand, clearly is not an eigenstate of L z . By introducing ψ l,s , spin polarised in the z-direction with spin up (s = + 1 2 ) spinor χ = (1,0) T or spin down (s = − 1 2 ) spinor χ = (0,1) T , we have an eigenstate of J z = L z + S z , with J z ψ l ,s = ¯h(l + s)ψ l ,s where the spin operator is
S = ¯h 2 ΣΣΣ = ¯h
2
σσσ 0 0 σσσ
. (2.8)
For a state ψ l ,s the latter terms in Eq. 2.4 are proportional to Ψ l ±2s . As pointed out by Bliokh et al. [56], this can be considered as the result of an intrinsic spin-orbit interaction which vanishes both in the non-relativistic (E → mc 2 ) and paraxial ( k k
⊥z
→ 0) limits.
Another case which is interesting to consider is that of spherically symmet- ric potentials (V (r) = V(r)), such as the Coulomb potential for hydrogen-like atoms. The solution in this case is [54]
ψ k j ,m (r,θ,φ) =
f k (r)Y j,m k (θ,φ) ig k (r)Y j,m −k (θ,φ)
, (2.9)
where f k and g k are radial functions, Y j,m k (θ,φ) are generalised spherical har- monics
Y j,m k (θ,φ) = −sgnk
k + 1 2 − m 2k + 1
1 0
Y l,m−
1 2+
k + 1 2 + m 2k + 1
0 1
Y l,m+
1 2. (2.10) The indices j and m denote the total angular momentum quantum numbers and
k =
l if l = j + 1 2
−(l + 1) if l = j − 1 2 (2.11) is a quantum number related to the parity of the solution. The radial functions are solutions to
d f k (r)
d = − 1 + k
r f k (r) + 1 c¯h
E + mc 2 + eV(r)
(2.12) d g k (r)
d = k − 1
r g k (r) − 1 c¯h
E − mc 2 + eV(r)
. (2.13)
Here can be noted that the orbital or spin angular momentum operators in- dividually do not commute with the Dirac Hamiltonian while total angular momentum and parity do. Typically, in condensed matter, those electrons for which relativistic effects tend to be most important are tightly bound core states. These are also, to a good approximation, in a spherical potential so that it is appropriate to describe them with solutions of the form given in Eq. 2.9.
2.1.1 Non-Relativistic Limit and the Scalar Relativistic Approximation
If one does not wish to work with the full four-component Dirac formalism, introduced in the previous section, but still wishes to retain certain relativistic effects, it is appropriate to make an expansion in the non-relativistic limit,
v
c 1, and only keep terms up to a certain order. The first step in doing so is to assume a solution of the form [54]
ψ(r) = χ(r)
η(r)
, (2.14)
where χ and η each has two components. A useful next step is to perform a Foldy-Wouthuysen transformation, where one introduces a unitary operator
U = Aβ + ααα · p
2mc A =
1 − p 2
4m 2 c 2 . (2.15) Transforming the Dirac equation according to H = UHU −1 and ψ = Uψ, performing some algebra and eventually only keeping terms to order
v c
2
leads to a decoupling of χ and η and a Hamiltonian H = (p + eA) 2
2m − eV + e
m S · B − (p + eA) 4 8c 2 m 3 −
− e¯h 2
8m 2 c 2 ∇ 2 V − e 2m 2 c 2 S ·
∇V × (p + eA)
, (2.16)
with spin operator S = ¯h 2 σσσ. The first terms in this equation make up the non- relativistic Schrödinger Hamiltonian and then comes the Zeeman term
H Zeeman = − e
m S · B, (2.17)
where B = ∇ × A is the magnetic flux density. After that comes a relativis- tic momentum correction, the Darwin term and finally the spin-orbit cou- pling (SOC). If one assumes a spherically symmetric scalar potential, the SOC (without the eA part) takes on the well known form
H SOC = − e 2c 2 m 2 r
dV (r)
dr S · L = ξ(r) L · S
¯h 2 , (2.18)
where L = r × p is the orbital angular momentum operator and ξ(r) = − e¯h 2
2c 2 m 2 r dV (r)
dr (2.19)
is the spin-orbit coupling constant. For the spherical potential V (r) = 4 πε eZ
0r of a hydrogen-like atom, the expectation value of the SOC constant, with respect to the non-relativistic eigenstates |n,l, is 1
ξ n ,l =
ξ(r) = n,l|ξ(r)|n,l = Z 4 α 4 mc 2
2n 3 l (l + 1 2 )(l + 1) , (2.20) where Z is the atomic number, α = 4 πε 1
0e ¯hc
2is the fine structure constant and n and l denote principal and angular momentum quantum numbers, respectively.
From this expression it is clear that the SOC becomes particularly important for states with low angular momentum in heavy atoms with large Z. For more realistic many electron atoms or solids, the SOC constant can be calculated using various methods of electronic structure theory. Results of such calcu- lations are shown in Fig. 2.1 where it is again clear that ξ increases with Z and, in a given series of the periodic table, the increase is approximately pro- portional to Z 2 . As is discussed in the coming section, the magnetocrystalline anisotropy is a result of SOC and therefore tends to be stronger in materials with large ξ. This is highly relevant for the part of this work which deals with
1
Easily evaluated using n,l|
r13|n,l =
m3c3α3Z3¯h3n3l(l+1)(l+12)
[57], where |n,l are non-relativistic
eigenstates of the hydrogen-like atom.
Z 2
0 2000 4000 6000 8000
(meV)
0 50 100 150 200 250
Figure 2.1. Calculated SOC constants for various elements. Reproduced from Ref. [58].
finding transition metal magnets with large magnetocrystalline anisotropy and it is the source to one of the main challenges in obtaining magnetic materials with large magnetocrystalline anisotropy, without the use of scarcely available and expensive elements. Elements with Z significantly larger than the value of Z = 26 for Fe tend to be less abundant than those with smaller Z.
The Hamiltonian in Eq. 2.16 acts on a two-component spinor
ψ(r) = ψ ↑
ψ ↓
, (2.21)
where ψ ↑ and ψ ↓ represent spin up and spin down electrons, respectively. The
SOC is the only term in Eq. 2.16 containing off-diagonal elements and hence
coupling the spin up and spin down electrons to each other. Ignoring the SOC
and using only the diagonal terms in that Hamiltonian is sometimes referred
to as the scalar relativistic approximation.
2.1.2 Spin-Orbit Coupling and the Magnetocrystalline Anisotropy
Magnetocrystalline anisotropy is the free energy dependence on magnetisa- tion direction, i.e. F = F( ˆM), where ˆM = (sinθ cosφ,sinθ sinφ,cosθ) is the direction of the magnetisation (spin quantisation axis) relative to the crys- tal lattice. This effect was first experimentally observed and described phe- nomenologically, based on anisotropy constants and crystal symmetries, with the requirement that the dependence of the free energy on the magnetisation direction should have the same symmetries as the crystal lattice [59]. Further- more, time reversal symmetry dictates that F ( ˆM) = F(− ˆM), whereby only even powers of sin θ are allowed. For example, in a uniaxial (e.g. tetragonal or hexagonal) crystal the leading contributions are [28, 59]
F = F 0 + K 1 sin 2 θ + K 2 sin 4 θ + ..., (2.22) where F 0 contains all isotropic energy contributions and K i are the anisotropy constants. Further terms will depend on the particular uniaxial crystal sym- metry and also contain φ dependence. For a tetragonal crystal a term of the form K 3 sin 6 θ cos4φ appears. For a hexagonal crystal with six-fold rota- tional symmetry a K 3 sin 6 θ cos6φ term appears, while for hexagonal crystals with three-fold rotational symmetry (for example the Laves phase structure of Fe 2 Ta 1 −x W x studied in Paper XI) an additional K 3 sin 6 θ cos3φ is allowed.
For a cubic structure on the other hand, the leading contribution is of fourth order and the energy is
F = F 0 + K 1 (α x 2 α y 2 + α x 2 α z 2 + α y 2 α z 2 ) + K 2 α x 2 α y 2 α z 2 + ..., (2.23) where α i are the directional cosines of the magnetisation direction ( α x = ˆx· ˆM and similarly for y and z).
That the microscopic origin of this anisotropy is related to the SOC was suggested by Van Vleck [60], since this is the link coupling the spin to the real space crystal symmetry via the orbital angular momentum. As described in the previous section, the spin-orbit Hamiltonian is H SOC = ξL · S, which is often conveniently rewritten using
L · S = 1
2 (L + S − + L − S+) + L z S z , (2.24) where we have introduced the ladder operators L ± = L x ± iL y and S ± = S x ± iS y . If one is mainly interested in the transition metal d-electron magnetism, then the SOC can be treated as a perturbation. This is motivated by the size of the SOC constant ξ being much smaller (less than 100 meV) than the band- width (several eV) in the relevant magnetic 3d-metals 2 , so that the size of
2
It is interesting to note that the size of the SOC constant determines an upper limit for the
MAE. At most one could therefore expect an MAE of 50-100 meV in 3d magnets. In practice it
the perturbation is much smaller than the typical separation of energy states under consideration. A study based on perturbation theory was done in sem- inal work by Brooks [62], who attempted to describe the anisotropy in cubic iron and nickel, but did not have access to an accurate description of the elec- tronic structure. Important contributions in this line of work was also done by Kondorskii and Straube [63], who used a Hartree-Fock band structure with perturbative SOC to calculate and analyse the MAE of fcc Ni. They reached the important conclusion that regions in the Brillouin zone which allow for coupling between occupied and unoccupied states very near the Fermi energy are crucial for the MAE, as will be discussed further below, while they also emphasised the importance of taking into account deformations of the Fermi surface. A perturbative treatment of SOC also allowed Bruno [31] to find the simple relation that the MAE is proportional to the orbital magnetic moment anisotropy, as will be discussed further in a later part of this section.
The SOC energy shift, to second order, of a particular energy eigenvalue E n is
ΔE n = ξ n|L · S|n + ξ 2 ∑
k =n
!!n|L·S|k!! 2
E n − E k , (2.25) where |n and |k denote eigenstates of the unperturbed Hamiltonian and E n
and E k are the associated energy eigenvalues. The unperturbed states have a well defined spin character (in contrast to the perturbed ones) and it is suitable to consider states such as
|n = ∑
i
c n,i |k,d n,i ,σ n , (2.26)
where σ denotes the spin, the index i runs over the d-orbitals (xy, yz, z 2 , xz, x 2 − y 2 ) and in the case of a periodic system k denotes a point in the Bril- louin zone. In the ten-dimensional space which is a direct product of the two- dimensional spin space and the five-dimensional space of d-states, the spin- orbit coupling operator is a 10 × 10 hermitian matrix with elements which are straightforward to evaluate 3 and listed in Table 2.1. The angles θ and φ are the angular spherical coordinates describing the spin quantisation axis and this dependence on magnetisation direction of the spin-orbit coupling matrix is the source of the magnetocrystalline anisotropy energy. Inserting Eq. 2.26 and Eq. 2.24 into the first term of Eq. 2.25 and noting that all diagonal elements in Table 2.1 are zero, as well as that d i |L z |d i = 0, one finds that the first or- der perturbation contribution of the SOC is zero. Consequently, the spin-orbit coupling is at most a second order perturbation. This can also be related to the
tends to be much smaller, usually less than 1 meV (around 1 μeV in bcc Fe). For single atoms on surfaces, fulfilling certain symmetry requirements, magnetic anisotropy of similar size as that of the SOC constant has been observed [61].
3
For example by first introducing the spin states |↑
ˆn= cos
θ2|↑
ˆz+ e
iφsin
θ2|↓
ˆzin arbitrary
direction ˆn.
Table 2.1. Matrix elements σ i ,d i |L · S|σ j ,d j of the spin-orbit coupling opera- tor with respect to spin states in direction ˆn = (sinθ cosφ,sinθ sinφ,cosθ) and d- orbitals, in units of ¯h 2 . Reproduced from Ref. [67, 68].
|↑,d xy |↑,d yz |↑,d z
2|↑,d xz |↑,d x
2−y
2↑,d xy | 0
12i sin θ sinφ 0 −
12i sin θ cosφ i cos θ
↑,d yz | -
12i sin θ sinφ 0 −
√23i sin θ cosφ
2icos θ
−i2sin θ cosφ
↑,d z
2| 0
√23i sin θ cosφ 0 −
√23i sin θ sinφ 0
↑,d xz |
12i sin θ cosφ −
2icos θ
√23i sin θ sinφ 0 −
12i sin θ sinφ
↑,d x
2−y
2| −icosθ
−i2sin θ cosφ 0
12i sin θ sinφ 0
↓,d xy | 0 −icosθ sinφ) −
12(cosφ 0 −
12(sinφ
+icosθ cosφ) −isinθ
↓,d yz | −icosθ sinφ)
12(cosφ 0 −
√23(sinφ
+icosθ cosφ) −
2isin θ −
12(sinφ +icosθ cosφ)
↓,d z
2| 0 +icosθ cosφ)
√23(sinφ 0
√3 2
(cosφ
−icosθ sinφ) 0
↓,d xz | +icosθ cosφ)
12(sinφ
2isin θ −
√23(cosφ
−icosθ sinφ) 0
1 2
(cosφ
−icosθ sinφ)
↓,d x
2−y
2| i sin θ −
12(sinφ
+icosθ cosφ) 0 −
12(cosφ
−icosθ sinφ) 0
so called quenching of orbital angular momentum, according to which orbital magnetism vanishes in crystal fields, when neglecting SOC 4 .
Looking at Table 2.1 and Eq. 2.25 one can deduce that the nth order per- turbation term will be a linear combination of l = n spherical harmonics. For comparison, the leading order anisotropy term in Eq. 2.22 contains l = 2 spher- ical harmonics while that in Eq. 2.23 contains l = 4 spherical harmonics. From this one can conclude that in uniaxial crystals the second order perturbation term is non-zero and the MAE is of order ξ 2 , while for a cubic crystal, the second and third order terms are zero and it is necessary to go to fourth or- der perturbation theory to find non-zero contributions to the MAE. This fact is crucial for applications where a large MAE is needed because it causes cu- bic crystals to typically have orders of magnitude smaller MAE than uniaxial ones, which explains why the MAE of bcc Fe is so much smaller than that of hcp Co, as was seen in Table 1.1. In searching for 3d-based materials with large MAE, one should therefore focus strictly on materials with non-cubic crystal structures.
It is worth mentioning that the discussion here breaks down in materials with stronger SOC (containing large Z atoms in the lower part of the periodic table), for which a perturbative approach is invalid. Thus, for example, the actinide compound US in cubic rock salt structure exhibits an enormous MAE
4
When spherical symmetry is broken by a crystal field, it is suitable to describe orbitals in terms
of real spherical harmonics. These can be considered as superpositions of states with opposite
orbital angular momentum, so that the expectation value of the orbital angular momentum op-
erator vanishes [1]. It has been suggested that non-collinear spin arrangements can give rise to
orbital magnetism without SOC [64–66].
in the order of 10 9 J/m 3 [69], i.e., orders of magnitude larger than that of Nd 2 Fe 14 B, albeit being in a cubic crystal structure.
From swapping the indices n and k in Eq. 2.25, it is clear that two given energy levels will couple to each other in a way so that they are both shifted by an equal amount but in opposite directions. This is illustrated in Fig. 2.2, with respect to various positions of the Fermi energy. In Fig. 2.2a), the Fermi energy is above both energy levels, with or without perturbation. In calculating the total energy, a summation over the two states will yield E n + E k = E n + E k and the total energy is unaffected by the perturbation. The coupling between such states is therefore not important for the MAE. In Fig. 2.2b), the Fermi energy is below both states before and after the perturbation, whereby they do not contribute to the total energy and such states are not important for the MAE either. In Fig. 2.2c), on the other hand, E n is occupied both with and without the perturbation, while E k is not. In this situation the perturbation will change the total energy by an amount
ΔE n ,k = ξ 2 !!n|L·S|k!! 2
E n − E k . (2.27)
These states are crucial for the MAE, in particular if E n and E k are located near the Fermi energy so that the denominator in Eq. 2.27 is small, which allows the energy shift to be relatively large. Finally, Fig. 2.2d) illustrates a situation where the perturbation shifts an energy eigenvalue across the Fermi energy. This situation will also contribute to the MAE and gives rise to the deformations of the Fermi surface discussed by Kondorskii and Straube [63].
Since this can only happen to states near the Fermi energy, the important con- clusion remains; the MAE, in systems with SOC which is weak enough for perturbation theory to be relevant, is determined by the electronic states near the Fermi energy. This insight is very important for the task of engineering new materials with large MAE without the use of very heavy elements, as it tells us that the key lies in engineering the electronic states near the Fermi energy. This result was used by Burkert et al. [70] to explain the unusually
