theoretical and computational studies on the physics of applied magnetism...
TRANSCRIPT
ACTAUNIVERSITATIS
UPSALIENSISUPPSALA
2016
Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1440
Theoretical and ComputationalStudies on the Physics of AppliedMagnetism
Magnetocrystalline Anisotropy of Transition MetalMagnets and Magnetic Effects in Elastic ElectronScattering
ALEXANDER EDSTRÖM
ISSN 1651-6214ISBN 978-91-554-9753-8urn:nbn:se:uu:diva-304666
Dissertation presented at Uppsala University to be publicly examined in Häggsalen,Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 25 November 2016 at 13:15 forthe degree of Doctor of Philosophy. The examination will be conducted in English. Facultyexaminer: Professor Patrick Bruno (European Synchrotron Radiation Facility).
AbstractEdström, A. 2016. Theoretical and Computational Studies on the Physics of AppliedMagnetism. Magnetocrystalline Anisotropy of Transition Metal Magnets and MagneticEffects in Elastic Electron Scattering. Digital Comprehensive Summaries of UppsalaDissertations from the Faculty of Science and Technology 1440. 109 pp. Uppsala: ActaUniversitatis Upsaliensis. ISBN 978-91-554-9753-8.
In this thesis, two selected topics in magnetism are studied using theoretical modelling andcomputational methods. The first of these is the magnetocrystalline anisotropy energy (MAE) oftransition metal based magnets. In particular, ways of finding 3d transition metal based materialswith large MAE are considered. This is motivated by the need for new permanent magnetmaterials, not containing rare-earth elements, but is also of interest for other technologicalapplications, where the MAE is a key quantity. The mechanisms of the MAE in the relevantmaterials are reviewed and approaches to increasing this quantity are discussed. Computationalmethods, largely based on density functional theory (DFT), are applied to guide the searchfor relevant materials. The computational work suggests that the MAE of Fe1-xCox alloys canbe 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 theelectronic structure around the Fermi energy and thus also the MAE, for example in thetetragonal compound (Fe1-xCox)2B. Additionally, it is shown that small amounts (2.5-5 at.%) ofvarious 5d dopants on the Fe/Co-site can enhance the MAE of this material with as much as70%. The magnetic properties of several technologically interesting, chemically ordered, L10
structured binary compounds, tetragonal Fe5Si1-xPxB2 and Hexagonal Laves phase Fe2Ta1-xWx arealso investigated. The second topic studied is that of magnetic effects on the elastic scattering offast electrons, in the context of transmission electron microscopy (TEM). A multislice solutionis implemented for a paraxial version of the Pauli equation. Simulations require the magneticfields 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 amagnetic signal, defined as a difference in intensity for opposite orbital angular momentumbeams, integrated over a disk in the diffraction plane, is observed. For nanometer sized electronvortex beams carrying orbital angular momentum of a few tens of ħ, a relative magnetic signalof order 10-3 is found. This is considered realistic to be observed in experiments. In addition toelectron vortex beams, spin polarised and phase aberrated electron beams are considered andalso for these a magnetic signal, albeit weaker than that of the vortex beams, can be obtained.
Keywords: Magnetism, Magnetic anisotropy, DFT, Permanent magnets, Electron vortexbeams, 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 1651-6214ISBN 978-91-554-9753-8urn: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 FexCo1−x alloys by Cimpurities - a potential new permanent magnetE. K. Delczeg-Czirjak, A. Edström, M. Werwinski, J. Rusz, N. V.
Skorodumova, L. Vitos, O. Eriksson
Physical Review B 89, 144403 (2014)
II Electronic structure and magnetic properties of L10 binary alloysA. Edström, J. Chico, A. Jakobsson, A. Bergman, J. Rusz
Physical Review B 90, 014402 (2014)
III Increased magnetocrystalline anisotropy in epitaxial Fe-Co-C thinfilms with spontaneous strainL. 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 CombinatorialApproach Exploiting the Possibilities of Modeling, ShapeAnisotropy in Elongated Nanoparticles, and CombinatorialThin-Film ApproachD. 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: acombined first principles and thin film studyL. Reichel, L. Schultz, D. Pohl, S. Oswald, S. Fähler, M. Werwinski,
A. Edström, E. K. Delczeg-Czirjak, J. Rusz
Journal of Physics: Condensed Matter 27, 476002 (2015)
VI Magnetic properties of (Fe1−xCox)2 B alloys and the effect ofdoping by 5d elementsA. Edström, M. Werwinski, D. Iusan, 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 (Fe1−xCox)2 B alloys and theeffect of doping by 5d elementsA. Edström, M. Werwinski, D. Iusan, 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 Fe5SiB2 and its alloys with P, S and CoM. Werwinski, 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 TetragonallyStrained Fe-Co-B FilmsR. 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 strainedFe-Co(-X) filmsL. 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 Fe5SiB2-Fe5PB2 alloysystemD. Hedlund, J. Cedervall, A. Edström, S. Kontos, O. Eriksson, J. Rusz,
P. Svedlindh, M. Sahlberg, K. Gunnarsson
Manuscript
XI Magnetocrystalline anisotropy of Laves phase Fe2Ta1−xWx fromfirst principles - the effect of 3d-5d hybridisationA. Edström
Manuscript
XII New permanent magnets; what to look for, and whereL. 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 PermalloyD. 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 MatterA. Edström, A. Lubk, J. Rusz
Physical Review Letters 116, 127203 (2016)
XV Magnetic effects in the Paraxial Regime of Elastic ElectronScatteringA. 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 authorIn 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 Fe1−xCox Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.2 (Fe1−xCox)2B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1.3 L10 Binary Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.1.4 Fe5Si1−xPxB2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.1.5 Fe2Ta1−xWx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 drives1. 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 power2. 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/in2 mag-
netic storage capabilities3 with bit patterned media [13, 14] and related tech-
nologies, and because solid state magnetism is of atomic origin, experimental
1It was estimated that the world’s total information storage exceeded 1020 bytes, 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.2The 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].3Current magnetic storage technologies, approaching 1 Tb/in2, represent a 500 million fold in-
crease in storage density compared to the first IBM hard disk drive with a capability of 2 kB/in2
in 1956 [13]. Further increases call for novel magnetic technology in the nano regime.
9
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 PermanentMagnets
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 Nd2Fe14B 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. Nd2Fe14B 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/m3 [17]. From the discovery and over the coming
twenty years, Nd2Fe14B 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 SmCo5 and Nd2Fe14B 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 elements4. 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 Ms as [20]
(BH)max <1
4μ0M2
s , (1.1)
which shows that a large saturation magnetisation is desirable for a strong
permanent magnet. However, a large coercive field Hc, which describes how
4Although 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].
10
1900 1920 1940 1960 1980 2000
(BH
) max
(kJ/
m3)
0
100
200
300
400
500
Steel
AlnicoFerrite
Sm-CoNd-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
US
D/k
g
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 μ0MsHc, while the coercivity is
bounded by the anisotropy field Hc < Ha. For the important case of a uniaxial
crystal, the anisotropy field is
Ha =2K
μ0Ms, (1.2)
where K is the uniaxial magnetic anisotropy. This implies that a large satu-
ration magnetisation is detrimental to the hard5 magnet properties, unless the
anisotropy is correspondingly large. These arguments have been discussed in
5Hard magnets have large anisotropy and coercivity while the opposite is true for soft magnets.
11
terms a hardness parameter
κ =
√K
μ0M2s
, (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 > μ0M2s , i.e. that the anisotropic
energy density is greater than the magnetisation energy density. Together with
the requirement that the Curie temperature TC 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 MS, MAE and TC. The saturation magnetisation
and Curie temperature should be as large as possible, with the constraint that
K > μ0M2s . 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 (TC, Ms, MAE and κ) and
extrinsic properties (Hc and (BH)max) of RE and ferrite permanent magnetscompared to transition metals bcc Fe and hcp Co. From this table we see thatthe simple and abundant material bcc Fe has a higher Curie temperature andsaturation magnetisation than the best performing RE-transition metal basedmagnets. However, the MAE is two orders of magnitude smaller, whereby asizeable coercivity and energy product can never be obtained. It is enough togo to hcp Co to gain one order of magnitude in the MAE, which is becauseof 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 oneorder of magnitude smaller than the RE-transition metal magnets, togetherwith a quite high saturation magnetisation. This results in a small value forκ , which makes the material useless as a permanent magnet. The physics ofmagnetism in RE elements is quite different from that in transition metal ele-ments [9]. In particular, RE elements typically have localised f-electrons withsignificant 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 relevantmaterials. In contrast, transition metal magnets usually have a weak SOC andlargely quenched orbital magnetic moments. This leads to the grand challengein 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 magnetsis therefore reviewed in Sec. 2.1.2 and possible paths towards large MAE intransition metal magnets are discussed. The conclusions from that discussionare 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
12
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 TC, coercivity Hc, energyproduct (BH)max, saturation magnetisation Ms, 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 beconsidered as an estimate of realistic values. An upper limit for the energy product ofbcc Fe and hcp Co was estimated from μ0MsHc.
Nd2Fe14B SmCo5 BaFe12O19 bcc Fe hcp Co
TC (K) 588 1020 740 1043 1388
μ0Hc (T) 1.21 0.90 0.15 7 ·10−5 5 ·10−3
(BH)max (kJ/m3) 512 231 45 < 0.1 < 7
μ0Ms (T) 1.61 1.22 0.48 2.21 1.81
MAE (MJ/m3) 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 Ji j (see Sec. 2.2 and Sec. 3.1.5). With the Ji 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 ElectronMicroscope
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
13
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].
14
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 Bhom can be written (in Coulomb gauge)[p2
2m+
em(L+2S) ·Bhom− 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.
15
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][γμ (
i∂μ + eAμ)
c−mc2]
ψ = 0, (2.1)
where−e is the electron charge, c the speed of light, γμ are the Dirac matrices,
Aμ is the electromagnetic potential and m is the electron mass. Alternatively
it might, after separating out the time dependence, be written[ααα · (−ih∇+ eA)c− eV +βmc2
]ψ = Eψ, (2.2)
16
where A is the magnetic vector potential, V is the scalar potential,
β = γ0 =
(I2×2 0
0 −I2×2
)and ααα =
(0 σσσσσσ 0
), (2.3)
where σσσ = (σx,σy,σz) 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 (Aμ = 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 =√
x2 + y2, ϕ = tan−1 yx , z = z), where it reads
ψl(r) =e−iEt/heikzz
⎡⎢⎢⎣⎛⎜⎝
√1+ mc2
E χ√1− mc2
E cosθσzχ
⎞⎟⎠eilϕJl(k⊥r)+
i
√1− mc2
E
⎛⎜⎜⎝
0
0
−β sinθ0
⎞⎟⎟⎠ei(l−1)ϕJl−1(k⊥r)+
i
√1− mc2
E
⎛⎜⎜⎝
0
0
0
α sinθ
⎞⎟⎟⎠ei(l+1)φ Jl+1(k⊥r)
⎤⎥⎥⎥⎦ , (2.4)
where Jl(x) are Bessel functions,
χ =
(αβ
)(2.5)
17
and θ is the angle defined by
cosθ =k⊥√
k2⊥+ k2
z
, (2.6)
with kz 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 E2−m2c4 = c2h2(k2⊥+ k2
‖). 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
Ψl(r) = e−iEt/heikzzeilϕJl(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 →mc2. The lower two components
are, however, different from the non-relativistic case as they carry additional
contributions proportional to Ψl±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 lh since LzΨl =−ih ∂∂ϕ Ψl = lhΨl . The relativistic state ψl , on
the other hand, clearly is not an eigenstate of Lz. By introducing ψl,s, spin
polarised in the z-direction with spin up (s = +12 ) spinor χ = (1,0)T or spin
down (s =−12 ) spinor χ = (0,1)T, we have an eigenstate of Jz = Lz +Sz, with
Jzψl,s = h(l + s)ψl,s where the spin operator is
S =h2
ΣΣΣ =h2
(σσσ 0
0 σσσ
). (2.8)
For a state ψl,s the latter terms in Eq. 2.4 are proportional to Ψl±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 → mc2)
and paraxial (k⊥kz→ 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]
ψkj,m(r,θ ,φ) =
(fk(r)Y k
j,m(θ ,φ)igk(r)Y −k
j,m (θ ,φ)
), (2.9)
18
where fk and gk are radial functions, Y kj,m(θ ,φ) are generalised spherical har-
monics
Y kj,m(θ ,φ) =−sgnk
√k+ 1
2 −m2k+1
(1
0
)Yl,m− 1
2+
√k+ 1
2 +m2k+1
(0
1
)Yl,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− 12
(2.11)
is a quantum number related to the parity of the solution. The radial functions
are solutions to
d fk(r)d
=−1+ kr
fk(r)+1
ch
(E +mc2 + eV (r)
)(2.12)
dgk(r)d
=k−1
rgk(r)− 1
ch
(E−mc2 + 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 RelativisticApproximation
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,vc � 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 ·p2mc
A =
√1− p2
4m2c2. (2.15)
Transforming the Dirac equation according to H ′ = UHU−1 and ψ ′ = Uψ ,
performing some algebra and eventually only keeping terms to order(
vc
)2
19
leads to a decoupling of χ and η and a Hamiltonian
H =(p+ eA)2
2m− eV +
em
S ·B− (p+ eA)4
8c2m3−
− eh2
8m2c2∇2V − e
2m2c2S · (∇V × (p+ eA)
), (2.16)
with spin operator S = h2 σσσ . The first terms in this equation make up the non-
relativistic Schrödinger Hamiltonian and then comes the Zeeman term
HZeeman =− em
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
HSOC =− e2c2m2r
dV (r)dr
S ·L = ξ (r)L ·Sh2
, (2.18)
where L = r×p is the orbital angular momentum operator and
ξ (r) =− eh2
2c2m2rdV (r)
dr(2.19)
is the spin-orbit coupling constant. For the spherical potential V (r) = eZ4πε0r of
a hydrogen-like atom, the expectation value of the SOC constant, with respect
to the non-relativistic eigenstates |n, l〉, is1
ξn,l =⟨ξ (r)
⟩= 〈n, l|ξ (r) |n, l〉= Z4α4mc2
2n3l(l + 12)(l +1)
, (2.20)
where Z is the atomic number, α = 14πε0
e2
hc is the fine structure constant and nand 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 Zand, in a given series of the periodic table, the increase is approximately pro-
portional to Z2. 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
1Easily evaluated using 〈n, l| 1r3 |n, l〉 = m3c3α3Z3
h3n3l(l+1)(l+ 12)
[57], where |n, l〉 are non-relativistic
eigenstates of the hydrogen-like atom.
20
Z20 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.
21
2.1.2 Spin-Orbit Coupling and the MagnetocrystallineAnisotropy
Magnetocrystalline anisotropy is the free energy dependence on magnetisa-
tion direction, i.e. F = F(M), where M = (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(M) = F(−M), whereby only
even powers of sinθ are allowed. For example, in a uniaxial (e.g. tetragonal
or hexagonal) crystal the leading contributions are [28, 59]
F = F0 +K1 sin2 θ +K2 sin4 θ + ..., (2.22)
where F0 contains all isotropic energy contributions and Ki 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 K3 sin6 θ cos4φ appears. For a hexagonal crystal with six-fold rota-
tional symmetry a K3 sin6 θ cos6φ term appears, while for hexagonal crystals
with three-fold rotational symmetry (for example the Laves phase structure
of Fe2Ta1−xWx studied in Paper XI) an additional K′3 sin6 θ cos3φ is allowed.
For a cubic structure on the other hand, the leading contribution is of fourth
order and the energy is
F = F0 +K1(α2x α2
y +α2x α2
z +α2y α2
z )+K2α2x α2
y α2z + ..., (2.23)
where αi are the directional cosines of the magnetisation direction (αx = x ·Mand 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 HSOC = ξ L ·S, which is
often conveniently rewritten using
L ·S =1
2(L+S−+L−S+)+LzSz, (2.24)
where we have introduced the ladder operators L± = Lx± iLy and S± = Sx±iSy. 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-metals2, so that the size of
2It 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
22
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 Enis
ΔEn = ξ 〈n|L ·S |n〉+ξ 2 ∑k �=n
∣∣〈n|L ·S |k〉∣∣2En−Ek
, (2.25)
where |n〉 and |k〉 denote eigenstates of the unperturbed Hamiltonian and Enand Ek 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
cn,i |k,dn,i,σn〉 , (2.26)
where σ denotes the spin, the index i runs over the d-orbitals (xy, yz, z2, xz,x2− y2) 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 arestraightforward to evaluate3 and listed in Table 2.1. The angles θ and φ arethe angular spherical coordinates describing the spin quantisation axis and thisdependence on magnetisation direction of the spin-orbit coupling matrix is thesource of the magnetocrystalline anisotropy energy. Inserting Eq. 2.26 andEq. 2.24 into the first term of Eq. 2.25 and noting that all diagonal elementsin Table 2.1 are zero, as well as that 〈di|Lz |di〉= 0, one finds that the first or-der perturbation contribution of the SOC is zero. Consequently, the spin-orbitcoupling 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].3For example by first introducing the spin states |↑〉n = cos θ
2 |↑〉z + eiφ sin θ2 |↓〉z in arbitrary
direction n.
23
Table 2.1. Matrix elements 〈σi,di|L · S |σ j,d j〉 of the spin-orbit coupling opera-tor with respect to spin states in direction n = (sinθ cosφ ,sinθ sinφ ,cosθ) and d-orbitals, in units of h2. Reproduced from Ref. [67, 68].
|↑,dxy〉 |↑,dyz〉 |↑,dz2〉 |↑,dxz〉 |↑,dx2−y2〉〈↑,dxy| 0 1
2 isinθ sinφ 0 − 12 i sinθ cosφ i cosθ
〈↑,dyz| - 12 i sinθ sinφ 0 −
√3
2 i sinθ cosφ i2 cosθ −i
2 sinθ cosφ
〈↑,dz2 | 0√
32 i sinθ cosφ 0 −
√3
2 i sinθ sinφ 0
〈↑,dxz| 12 i sinθ cosφ − i
2 cosθ√
32 i sinθ sinφ 0 − 1
2 i sinθ sinφ〈↑,dx2−y2 | −i cosθ −i
2 sinθ cosφ 0 12 isinθ sinφ 0
〈↓,dxy| 0− 1
2 (cosφ−i cosθ sinφ) 0
− 12 (sinφ
+i cosθ cosφ) −i sinθ
〈↓,dyz|12 (cosφ
−i cosθ sinφ) 0−√
32 (sinφ
+i cosθ cosφ)− i
2 sinθ − 12 (sinφ
+i cosθ cosφ)
〈↓,dz2 | 0
√3
2 (sinφ+i cosθ cosφ)
0
√3
2 (cosφ−i cosθ sinφ)
0
〈↓,dxz|12 (sinφ
+i cosθ cosφ)i2 sinθ −
√3
2 (cosφ−i cosθ sinφ)
012 (cosφ
−i cosθ sinφ)
〈↓,dx2−y2 | i sinθ − 12 (sinφ
+i cosθ cosφ) 0− 1
2 (cosφ−i cosθ sinφ) 0
so called quenching of orbital angular momentum, according to which orbitalmagnetism vanishes in crystal fields, when neglecting SOC4.
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
4When 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].
24
in the order of 109 J/m3 [69], i.e., orders of magnitude larger than that of
Nd2Fe14B, 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 En +Ek = E ′n +E ′kand 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, En is occupied both with and
without the perturbation, while Ek is not. In this situation the perturbation will
change the total energy by an amount
ΔEn,k = ξ 2
∣∣〈n|L ·S |k〉∣∣2En−Ek
. (2.27)
These states are crucial for the MAE, in particular if En and Ek 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
Figure 2.2. Schematic image showing the effect of SOC on two energy levels En and
Ek with various locations of the Fermi energy EF.
large MAE of certain compositions of tetragonally strained Fe1−xCox, which
provides an important background for the work in papers I, III, V and VIII-IX,
25
and it is discussed further in Sec. 4.1.1. Similar reasoning has also been used,
for example, by Costa et al. [71] to analyse the large MAE of Fe2P.
In addition to the separation of the states that appears in the denominator
of Eq. 2.27, the energy shift is determined by numerator, where the matrix
elements in Table 2.1 enter. In the important case described in Fig. 2.2c), with
En < EF < Ek, there is a negative energy shift ΔEn,k < 0 and hence a lowering
of energy whenever 〈n|L ·S |k〉 is non-zero. Thus, any coupling containing a
cosθ in Table 2.1 will contribute with an energy reduction for θ = 0, corre-
sponding to magnetisation along the z-axis, whereas sinθ terms will favour
θ = π/2, i.e. magnetisation in the xy-plane. Coupling between any two states
with the same d-orbital type is zero and does not contribute to the MAE. Fur-
ther analysis of the SOC matrix elements and assignment of the quantum num-
ber |m|= 0 to dz2 , |m|= 1 to dxz and dyz and |m|= 2 to dxy and dx2−y2 , leads to
the observation that |m|= 0 states do not couple to |m|= 2 states (which can be
understood since the ladder operators in Eq. 2.24 can only couple states that
differ by m= 1). Furthermore, coupling between states with the same spin and
|m| (e.g. 〈↑,dxy| coupled to |↑,dx2−y2〉, but not another |↑,dxy〉 since diagonal
elements are zero) contain cosθ and favour magnetisation along the z-axis (a
uniaxial magnetic anisotropy along the z-axis is often wanted in technological
applications), while states with same spin but |m| differing by 1 have a sinθcoupling, favouring magnetisation in the xy-plane. For opposite spin states
this situation is reversed. From this analysis it is possible to look at the unper-
turbed electronic structure near the Fermi energy and, by determining the spin
and orbital character of the important occupied and unoccupied states, one can
deduce how these states will contribute to the MAE. Often the band structure
is very complicated with many states contributing in competing ways, making
a useful analysis difficult, but in some simple cases one might be able to de-
duce, e.g., the easy axis of magnetisation by looking at the dominating states
near the Fermi energy.
Based on the discussion so far in this section and considering the band
structure of a solid, a large MAE might appear if there are many occupied
and unoccupied states with energies very near the Fermi energy. A schematic
illustration of a such a band structure is shown in Fig. 2.3. The emphasised
region contains relatively flat bands just above and below the Fermi energy.
This allows for many pairs of occupied and unoccupied states to be near each
other in energy and couple strongly via SOC. If these states have the right
spin and orbital character, they will contribute significantly to the MAE. For
example, if k denotes a spin up dxy state while n denotes a spin up dx2−y2 state,
there is a strong contribution towards an easy magnetisation axis along the
z-direction. A problem in many real materials is that there are few such flat
bands near the Fermi energy and additionally there are often different regions
in k-space yielding opposite contributions to the MAE, resulting in a large
degree of cancellation as an integration is performed over the Brillouin zone.
26
Figure 2.3. Schematic band structure with a region containing flat bands with many
occupied energies Ek,n and unoccupied energies Ek,k near each other.
The perturbation theory that has been applied here assumes that all unper-
turbed states are non-degenerate and that the energy eigenvalues are separated
by much more than the SOC constant ξ . As described above, strong contribu-
tions to the MAE are expected in regions with occupied and unoccupied states
with similar energy. Quantitatively this situation might not be well described
by perturbation theory but it is still useful for a qualitative description and to
obtain an understanding of the origin of the MAE. For degenerate states the
perturbation theory described so far completely breaks down. From the discus-
sion around Fig. 2.2, degenerate states are only expected to be of importance
in the situation of Fig. 2.2d), i.e., when they are located very near the Fermi
energy so that they end up on different sides of the Fermi energy after the
perturbation is applied. The effect of the perturbation on degenerate states is
straightforward to evaluate using degenerate perturbation theory [72], accord-
ing to which one should diagonalise the degenerate subspace. For example, in
the case of two degenerate levels, one should diagonalise the matrix(h11 h12
h21 h22
)=
(0 h12
h21 0
), (2.28)
where hnm = 〈n|ξ L ·S |m〉, |n〉 is a state such as that in Eq. 2.26 and the diago-
nal elements vanish, just as they did in Eq. 2.25. Diagonalisation of the matrix
in Eq. 2.28 results in an energy shift of
ΔE±n,m =±∣∣ξ 〈n|L ·S |m〉∣∣ . (2.29)
One of the energy eigenvalues moves up and the other down, by an amount
corresponding to the absolute value of the matrix element of the SOC operator
for the given degenerate states. Thus, we can conclude that in the degenerate
case an important conclusion still holds; the MAE is determined by states very
near the Fermi energy (they cannot be more than ξ from the Fermi energy to
allow the perturbation to push them across the Fermi energy) and the matrix
elements of the SOC operator with respect to these states, that is, the matrix
elements in Table 2.1. It is interesting to note that the energy shit in Eq. 2.29
27
is of first order in ξ , in contrast to that in Eq. 2.27, which was of second order
in ξ . Degenerate states could, in principle, therefore yield relatively strong
contributions to the MAE. In practice, however, they tend to yield limited con-
tributions because degenerate states near enough to the Fermi energy usually
only appear in a small region of the Brillouin zone.
Based on a perturbation expression such as that which has been discussed
here and ignoring spin-flip terms as well as deformations of the Fermi surface,
Bruno [31] found a simple and useful relation between the MAE and the or-
bital moment anisotropy. When Fermi surface deformations are neglected, the
change in energy due to SOC, for spin quantisation axis n, may be written
ΔESOC(n) = ξ 2 ∑{n,k; En<EF<Ek}
∣∣〈n|L ·S |k〉∣∣2En−Ek
(2.30)
and the MAE is the difference MAE = ΔESOC(n1)−ΔESOC(n2), where n1
and n2 should be the magnetisation directions of maximum (hard axis) and
minimum energy (easy axis), respectively. The expectation value of the orbital
angular momentum operator projected on the magnetisation direction Ln is
zero for the unperturbed states, that is 〈n|Ln |n〉= 0. The perturbed states are,
in first order,
|n′〉= |n〉+∑k
〈n|ξ L ·S |k〉En−Ek
|k〉 (2.31)
and the orbital angular momentum after the perturbation is, to leading order,
〈Ln〉= ∑{n′; En′<EF}
〈n′|Ln |n′〉=
= ∑{n,k; En<EF<Ek}
2Re{〈n|ξ L ·S |k〉En−Ek
〈k|Ln |n〉}, (2.32)
but, with μn denoting the orbital character of state |n〉,〈k|Ln |n〉= 〈μk,σk|Ln |μn,σn〉= δσn,σk 〈μk,σn|Ln |μn,σn〉==
1
σδσn,σk 〈μk,σ |LnSn |μn,σ〉= 1
σδσn,σk 〈μk,σ |L ·S |μn,σ〉 , (2.33)
where σ =±12 is the spin with respect to n. Thus, combining the above,
〈Ln〉= 2ξσ ∑{n,k; En<EF<Ek}
δσn,σk
∣∣〈μn,σ |L ·S |μk,σ〉∣∣2
En−Ek, (2.34)
which reminds of Eq. 2.30, except that coupling is only allowed between states
of same spin. If, in Eq. 2.30, one would also only allow for coupling between
states of the same spin (neglecting spin-flip terms), this leads to the well known
relation of Bruno:
MAE =±ξ4
ΔmL, (2.35)
28
where ΔmL =⟨Ln1
⟩− ⟨Ln2
⟩is the anisotropy in the orbital moment and the
sign depends on whether coupling between spin up (+) or spin down (-) states
is taken into account. Considering only coupling between states of one spin
channel is a good approximation for so called strong ferromagnets, that is ma-
terials with completely occupied majority spin bands. In such materials the
density of states at the Fermi energy is dominated by minority spin states,
resulting in the easy axis of magnetisation coinciding with the maximum or-
bital magnetic moment. One example of a strong ferromagnet is permalloy
(fcc Fe0.2Ni0.8) and in Paper XIII the local MAE of various atomic clusters in
permalloy is evaluated by applying Eq. 2.35.
Eq. 2.35 tends to give a qualitatively correct description in that there is a
proportionality between the anisotropy in the orbital moment and the MAE.
Also, the easy axis typically coincides with the direction where the orbital
moment has its maximum if coupling between minority spin states dominates,
while it is in the direction of minimal orbital moment if the majority spin
coupling dominates. It happens, however, that the relation breaks down, for
example due to hybridisation effects in complex materials [73]. On occasion
Eq. 2.35 has also been inaccurately applied in explaining the origin of MAE
in transition metal alloys, such as FeNi [74, 75], as being due to anisotropy
in the orbital moment. This way of looking at the situation is incorrect in the
sense that Eq. 2.35 does not provide causality in the relation between MAE
and ΔmL. The relation between the two quantities is rather due to the origin of
both being the SOC. The key to understanding the MAE of a crystalline solid
lies instead in the SOC and the details of the band structure near the Fermi
energy, as has been discussed thoroughly in this section.
Fig. 2.4 shows how the orbital and spin magnetic moments, as well as the
energy, vary with the angle between the magnetisation direction and the z-
direction, as the magnetisation direction varies from the [001] to the [100]-direction. Calculations were done with WIEN2k in the generalised gradient ap-
proximation [76]. From Fig. 2.4b), which shows the change in energy plotted
against the change in orbital moment, it is clear that there is a proportional-
ity between these two quantities as predicted by Eq. 2.35 and that the easy
axis of magnetisation coincides with direction where the orbital moment has
its maximum. In Fig. 2.4c) one can also observe that, as pointed out in refer-
ences [74, 75], the largest change in orbital moment is on the Fe atom while
that on the Ni atom is smaller and of opposite sign.
Andersson et al. found that the formula of Bruno breaks down in Co/Au
layers with significant hybridisation and also that the system has a large MAE.
They explained this using second order perturbation theory, such as that which
has been discussed here, but considering several atomic species q. The result-
ing equations are interesting for finding materials with large MAE and hence
worth describing here. One can consider unperturbed single particle states of
29
0 20 40 60 80
0
10
20
30
40
Angle (deg)
ΔE
SO
C (
μe
V/a
tom
)
(a) Change in total energy as function of angle
between magnetisation direction and z-axis.
−3 −2 −1 0
0
10
20
30
40
ΔmL (10
−3μ
B/atom)
ΔE
SO
C (
μe
V/a
tom
)
Calculated points
Linear fit
(b) Change in energy versus change in orbital
moment as angle is varied.
0 20 40 60 80
35
40
45
50
Angle (deg)
mL (
10
−3μ
B/a
tom
)
Fe
Ni
(c) Orbital moment as a function of angle.
0 20 40 60 800.5
1
1.5
2
2.5
3
Angle (deg)
mS (
μB/a
tom
)
Fe
Ni
(d) Spin moment as a function of angle.
Figure 2.4. Variations in energy and moments of FeNi as functions of the angle θbetween the direction of magnetisation and the z-axis.
the form
|kn〉= ∑q,μ
ck,n,q,μ |kqμσn〉 (2.36)
with corresponding energy eigenvalue Ekn. In the case where Fermi surface
deformations are again neglected and k-diagonal on-site SOC is considered
so 〈kqμσn|HSOC |k′q′μ ′σk〉 = δk,k′δq,q′ 〈kqμσn|ξqL · S |kqμ ′σk〉 with ξq be-
ing the SOC constant of atom q, inserting Eq. 2.36 into an equation such as
Eq. 2.30, leads to
ΔEsoc,k(n) = ∑{n,k; En<EF<Ek}
∑qq′
∑μμ ′μ ′′μ ′′′
nk,n,qμ,q′μ ′′′nk,k,q′μ ′′,qμ ′
· 〈qμσn|ξqL ·S|qμ ′σk〉〈q′μ ′′σk|ξq′L ·S|q′μ ′′′σn〉Ek,n−Ek,k
. (2.37)
Here, nk,n,qμ,q′μ ′′′ = c∗k,n,q,μck,n,q′,μ ′′′ is an occupation number matrix, which is
non-zero when there is q-q′ hybridisation. Eq. 2.37 differs from Eq. 2.30 in
30
that there is not a matrix element squared in the numerator, but instead a prod-
uct of a matrix element for q accompanying a factor ξq (spin-orbit constant of
atom q) and one for q′with an accompanying factor ξq′ . One can now combine
a magnetic 3d atom with strong exchange interaction but weak SOC with, for
example, an often non-magnetic 5d atom with strong SOC (see Fig. 2.1). If the
hybridisation is strong (nk,n,qμ,q′μ ′′′ is large), one can obtain a ferromagnetic
compound with the three desired properties of large saturation magnetisation,
high Curie temperature and strong MAE. This is the key to the very large MAE
in the chemically ordered state of the widely studied L10 structured FePt com-
pound [77–82], which has strong contributions from both Fe and Pt to the
density of states at the Fermi energy [82]. Pt is, however, notoriously expen-
sive, as are many other heavy elements with strong SOC from the 5d series or
elsewhere [20]. Luckily, there are exceptions such as W, whereby, for exam-
ple, uniaxial Fe-W based compounds can be a promising path towards finding
novel transition metal based compounds with large MAE. This idea is explored
in Paper VI, where an (Fe1−xCox)2B alloy is doped with 5d elements, as well
as in Paper XI, where hexagonal Fe2W1−xTax is studied. More examples of
possibly interesting 3d-5d combinations are also discussed in Sec. 4.1.6.
To summarise this section, the MAE has been thoroughly discussed in terms
of the SOC treated within second order perturbation theory. The ideas pre-
sented are of utmost importance for the part of this thesis which deals with the
electronic structure theory of MAE and its use in finding novel magnetic 3d-
based compounds with large MAE, as is crucial for permanent magnets and
other applications. Three important points are worth emphasising again:
1. In the relevant group of materials (d-electron transition metal magnets),
a large MAE is only obtainable in non-cubic crystal structures.
2. The MAE is completely determined by the band structure near the Fermi
energy and, in particular, the coupling between occupied and unoccupied
states. The MAE can thus be controlled by tuning the electronic struc-
ture at the Fermi energy, which is practically possible by, for example,
alloying or strain engineering.
3. Strong SOC (i.e. heavy) elements are often what one wishes to avoid
in technological applications. Nevertheless, reasonable amounts of the
right elements with large ξ might be highly advantageous in increasing
the MAE.
These points essentially encompass the main message of Paper XII.
Temperature dependence of magnetocrystalline anisotropyThe general problem of temperature dependent MAE is a complicated one.
In the transition metal magnets, of most interest here, we have seen that the
MAE depends on details in the electronic structure states near the Fermi en-
ergy and consequently the temperature dependent MAE should be obtained by
detailed temperature dependent electronic structure calculations. Methods for
performing such calculations are discussed in Sec. 3.1.4. Here, we focus on
31
providing a brief discussion regarding earlier theories for the temperature de-
pendence of MAE, not requiring advanced computational methods. A review
on the topic was provided by Callen and Callen [83].
Most somewhat general results regarding temperature dependent MAE have
been obtained by considering single ion anisotropy models, where localised
spins are assumed to each have an anisotropic energy with the symmetry of
the relevant crystal, i.e. they have an energy with directional dependence of
the form described in Eq. 2.22-2.23. It is useful to rewrite the directional
dependence in terms of spherical harmonics. In the case of a uniaxial crystal,
Eq. 2.22 becomes5
F =F0 +2
3K1 +
56
105K2︸ ︷︷ ︸
k0
− 4
21
√π5(7K1 +8K2)︸ ︷︷ ︸
k2
Y 02 (r)+
16
105
√π K2︸︷︷︸
k4
Y 04 (r)
= ∑l
kl ∑m
cmY ml (r) (2.38)
with new anisotropy constants kl as prefactors for Y ml (r) = Y m
l (θ ,φ) terms.
The possibly most well known result for the temperature dependence of MAE
is that, at low temperatures,
kl(T )kl(0)
=
[M(T )M(0)
]l(l+1)/2
. (2.39)
For uniaxial crystals this leads to a 3rd power dependence of anisotropy on
magnetisation, while for cubic crystals it is of the 10th power. Extensions of
the theory exists also for higher temperatures [83, 84].
The above results are expected to describe localised magnetic moments as
found, for example, in magnetic insulators well, while it can not be expected
to hold for itinerant magnetism. Indeed the theory has been found to provide a
good description of the magnetic insulator yttrium iron garnet [83, 85], as seen
in Fig. 2.5a), which contains a comparison between experiment [85] and the-
ory [83, 84] for this compound. Fig. 2.5a) also contains experimental data [86]
for the first anisotropy constant of bcc Fe. Since bcc Fe is an itinerant magnetic
metal, there is no reason for the theory above to provide a good description.
Nevertheless, it is seen that the temperature dependence of K1 is reasonably
well reproduced by a 10th power law, surprisingly even at temperatures up to
TC.
FePt is another itinerant ferromagnet for which the power laws of Eq. 2.39
are not expected to provide an accurate description and in this case they do
not, as is also seen in Fig. 2.5a). Interestingly, the anisotropy constant of FePt
5This is easily obtained using sin2 θ = 23 − 4
3
√π5 Y 0
2 (Yml (r)) and
sin4 θ = 1105
[56−160
√π5 Y 0
2 (Yml (r))+16
√πY 0
4 (Yml (r))
].
32
T/TC
0 0.2 0.4 0.6 0.8 1
K1(T)/K
1(0)
0
0.2
0.4
0.6
0.8
1 YIGYIG, theoryFeFe, [M(T)/M(0)]10
FePtFePt, [M(T)/M(0)] 3
FePt, [M(T)/M(0)] 2
(a) First anisotropy constant K1 for yttrium iron garnet (YIG), bcc Fe and
L10 FePt as well as comparison to various power laws. Data were taken
from Ref. [77, 85–87].
T (K)
0 200 400 600
K (
MJ/m
3)
-0.5
0
0.5
1K
1
K2
(b) Experimentally measured [88] temperature dependence of the anisotropy
constants, K1 and K2, for hcp Co.
Figure 2.5. Temperature dependence of magnetic anisotropy constants in various ma-
terials.
is instead well described by a power law with exponent of approximately two,
instead of three expected for a uniaxial single ion anisotropy. This situation
has been analysed in terms of an effective spin model by Mryasov et al. [80],
who identified an exponent of value 2.1 and related this to the delocalised
nature of the induced Pt moment and an anisotropic exchange model. The
unusual exponent of FePt has also been reproduced by Staunton et al. using
relativistic first-principles electronic structure calculations (see Sec. 3.1.4).
Although FePt deviates from Eq. 2.39, all examples shown in Fig. 2.5a) fol-
low some power law for the temperature dependence of the anisotropy in terms
of the magnetisation. This leads to the MAE being a monotonically decreasing
function of temperature, since the magnetisation is normally a monotonically
decreasing function of temperature. Indeed the most common behaviour seen
33
in the temperature dependence of MAE is that its magnitude decreases with
increasing temperature. There are, however, clear violations to this. One is
found in hcp Co, for which the temperature dependence of the first anisotropy
constants is shown in Fig. 2.5b). For Co there is a change of sign in K1 (spin
reorientation transition) at approximately 532 K, a behaviour which cannot be
explained in the single ion anisotropy theories. Carr [86] was able to relate
this with the thermal expansion of the lattice, deriving a relation
K1(T )K1(0)
=
(1− T
TSRT
)[M(T )M(0)
,
]3
, (2.40)
where TSRT is the temperature of the spin reorientation transition, which for Co
occurs at approximately 13 TC. Similarly, hexagonal MnBi also exhibits a spin
reorientation transition and non-monotonic variations of the MAE [89] and it
has been explained in terms of temperature dependence of the lattice param-
eters [90]. MnBi has a small negative MAE at low temperature while it has
a relatively large positive MAE at room temperature, whereby it has received
attention as a potential rare-earth free permanent magnet material [91–93].
Co and MnBi provide two examples of materials with MAE which can
in no way be understood in terms of only single ion anisotropy models, but
instead temperature variations drive some other change which affects the in-
trinsic anisotropy constants, in these cases the lattice expansion. There are yet
other examples of materials with non-monotonic temperature dependence of
the MAE that cannot be explained in terms of thermal lattice expansion. One
notable case is found in the (Fe1−xCox)2B compounds studied in Paper VI,
where such behaviour is found for various alloy concentrations, e.g. Fe2B [94]
which has a spin-reorientation transition at a temperature just above 500 K.
For this compound it has been concluded that the non-monotonic variations in
MAE as function of temperature can not be traced to lattice expansion [95].
The origin of this temperature dependence of the MAE is discussed in Pa-
per VI and explained in terms of a simple model which maps the MAE to
temperature via so called fixed spin moment calculations. The conclusion is
that temperature causes a reduction of the exchange splitting, which changes
the electronic structure around the Fermi energy and thus modifies the MAE.
The reduction in the exchange splitting of the bands is not explained in this
simplified approach but can be understood and reproduced from the more ad-
vanced first principles calculations discussed in Sec. 3.1.4.
Finally, it is worth mentioning that also the Fe5SiB2 compound studied in
Paper VII and Paper X has been suggested [96] to exhibit a spin reorientation
transition, although no direct observation of this has so far been reported based
on experimental MAE measurements. Recent neutron scattering experiments
support the hypothesis [97].
34
2.2 Exchange Interactions and the HeisenbergHamiltonian
The quantised spin and orbital angular momentum and the associated magnetic
moments allow us to understand the appearance of para- and diamagnetism.
To understand spontaneous magnetic ordering, such as ferro-, ferri- or anti-
ferromagnetism, we need to include also an interaction between the atomic
magnetic moments. Dipole-dipole interactions between atomic moments are
typically very small and would not allow magnetic ordering at the significant
temperatures where it is observed6. The relevant interaction is instead the
exchange interaction due to the Coulomb repulsion and the fermionic char-
acter of electrons. This can be seen, for example, in moving from Hartree to
Hartree-Fock theory, where the inclusion of antisymmetry of the wavefunction
leads to an exchange term, which results in a lowering of energy for parallel
spin ordering [98]. From, for example, the Heitler-London model one can see
that localised spins tend to interact in a way so that the energy is proportional
to the scalar product of the spin operators [1]. Even though the Heitler-London
model only describes a simple system consisting of two atoms with one lo-
calised electron each, the result regarding the form of the spin-spin interaction
turns out to be quite general and in many cases also describes magnetism in a
solid well [1, 98]. This result is represented by the Heisenberg Hamiltonian
HHeisenberg =−1
2∑i�= j
Ji jSi ·S j, (2.41)
where Si and S j are the atomic spins on sites i and j, respectively, and Ji j is
the exchange coupling parameter between these spins. For a magnetic system
described by Eq. 2.41, the magnetic ordering and its TC is now determined by
the exchange coupling parameters Ji j. For a given material, these parameters
can be obtained from the electronic structure as discussed in Sec. 3.1.5.
Once one has knowledge of the Ji j, one can then study the magnetic or-
dering and transition temperatures via, for example, Monte Carlo simulations
which will be discussed in Sec. 3.2. One can also estimate magnetic transition
temperatures via mean field theory, according to which [98, 99]
TC =J0S(S+1)
3kB, (2.42)
where S is the atomic spin, J0 = ∑i J0i is the sum of the exchange interactions
and kB is the Boltzmann constant. From Eq. 2.42 it is clear that the Curie tem-
perature is proportional to the strength of the exchange interactions. Eq. 2.42
overestimates the transition temperature by around twenty percent or more,
6The energy of two magnetic dipoles of magnitude μB separated by a distance of one Ångström
is approximately 0.2 meV, corresponding to a temperature of just over 2 K, which can be com-
pared to Curie temperatures around 1000 K, found among many materials.
35
depending on dimensionality and coordination number [98], but can be useful
in effortlessly establishing an upper limit for TC and it is applied and compared
with MC results in Paper II. Eq. 2.42 can be extended to systems with several
atomic sites by constructing a matrix JAB, with each element containing the
sum over exchange coupling parameters between A and B-sites. The Curie
temperature is then obtained by finding the largest eigenvalue of the matrix
JAB [100, 101]. In addition to the Curie temperatures, magnetic excitations
can be studied. For example, a lattice Fourier transform of the Ji j parame-
ters allows one to compute adiabatic magnon dispersion relations [102, 103].
The exchange coupling parameters can also be used as input for the Landau-
Lifshitz-Gilbert (LLG) equations to study atomistic spin dynamics [103, 104].
In ferromagnetic metals, which are the materials of main interest in this
thesis, the exchange coupling tends to be mediated by conduction electrons
and the coupling is said to be of RKKY-type. The typical form of the ex-
change coupling parameters in such a system is, asymptotically in the long
range limit [99],
JRKKYi j ∝ n4/3 sin(2kFRi j)−2kFRi j cos(2kFRi j)
(kFRi j)4, (2.43)
where n is the density of conduction electrons, kF is the Fermi wave vector and
Ri j is the distance between sites i and j. Eq. 2.43 shows how RKKY-type inter-
actions exhibit an oscillatory behaviour with a long range decay proportional
to R−3i j . Strictly speaking, Eq. 2.43 is derived assuming localised moments in a
metal and only in this type of system one can formally expect the Heisenberg
Hamiltonian with exchange coupling parameters given by Eq. 2.43 to be real-
istic. Hence, one would not necessarily expect it to be applicable to magnetic
3d metals, as these tend to exhibit itinerant ferromagnetism with an exchange
splitting of the conduction bands. However, it turns out that the same type of
behaviour is often found also in itinerant ferromagnets [101, 105]. This can
possibly be understood by the fact that even in itinerant ferromagnets, such as
bcc Fe, the spin density is often well localised around the atoms, as illustrated
in Fig. 2.6.
Fig. 2.7 shows the exchange coupling parameters for Fe and random alloy
Fe0.4Co0.6 in the bcc structure, calculated by the SPR-KKR method which is
described in the next chapter. Fig. 2.7b) and Fig. 2.7d) illustrate that the ex-
change coupling parameters decay approximately as R−3i j , as expected from
Eq. 2.43. In Fig. 2.7c) it is seen that the strength of the Fe-Fe interactions in-
crease as Co is alloyed into the material while also Fe-Co and Co-Co interac-
tions are strong. This explains the observed effect that the TC increases as one
alloys Co into bcc Fe [107]. Application of mean field theory on the exchange
coupling parameters in Fig. 2.7 results in TC = 1298 K and TC = 1554 K for
Fe and Fe0.4Co0.6, respectively. Looking at Eq. 2.43 it can be speculated that
the increase in the strength of the Ji j is due to an increase in the density of
conduction electrons as Co is added into the material.
36
Spi
nde
nsity
(1/Å
3)
-6
-4
-2
0
2
4
6
Figure 2.6. Spin density of bcc Fe in a 001-plane containing twelve Fe atoms. Cal-
culated with the WIEN2k [106] FP-LAPW method in the generalised gradient approx-
imation [76] (see Sec 3.1).
Rij / a
0 2 4
Jij (
me
V)
-5
0
5
10
15
(a) bcc Fe
Rij / a
0 2 4 6
Rij3 ·
Jij (
a3 ·
me
V)
-50
0
50
(b) bcc Fe
Rij / a
0 2 4
Jij (
me
V)
-10
0
10
20
30Fe-Fe
Fe-Co
Co-Co
(c) bcc Fe0.4Co0.6
Rij / a
0 2 4 6
Rij3 ·
Jij (
a3 ·
me
V)
-40
-20
0
20
40
(d) bcc Fe0.4Co0.6
Figure 2.7. Exchange coupling parameters, Ji j, for bcc Fe and bcc Fe0.4Co0.6, as a
function of interatomic distances. Fig. b) and d) show Ji j ·R3i j to illustrate the RKKY-
type long-range behaviour of the exchange interactions.
The Hamiltonian in Eq. 2.41 is rotationally invariant and hence does not
include any form of magnetic anisotropy. The Heisenberg Hamiltonian can be
37
expanded with a term taking into account magnetocrystalline anisotropy and
in the case of a uniaxial crystal, based on Eq. 2.22 if one keeps only the two
first anisotropy constants, such a Hamiltonian is
HMAE = ∑i
[K1(mi · ez)
2 +K2(mi · ez)4], (2.44)
where mi is the direction of the moment at site i and ez is the direction of the
crystal axis. The magnetic spin Hamiltonians can also be expanded to take into
consideration other forms of exchange interactions, such as Dzyaloshinskii-
Moriya interactions [103, 104].
2.3 Microscopic Magnetic Fields in a Solid
It is interesting to consider what the magnetic vector potential A, and corre-
sponding magnetic flux density B, look like in a magnetic solid and, further-
more, it is of practical importance for the work in Papers XIV-XV where such
a description is used to evaluate the magnetic interaction between an electron
beam and magnetic materials. Discussion of this topic, including more de-
tails, is also contained in the mentioned papers. In the case of A, the precise
form will of course depend on the gauge choice. At a macroscopic level, with
zero applied field H = 0, one expects a contribution to the flux density of
B = μ0M, proportional to the magnetisation, which is the volume average of
the magnetisation density, that is
M =1
V
∫V
m(r)dV, (2.45)
where the magnetisation density m(r) describes the magnetism on a micro-
scopic level and the integral is, for periodic systems (which are of focus here),
taken over a unit cell. On a microscopic level one should rather relate B to
m instead of M. This should be done in a way so that B fulfils Maxwell’s
equations
∇ ·B = 0 (2.46)
∇×B = μ0j, (2.47)
and physical boundary conditions. In Eq. 2.47, j is the current density. For the
periodic systems under consideration here, the boundary conditions should be
periodic and such that the volume average is Bavg = μ0M. These boundary
conditions are sufficient for a unique solution to Maxwell’s equations [108].
The vector potential A should fulfil the defining equation B = ∇×A together
with a gauge choice. Here Coulomb gauge, ∇×A = 0 will be used. This
still allows for an arbitrary additive constant which can be chosen to a suitable
value.
38
Before describing a practical scheme to arrive at a microscopic description
for A and B, starting from first principles electronic structure calculations,
an important point to mention is that while B is necessarily periodic in a peri-
odic system, this is not true for A, which does not need to satisfy such physical
boundary conditions. In fact, it is easy to show that regardless of gauge choice,
A can not be periodic unless Bavg = μ0M = 0, that is in a non-magnetic or an-
tiferromagnetic material. See the appendix of Paper XV for details regarding
this7. For this reason it is practical to consider a decomposition of the fields
such that A = Ap +Anp is a sum of a periodic (Ap) and a non-periodic (Anp)
part while B = Bp +Bavg is the sum of a periodic, but spatially non-uniform
part, with volume average zero (Bp) and a uniform part (Bavg) with a suitable
volume average, i.e. Bavg = μ0M in the case of no external field. These fields
are related by
Bavg = ∇×Anp (2.48)
and
Bp = ∇×Ap. (2.49)
Furthermore, in Coulomb gauge
Anp =1
2Bavg× r =
1
2μ0M× r. (2.50)
Thus, Anp and Bavg are the macroscopically relevant quantities and they are
readily available from knowledge of the magnetisation. The periodic fields,
on the other hand, are only available at the microscopic level as they average
to zero and vanish at the macroscopic scale.
From Eq. 2.47 the microscopic B-field is obtainable if a microscopic de-
scription of j is available. Equivalently, A (in Coulomb gauge) is readily ob-
tained by solving the Poisson equation, since
∇×B = ∇× (∇×A) = ∇(∇ ·A)−ΔA =−ΔA = μ0j︸ ︷︷ ︸Poisson eq.
. (2.51)
A Gordon decomposition [55, 109] allows one to separate the relativistic cur-
rent density into a spin part, an orbital current part as well as a relativistic
correction term which vanishes in the stationary case considered here [109].
In the case that magnetism is dominated by the spin magnetic moments, as it
is in magnetic 3d metals, it is reasonable to neglect the orbital currents and to
relate a spin current density to the magnetisation density according to
j(r) = ∇×m(r). (2.52)
7This is clear in reciprocal space where the relation between A and B is B(k) = ∇×A(k) =ik×A(k), which can only be fulfilled if B(k) = 0, that is if the volume average of B(r) is zero.
39
The magnetisation density m(r) can be computed from the density matrix ρas
m(r) = μB〈σσσ〉= μBTr[ρ(r)σσσ ] = (2.53)
= μB
(2Re
(ψ∗↑ψ↓
),−2Im
(ψ∗↓ψ↑
),∣∣ψ↑∣∣2− ∣∣ψ↓∣∣2) .
Here 〈σσσ〉 is the expectation value of vector containing the Pauli spin matrices.
The density matrix can be computed using first principles electronic struc-
ture calculations, for example DFT. In the case of considering collinear mag-
netism with a spin quantisation axis z, the magnetisation density is simply
m(r) = μB(∣∣ψ↑∣∣2− ∣∣ψ↓∣∣2)z, proportional to the spin density and parallel to
the spin quantisation axis. For collinear spin magnetism, j and A have zero
z-components, while B is non-zero in all components.
The scheme proposed to obtain a microscopic description of the magnetic
flux density and vector potential in a magnetic solid can be summarised as
follows:
1. Compute m(r), or in the case of collinear magnetism just∣∣ψ↑∣∣2− ∣∣ψ↓∣∣2,
for the system of interest using a suitable method (see for example Sec. 3.1).
2. Compute the current density using Eq. 2.52.
3. Compute the periodic part of the vector potential Ap by solving the Pois-
son equation with periodic boundary conditions for each component ac-
cording to Eq. 2.51.
4. Calculate Bp as the curl of Ap, according to the defining relation between
the flux density and vector potential, i.e. Eq. 2.49.
The above procedure provides the microscopic fields Bp and Ap to which
Bavg = μ0M and Anp = 12 Bavg× r should be added for a complete descrip-
tion of the fields. It is straightforward to perform the steps listed above in
real space, although for a periodic system it is conveniently done in recipro-
cal space, where the Poisson equation in Eq. 2.51 takes on the simple form
k2A(k) = μ0j(k). A(k = 0) can be set to zero or any other arbitrary value due
to the remaining gauge freedom.
The result of the procedure described above applied to bcc Fe is presented
in Fig 2.8. The first column contains the collinear spin density from a density
functional theory calculation in the generalised gradient approximation [76]
using WIEN2k (see Sec. 3.1). The following three columns contain the x-
components of the current density, magnetic vector potential and magnetic
flux density as obtained by the scheme outlined above. The y-components of
these fields differ from the x-components only by rotation of π/2 about the
z-axis due to the crystal symmetry. The z-components of the current density
and vector potential are both zero because collinear spin magnetism is consid-
ered. The final column contains the z-component of the magnetic flux density,
which shows clear resemblance to the spin density, although there are qual-
itative differences, out of which the most notable is that the flux density has
40
non-zero x- and y-components even with collinear magnetism, where m(r) ‖ z.
It is also of interest to note that Bz, as well as Bx, reach values well beyond
that of μ0M = 2.2 T, corresponding to the saturation magnetisation of bcc Fe.
Data such as that presented in Fig. 2.8 is used as input for calculations in Paper
a/2
-0.5 Å -3 6 Å-3 5 ÅT -4 T 45 T10 T6 mA/Å2
2a/5
3a/1
0a/
5a/
10
spin
z=0
Ax BzBxjx
Figure 2.8. Spin density, x-components of current density, vector potential and flux
density and z-component of the flux density in various planes with fixed z in one unit
cell of bcc Fe. The lattice parameter of bcc Fe is a = 2.87 Å.
XIV. Similar data for FePt and the antiferromagnet LaMnAsO are presented
and used in Paper XV.
41
3. Computational Methods
This chapter provides a brief description of the computational methods used
in the work behind this thesis. Density functional theory (DFT) [10, 109, 110]
is used to calculate ground state properties of materials and it is described in
Sec. 3.1. In particular, the full potential linearised augmented plane waves and
the spin polarised relativistic KKR methods are used to solve the Kohn-Sham
equations of DFT and an introduction to these methods is given in Sec. 3.1.1
and Sec. 3.1.2. As we are interested in disordered alloys, models to describe
these are discussed in Sec. 3.1.3, while Sec. 3.1.4 provides specifics regarding
the computation of MAE. A brief discourse about the calculation of the tem-
perature dependence of MAE, using the so called relativistic disordered local
moments method, is provided in Sec. 3.1.4. In order to calculate Curie temper-
atures, Monte Carlo simulations [111] are employed as discussed in Sec. 3.2.
Multislice methods for TEM simulations are used in Papers XIV-XV and dis-
cussed in Sec. 3.3.
3.1 Density Functional Theory
Density functional theory (DFT) is our method of choice for finding the ground
state solution to an N-electron Schrödinger equation, which (within the Born-
Oppenheimer approximation [10]) reads⎛⎝− h2
2m
N
∑i=1
∇2i +
N
∑i
Vext(ri)+1
2
N
∑i�= j
w(∣∣ri− r j
∣∣)⎞⎠Ψ = EΨ, (3.1)
where Vext(r) is an external potential, w(∣∣ri− r j
∣∣) is a Coulomb interaction
between an electron at ri and one at r j, and Ψ is an N-electron wavefunction.
The first key ingredients of DFT are the Hohenberg-Kohn theorems [112],
which allow us to focus on electron densities, rather than wavefunctions. They
guarantee that all information about a many-electron system is contained in
the ground state density, which can, in principle, be found by a functional
minimisation of the total energy with respect to the density. Clearly this is
a huge simplification for systems with many electrons, since it reduces the
problem of dealing with a wavefunction depending on 3N spatial coordinates,
to a problem of dealing with a density depending on three spatial coordinates.
Unfortunately, the precise form of the energy as a functional of the density is
42
unknown, so far rendering these potentially very powerful theorems useless.
This situation changes with the Kohn-Sham approach [113], which introduces
an expression for the energy functional together with non-interacting single
particle orbitals ψi. Variational minimisation of the Kohn-Sham functional
allows the many-body problem in Eq. 3.1 to be simplified to a number of
single particle problems,(− h2
2m∇2 +Veff(r)
)ψi(r) = εiψi(r), (3.2)
where εi are the Kohn-Sham energy eigenvalues. These eigenvalues, as well
as the orbitals, in general, individually lack clear physical interpretation, al-
though the ground state density is
n(r) =N
∑i
∣∣ψi(r)∣∣2 , (3.3)
with summation over the N eigenstates with lowest energy eigenvalues. In
contrast, the total energy is not the sum over the N lowest energy eigenvalues,
but instead it is
E =N
∑i
εi− 1
2
e2
4πε0
∫ n(r)n(r′)|r− r′| drdr′ −
∫Vxcn(r)dr+Exc. (3.4)
The Kohn-Sham equations in Eq. 3.2, are essentially on the form of single par-
ticle Schrödinger equations with an effective potential Veff. Thus, we have re-
duced the problem of N interacting electrons to a problem of N non-interacting
Kohn-Sham particles in an effective potential
Veff(r) =Vext(r)+∫
dr′n(r)|r− r′| +
δExc[n(r)]δn(r)
, (3.5)
which depends on the density. In doing so, we have also included the exchange-
correlation potential Vxc =δExc[n(r)]
δn(r) and exchange-correlation energy Exc[n(r)],which contain the complicated many-body effects. This unknown quantity,
Exc[n(r)], is the cost of our simplification from a many-electron wavefunction
into a single particle description and finding good approximations for Exc[n(r)]is the grand challenge in making DFT accurate and useful. For some very sim-
ple model systems it might be possible to find an exact exchange-correlation
functional [114], but realistic systems must be treated by approximations.
The first and still widely used approximation for the exchange and cor-
relation was the local density approximation [113, 115–118] (LDA), which
approximates the exchange-correlation functional at a given point with that of
a homogeneous electron gas (HEG), with the same density. This should be
accurate if variations in the density are slow. Even in the case of the HEG,
43
evaluation of the exchange-correlation energy is far from trivial and, in the
early days, simplified approximate expressions were used [113, 115, 116].
These were enough to demonstrate the great usefulness of the LDA in predict-
ing the properties of real materials and early successes included calculations
of the cohesive properties of various metals [119]. Later on, accurate quan-
tum Monte Carlo simulations for the HEG appeared [120] and allowed for
parametrisations of the LDA [117, 118, 121]. With these, DFT has devel-
oped into an accurate tool for calculations and predictions of the properties of
realistic materials, in a way that no other method so far could compete with.
After the LDA, a plethora of functionals have been developed to describe
exchange and correlation in DFT [122], some striving to be ab initio, while
others introduce parameters which are fitted to experiments. The perhaps most
famous extension beyond the LDA is the generalised gradient approximation
(GGA) [76, 123, 124], which takes into account gradients in the density. In
this thesis, we are mainly interested in calculating the magnetic properties of
3d transition metals and their alloys and compounds. For these systems, the
GGA tends to accurately describe the desired properties [125–128]. For ex-
ample, the GGA correctly predicts the ferromagnetic bcc ground state of Fe,
in contrast to the LDA [127]. Among GGA functionals, the PBE form [76]
has been suggested to perform particularly well [122, 125] and, for example,
reproduces the lattice parameter of bcc Fe better than alternative GGAs [128].
Hence, the PBE [76] form of the GGA is the main exchange-correlation func-
tional employed in the DFT calculations which are part of this work. If nothing
else is specified, all DFT calculations can be assumed to have been performed
with the PBE GGA.
With a useful functional to treat exchange and correlation at hand, the next
step in DFT is to solve the equations in Eq. 3.2. Many methods have been
developed for doing this and, as usual in numerical problem solving, one typ-
ically needs to weigh computational speed against accuracy and generality.
Those methods of solving the Kohn-Sham equations which are used in this
thesis will be briefly described in the coming two sections. Since the density,
which is calculated from the solutions ψi, is also needed to calculate the poten-
tial Veff which appears in the equations, the problem is solved self-consistently
by iteration until a solution is converged with required numerical accuracy. As
we are mainly interested in periodic systems, it is appropriate to describe them
in reciprocal space, with the help of Bloch’s theorem [98].
Often DFT in the LDA or GGA has been extremely successful in reproduc-
ing and predicting ground state properties with high accuracy. Being based on
the HEG, the exchange-correlation functionals are expected to be particularly
accurate for delocalised electrons with slowly varying density. One then an-
ticipates problems in describing systems with localised electrons. Indeed, the
LDA or GGA often fails in describing transition metal oxides with localised d
states or rare earth compounds with localised f states. Hence, various methods,
such as the so called LDA+U [129] method or dynamical mean field theory
44
(DMFT) [110, 130, 131] have been developed to improve on the shortcomings
of approximations based on the HEG. In Paper VI, the GGA is found to fail
at describing the magnetic properties of Co2B and DMFT is seen to be able to
improve the calculated magnetic moment.
In the discussion above, no spin dependence was included. In order to de-
scribe magnetism, spin polarised DFT must be used, as first introduced by
Von Barth and Hedin [116]. The density should then be split up into spin up
and spin down parts, so n(r) = n↑(r)+ n↓(r) and spin dependence should be
included into the effective potential. Furthermore, as discussed in Sec. 2.1,
relativistic effects are often important and can be taken into account by solv-
ing the Dirac equation rather than the Schrödinger equation, or by using the
scalar relativistic approximation discussed in Sec. 2.1.1. In particular, spin-
orbit coupling is essential for calculating magnetocrystalline anisotropy and
specifics regarding this will be discussed in Sec. 3.1.4. DFT can be gener-
alised to a relativistic form [55, 109] by considering a four-current in place of
the density and it leads to a set of Kohn-Sham-Dirac equations, the relativistic
equivalence of Eq. 3.2, where the single particle equations are essentially on
the form of Eq. 2.2. Truly relativistic DFT, however, remains challenging since
relativistic functionals for the exchange and correlation energy are not as well
developed as the non-relativistic versions, so even in relativistic calculations
the common non-relativistic functionals are usually used. Furthermore, the
vector potential A appears in the Dirac equation, posing a problem for periodic
systems. As was mentioned in Sec. 2.3, there is no gauge choice which allows
for a periodic vector potential unless the volume average of the flux density is
zero. The pragmatic solution to this problem is found in applying the Gordon
decomposition [109] and neglecting the orbital currents, which allows for a
description without the vector potential. Unfortunately, this leaves us with an
incomplete description of relativistic effects. As discussed in Sec. 2.1.2, there
is usually a quenching of the orbital magnetic moment in the non-relativistic
description of solid transition metal magnets and the small orbital magnetic
moment which exists can be considered to be of entirely relativistic origin. In
this context it appears problematic to justify the neglect of orbital currents and
seems difficult to determine whether the poor description of orbital magnetic
moments [132] in the DFT calculations with the usual exchange-correlation
functionals is due to inaccuracy in the treatment of exchange and correlation
or the result of a poor treatment of relativistic effects.
The powerful ideas of Hohenberg and Kohn and the scheme of Kohn and
Sham, which makes these ideas useful, together with the development of ac-
curate approximations for the exchange and correlation, have allowed DFT
to become a powerful method for solving the many-body Schrödinger (or
Dirac) equation. It has been able to predict and explain the properties of
many real materials and the power of the theory deserves to be emphasised.
Continuous developments in practical schemes for solving the Kohn-Sham
equations, computer codes and modern large scale parallel computers have
45
allowed DFT to flourish and become a widely used method for first princi-
ples studies of a wide variety of complex materials, within condensed matter
physics, chemistry and materials science and engineering. Recent work has
confirmed that the wide variety of different methods produce coherent repro-
ducible results [133]. The success and popularity of the theory has led to a
situation where several of the most cited scientific works of all time are re-
lated to DFT [134] and it appears clear that it should be considered as one of
the greatest achievements in modern theoretical physics.
3.1.1 FP-LAPW
One of the computational methods used to solve the Kohn-Sham equations is
the full potential linearised augmented plane waves [135] (FP-LAPW) method
as implemented in the WIEN2k code [106]. Full potential implies that no shape
approximation is applied for the potential or charge density. This is in con-
trast to the commonly used atomic sphere approximation (ASA), where the
potentials are assumed to be spherically symmetric around atoms. LAPW is
the linearised [136] version of Slater’s augmented plane wave [137] (APW)
method, in the sense that energy dependence is removed from the basis func-
tions. Space is partitioned into muffin-tin (MT) regions of atomic spheres Sαand an interstitial region I, whereupon the Kohn-Sham orbitals in Eq. 3.2 are
expanded in basis functions consisting of radial solutions uαl (r
′,Eαl ) to the
Schrödinger equation of a free atom with energy Eαl and its energy derivative
uαl (r
′,Eαl ) within Sα , while in I, plane waves are used according to
ϕk,K(r) =
⎧⎨⎩
1√V
ei(k+K)·r r ∈ I
∑l,m
(Aα,k+K
l,m uαl (r
′,Eαl )+Bα,k+K
l,m uαl (r
′,Eαl )
)Y m
l (r′) r ∈ Sα.
(3.6)
Here k is a point in the Brillouin zone, K is a reciprocal lattice vector, V is
the volume of the unit cell, Y ml (r′) are spherical harmonics and r′ is the posi-
tion relative to the position coordinate of atomic sphere Sα . On the boundary
of the atomic spheres a matching is done so that ϕk,K(r) is continuous and
differentiable in all space. The number of basis functions used are usually
determined so that one basis vector is included for each vector K such that
|K| < Kmax with RMTKmax being a convergence parameter while RMT is the
radius of the smallest atomic sphere. The Kohn-Sham equations can then be
solved as an eigenvalue problem for a dense enough grid of k-vectors to obtain
an accurate solution to the problem. An integration must be performed over
the Brillouin zone and the standard way of doing this is by the modified tetra-
hedron method [138], although alternatives are available [106]. RMTKmax is a
good parameter to describe the accuracy of the number of basis functions used
since smaller radii of the atomic spheres will require more basis functions to
be included to describe the more rapid variations closer to the nuclei.
46
In the WIEN2k code, core states are treated fully relativistically by solv-
ing the spherically symmetric Dirac equation, while valence states in atomic
spheres are treated within the scalar relativistic approximation discussed in
Sec. 2.1.1. In order to calculate MAE, one needs to include the SOC also for
valence states which can be done in a second variational approach [139, 140].
The effect of SOC is only included within the atomic muffin-tin spheres.
3.1.2 SPR-KKR
The spin-polarised relativistic KKR method [141, 142] relies on the method of
Korringa [143] and Kohn and Rohstocker [144] (KKR) for solving the Kohn-
Sham (or Kohn-Sham-Dirac) equations and constitutes a somewhat different
approach than that discussed in the previous section. The SPR-KKR method
evaluates the Green’s function [145] (GF), G(r,r′,E), defined according to
(E−H )G(r,r′,E) = δ (r− r′), (3.7)
where H is the Hamiltonian of the system. With a free electron GF G0(r,r′,E),the single-site GF can be introduced via a Dyson equation1
Gn(r,r′,E) = G0(r,r′,E)+G0(r,r′,E)tnG0(r,r′,E), (3.8)
where tn is the single site t-matrix. When there are multiple scatterers, the full
Green’s function G is [142, 146]
G(r,r′,E) = G0(r,r′,E)+G0(r,r′,E)T G0(r,r′,E), (3.9)
where
T = ∑n,m
τnm (3.10)
1With a Hamiltonian H =H0+V and Green’s functions according to (E−H0)G0(E) = I and
(E−H )G(E) = I (the real space representation is G(r,r′,E) = 〈r|G(E) |r′〉), the Dyson equa-
tion is G(E) = G0(E)+G0(E)V G0(E)+G0(E)V G0V G0(E)+ ...= G0(E)+G0(E)V G(E) =G0(E)+G0(E)t(E)G0(E), where the t-matrix is t(E) =V +V G(E)V =V + t(E)G0(E)V . Of-
ten the term multiple scattering theory is used in the context of KKR methods, which can be
understood in connection to the (Born) series for G(E), which describes a series of scattering
events on the potential V . Truncation of the series after the term G0(E)V G0(E) would describe
single scattering and amounts to the first order Born approximation. This can be related to the
discussion of electron scattering and TEM simulations in Sec. 3.3. In the context of TEM, the
terms dynamical or kinematic scattering are often used to distinguish between consideration of
multiple or single scattering events, where the latter can again be related to the first order Born
approximation. What is called multiple scattering theory in relation to KKR electronic structure
methods is then essentially equivalent to what is called dynamical diffraction in the context of
TEM simulations.
47
and τnm is the scattering path operator
τnm =tnδnm + tnG0(1−δnm)tm+
∑k
tnG0(1−δnk)tkG0(1−δkm)tk + ...
=tnδnm +∑k
τnkG0(1−δkm)tm, (3.11)
which brings an incoming wave at site m to an outgoing at site n. In SPR-
KKR, an angular momentum representation is employed, allowing the opera-
tors appearing in Eq. 3.11 to be represented as matrices, whereby Eq. 3.11 can
be solved by a matrix inversion. For periodic systems, an additional Fourier
transform is required. Further details about the method are available in liter-
ature [142, 145, 146]. For periodic calculations in SPR-KKR, Brillouin zone
integration is typically done with the special points method [147].
KKR calculations often rely on the atomic sphere approximation (ASA),
in which space is partitioned into atomic spheres centred at atomic sites with
spherically symmetric potentials within the spheres. This is done in a way so
that the sum of the volume of the spheres corresponds to the volume of the unit
cell, leading to regions with overlap of atomic spheres as well as empty voids.
The approximation is expected to be more reliable for close packed structures.
However, SPR-KKR also allows for full potential calculations where the con-
straint of spherical symmetric potentials is removed [142]. In Paper VI and
Paper VII the ASA is found to be insufficient for an accurate description of
the magnetic properties of the studied compounds, namely (Fe1−xCox)2B and
Fe5Si1−xPxB2.
From the Green’s function, the density of states n(E) can be computed
as [142, 145]
n(E) =− 1
πImTrG(E) =− 1
πImTr
∫drG(r,r,E). (3.12)
At a point k in the Brillouin zone, the so called Bloch spectral function can
also be calculated as
A(k,E) =− 1
πNImTr
N
∑nm
eik·(Rn−Rm)∫
drG(r+Rn,r+Rm,E), (3.13)
which can be considered as a k-resolved version of the density of states, in
analogy with the energy dispersion relations (electronic band structures). For
an ordered system Eq. 3.13 reduces to the usual band structure. However,
one of the main advantages with using the SPR-KKR method, and one of the
main motivations for using it in this work, is that it is well suited for use with
the coherent potential approximation to disordered systems, discussed in the
coming section. In that case, the Bloch spectral functions provide a useful
description of the electronic structure. An example of A(k,E) for FeCo alloys
will be provided and compared to band structures from more simplified alloy
theory in Fig. 3.3.
48
3.1.3 Models to Treat Disorder
We will be interested in studying disordered alloys where, for example, one
might be interested in the magnetic (or other) properties of Fe1−xCox as a func-
tion of the concentration x, with Fe and Co atoms randomly occupying lattice
sites with the constraint of having the correct concentration. Hence, we need
models to describe this type of chemical disorder and there are various meth-
ods available [148]. The perhaps most accurate approach is to consider super-
cell calculations, where the smaller unit cell of the original periodic system
is replaced with a larger supercell in which the atomic sites are occupied in a
suitable manner. One might randomly occupy positions, perform calculations
for several such configurations and finally calculate an average result. Alterna-
tively, one can use stochastic methods, such as special quasirandom structures
(SQS) [149] to generate configurations which mimic the average random con-
figuration. It has been shown, in recent supercell calculations [95, 150], that
the MAE can be very sensitive to the atomic configurations, while averaging
over several configurations yields results in agreement with other computa-
tional descriptions of alloys or experiments. This is problematic for the SQS
approach, since it indicates that one configuration is insufficient for an ac-
curate description of the MAE. This can also be related to the results of Pa-
per XIII, where the local magnetic anisotropy in permalloy is found to vary
strongly among different clusters and also to be much greater than the aver-
age total magnetic anisotropy of the alloy. One advantage of supercells is also
the possibility of describing clusters and local ordering. Unfortunately, su-
percell techniques quickly become unreasonably computationally expensive.
It is therefore highly advantageous to apply so called single site approxima-
tions. The perhaps simplest such method is the rigid band model, where the
band structure is assumed constant in the alloy, while only the Fermi energy
changes. This model is not used here. Instead, mainly two different single site
approximations are used in this thesis, namely the virtual crystal approxima-
tion (VCA) and coherent potential approximation (CPA), which are discussed
further below.
Single site approaches to describe alloys rely on keeping the small unit cell
and artificial periodicity, while introducing some average quantity to describe
the random occupancy of constituent atoms. Looking at the Schrödinger equa-
tion, there are two possible quantities to average over, either the potential V (r)or the wavefunction ψ(r). The wavefunction is not a meaningful quantity to
average [148], which leaves the potential. This leads to the virtual crystal ap-
proximation (VCA). In the VCA, one introduces a virtual atom C to describe
the binary alloy A1−xBx and this virtual atom should have a (possibly non-
integer) atomic number ZC = (1− x)ZA + xZB. This simple model has been
confirmed to yield a correct behaviour for various properties when alloying
elements, such as Fe and Co or Co and Ni [151], which are neighbours in
the periodic table [148]. For example, it reproduces the Slater-Pauling max-
49
imum in the magnetic moment for FeCo alloys [152]. On the other hand, it
breaks down for elements further away from each other [148], such as Fe and
Ni [153]. Alternatively formulated, the potential
VC = (1− x)VA + xVB, (3.14)
together with a correct average number of electrons, yields a correct descrip-
tion of the random alloy consisting of atoms A and B with potentials VA and
VB, respectively, in the limit where VA =VB [148]. However, the MAE which
is one of the key properties studied in this thesis, tends to be quantitatively
severely overestimated by VCA. In particular, this has been observed in FeCo
alloys, where VCA calculations have predicted a huge MAE [70], which has
been found to be significantly overestimated but qualitatively reasonably cor-
rect in comparison with more advanced single site approaches [154, 155], su-
percell techniques [150, 156] or experiments [157, 158]. This is highly rel-
evant for Papers I, Paper III and Papers VI-IX where the VCA is used for
calculations regarding FeCo-based alloys.
Writing the Schrödinger equation in terms of the Green’s function (see
Eq. 3.7) introduces a new possible quantity to average over, namely the Green’s
function2. This leads to the more sophisticated single site model of disorder
in the form of the CPA [159, 160]. In this approach, an impurity of each atom
type, A or B, is placed in an effective CPA medium. One then considers the
alloy to be described by the weighted average of the two different impurity
solutions, as illustrated in Fig. 3.1. This should be done in a self consistent
manner so that the CPA medium corresponds to weighted average of the indi-
vidual components according to the CPA equations
(1− x)τAnn + xτB
nn = τCPAnn (3.15)
and
ταnn = [(tα)−1− (tCPA)−1− (τCPA)−1]−1, α = A, B . (3.16)
Finally, an average GF,
G(r,r′,E) = (1− x)GA(r,r′,E)+ xGB(r,r′,E), (3.17)
is obtained.
Making use of the CPA, various other forms of disorder than substitutional
chemical disorder can be described in so called alloy analogy models. This
has, for example, been applied for finite temperature linear response calcula-
tions of Gilbert damping and electrical conductivity [161, 162]. This is also
an important basis for the finite temperature calculations of MAE discussed in
Sec. 3.1.4.
2In principle, the single-site t-matrix could also be averaged over, which would lead to the aver-
age t-matrix approximation (ATA). This approximation has problems, in particular a possibility
of negative DOS, which are overcome by the CPA [148]. The ATA will not be discussed further
here.
50
Figure 3.1. In the CPA, each atomic type is embedded as an impurity in an effective
CPA medium, which should correspond to a weighted average of the components.
Fig. 3.2 shows a comparison between VCA calculations in WIEN2k and CPA
calculations using SPR-KKR. The MAE and magnetic moments have been
computed as functions of the tetragonal strain c/a for a bct alloy Fe0.4Co0.6,
similarly as has been done previously in Ref. [154]. The MAE has been eval-
uated by total energy difference and magnetic force theorem with the VCA
in WIEN2k and with total energy difference and the torque method in SPR-
KKR ASA mode with the CPA. The different methods for computing the
MAE will be described in the coming Sec 3.1.4. For the SPR-KKR-CPA
calculations, an LDA calculation is also presented, as well as a full poten-
tial calculation. Fig. 3.2a) illustrates how the VCA qualitatively describes the
c/a1 1.1 1.2 1.3 1.4
MAE
(μeV
/u.c.)
0
200
400
600VCA, E-diff.VCA, FTCPA, E-diff.CPA, TorqueCPA, LDACPA, FP
(a) MAE(c/a)
c/a1 1.1 1.2 1.3 1.4
Magnetic
moment(μ
B)
1.6
1.8
2
2.2
2.4
2.6
2.8
mavg, VCAmavg, CPAmFe, CPAmCo, CPA
(b) m(c/a)
Figure 3.2. MAE and magnetic moments of Fe0.4Co0.6 as functions of c/a, calculated
by various methods. All calculations were done with the exchange-correlation treated
with the GGA, except the MAE calculation marked LDA, which was performed with
SPR-KKR, CPA and total energy difference calculation. The full potential calculation
was performed with the torque method. In the SPR-KKR calculations, a discretisation
over 40 energy points was considered on a semi-circular path in the complex energy
plane and Brillouin zone integration was performed with 80000 k-points (in the full
Brillouin zone, corresponding to approximately 5000 in the irreducible wedge) and
the special points method [147]. In WIEN2k, RMTKmax was set to 9, while Brillouin
zone integration was performed over 35000 k-points with the modified tetrahedron
method [138].
correct behaviour of the MAE, although it overestimates the maximum values
significantly compared to the CPA. It can also be noted that ASA reduces the
51
MAE compared to the FP calculations. GGA and LDA yield nearly identical
results. From Fig. 3.2b), it is clear that the moment provided by the VCA co-
incides well with the average moment provided by CPA, but the CPA yields
more information as it also provides the atom specific moments, not only the
average.
Fig. 3.3 illustrates the Bloch spectral functions around the Fermi energy
for the disordered tetragonal alloy Fe0.4Co0.6 with different tetragonal strains.
Also the band structure calculated in scalar relativistic, spin polarised, VCA
calculations with WIEN2k are shown. The most pronounced difference be-
tween the CPA spectral functions and the VCA band structures is the smear-
ing observed in the spectral functions. This smearing should be considered the
main reason that the magnitude of MAE is smaller in CPA calculations than
the VCA calculations, as it reduces the coupling between the important states
near the Fermi energy. The large MAE in tetragonally strained FeCo was dis-
cussed by Burkert et al. [70] and related to states at the Γ point which turn out
to be near the Fermi energy and on opposite sides of it in the region of large
MAE. Looking at the Γ-point in Fig. 3.3, one can see two spin down bands
which approach the Fermi energy as the tetragonal strain is increased from
c/a = 1. At c/a = 1.2, near the maximum MAE according to Fig. 3.2, these
bands are very near the Fermi energy and one is occupied while the other is
unoccupied, precisely according to the requirement for the states to contribute
strongly to the MAE. As the strain is increased further, these bands move away
from the Fermi energy and consequently should couple more weakly via the
SOC. By analysing the orbital contributions to these bands one can conclude,
in agreement with the analysis by Burkert et al., that the occupied band is
mainly of dxy character while the unoccupied one is dx2−y2 . Going back to the
matrix elements in Table 2.1 and related discussion, spin diagonal coupling
between a dxy and a dx2−y2 orbital yields a positive (uniaxial) contribution to
the MAE, thus explaining the MAE maximum in Fig. 3.2.
3.1.4 Computing the MAE
When defining the MAE as the largest possible energy difference between two
different magnetisation directions (the easy and the hard axes), it is clear that
this can be calculated by performing total energy calculations, including SOC,
for a magnetisation in each of the two directions n1 and n2 and taking the
difference according to
MAE = E(n1)−E(n2). (3.18)
The first difficulty here is determining which axis is easy and which is hard
when there are infinitely many directions to probe. This, however, does not
tend to be a problem since the directions of interest are typically some of the
high symmetry directions which can be seen in the phenomenological expres-
52
Z Γ N P X
E-E
F (
eV
)
-2
0
2
(a) c/a = 1.0, MAE = 0.0 μeV/atom
Z Γ N P X
E-E
F (
eV
)
-2
0
2
(b) c/a = 1.05, MAE = 34.7 μeV/atom
Z Γ N P X
E-E
F (
eV
)
-2
0
2
(c) c/a = 1.1, MAE = 73.6 μeV/atom
Z Γ N P X
E-E
F (
eV
)
-2
0
2
(d) c/a= 1.15, MAE= 113.1 μeV/atom
Z N P X
E-E
F(e
V)
-2
0
2
(e) c/a = 1.2, MAE = 179.2 μeV/atom
Z Γ N P X
E-E
F (
eV
)
-2
0
2
(f) c/a = 1.25, MAE = 172.0 μeV/atom
Z Γ N P X
E-E
F (
eV
)
-2
0
2
(g) c/a = 1.3, MAE = 94.5 μeV/atom
Z Γ N P X
E-E
F (
eV
)
-2
0
2
(h) c/a = 1.4, MAE = 10.9 μeV/atom
Figure 3.3. Bloch spectral functions around the Fermi energy for bct Fe0.4Co0.6 with
various tetragonal strains, calculated in fully relativistic full potential mode using
SPR-KKR, with magnetisation in the 001-direction and disorder treated in CPA. The
spin polarised band structures from scalar relativistic WIEN2k calculations are also
shown with dashed black lines indicating spin down bands and solid black lines in-
dicating spin up bands. The MAE values listed are from FP CPA calculations (see
Fig. 3.2).
53
sions in Sec. 2.1.2. Also, by calculating the energy for a several magnetisa-
tion directions, one can calculate the anisotropy constants by fitting and from
these the easy and hard axes can be found. In a uniaxial crystal, it is usu-
ally enough to evaluate K1 and K2 since variations within the plane tend to
be orders of magnitude smaller and also the second anisotropy constant is of-
ten smaller than the first one. For example, in the tetragonal Fe0.4Co0.6 alloy
with c/a = 1.2, the energy difference between magnetisation directions along
z-axis or in the xy-plane is MAE = 1.2 ·10−4 eV/atom, while the energy vari-
ations within the plane are too small to be resolved with a numerical accuracy
of 10−8 Ry/atom = 1.36 ·10−7 eV/atom. What is, on the other hand, a prob-
lem is that the MAE tends to be a very small energy difference, compared
to cohesive energies or even magnetic exchange interactions. Hence, one is
required to compute a small difference between two large energy values and
this makes the MAE difficult to evaluate with high numerical accuracy and
thus also computationally costly. Fig. 3.4 shows the convergence of the MAE
as a function of the number of k-points used in integration over the Brillouin
zone for calculations of the MAE in FeNi, which is one of the materials stud-
ied in Paper II. Calculations are performed with either WIEN2k and Brillouin
zone integration with the modified tetrahedron method [138] or SPR-KKR in
either ASA or FP mode and Brillouin zone integration with the special points
method [147]. At least 104 k-points should be sampled for an accuracy within
a few percent and it appears motivated to use around 105 or higher.
Nr of k-points (104)0 5 10 15
MA
E(
eV/f.
u.)
0
50
100
150
W2k
KKR-ASA
KKR-FP
Figure 3.4. MAE as a function of the number of k-points used for numerical integra-
tion over the full Brillouin zone for L10 structured FeNi.
Fig. 3.4 also shows that the SPR-KKR-ASA calculation overestimates the
MAE of FeNi relative to SPR-KKR-FP. This partly explains the notable dis-
crepancy between FP-LAPW and SPR-KKR calculations seen in Paper II.
Due to the computational challenges involved in evaluating the MAE from
total energy differences, various approximation methods have been developed
to compute the MAE more efficiently. Two such methods, which have been
used in this work and will be described further below, are the force theorem
used in Papers I and III and the torque method, which is used to a large extent
in Paper II and Paper VI.
54
Force TheoremWhen calculating the MAE as a difference of total energies as described above,
one needs to perform highly accurate self consistent calculations, including
SOC, in each of the two magnetisation directions. This needs to be done
with a very dense k-point sampling making the calculations computationally
expensive. Early attempts at calculating the MAE for elemental transition
metal magnets [163], therefore, employed the so called magnetic force the-
orem. Such calculations failed at describing the correct sign of the MAE in
Co and Ni. In the case of Co, this was corrected in later total energy calcula-
tions [164], while the case of Ni remains an unsolved problem. Nevertheless,
the force theorem often remains a useful approximation. By considering the
change in energy due to some perturbation, the magnetic force theorem [165]
tells us that, to leading order, it is enough to consider the change in the sin-
gle particle Kohn-Sham eigenvalues, rather than the total energy (compare
Eq. 3.4). The MAE can then be evaluated as [163]
MAE≈ Es.p.(n1)−Es.p.(n2), (3.19)
where
Es.p.(n) = ∑{k,i; εi,k(n)<EF(n)}
εi,k(n) (3.20)
is the sum over occupied single particle Kohn-Sham energy eigenvalues. The
approximation of considering the change in the total energy as the change in
the single particle eigenvalues should be accurate if the perturbation does not
significantly alter the charge density and potential. This method allows one
to first perform only one full self consistent calculation without SOC. In the
next step one diagonalises the Hamiltonian including SOC only once for each
magnetisation direction and evaluates εi. This saves approximately half or
more of the computational effort. Fig. 3.2 contains a comparison of the MAE
calculated by total energy difference or using the force theorem and illustrates
that, within the limitations of the VCA, the force theorem provides a good
approximation of the MAE for the given system.
Another useful property of the force theorem is that it allows the MAE
to be analysed in a k-point resolved manner, by restricting the summation in
Eq. 3.20 to the band index i. This is done in Paper VI, Paper VII and Paper XI.
An example for the bct Fe0.4Co0.6 alloy with c/a = 1.2, which was considered
also in Fig. 3.3e), is shown in Fig. 3.5. The dominating positive contribution to
the MAE originates from a region near the Γ-point, where unperturbed |m|= 2
bands are near but on opposite sides of EF, so they can couple strongly via
SOC, according to the discussion in connection to Fig. 3.3 and the analysis in
Ref. [70]. The positive MAE contribution corresponds to the highest occupied
band being pushed down for magnetisation along the 001-direction, whereby
this magnetisation direction is energetically favoured. It can also be pointed
out that the MAE contributions are negligible in regions where there are not
55
occupied and unoccupied bands near the Fermi energy. Furthermore, dras-
tic changes in the MAE contributions always appear in connection to bands
crossing the Fermi energy.
Z Γ N P X
E-E
F (
eV
)
-1
0
1
MA
E (
10
-2e
V/k
-po
int)
-5
-2.5
0
2.5
5
Figure 3.5. Band structure of bct Fe0.4Co0.6 with c/a = 1.2 including SOC and mag-
netisation along the 001-direction as blue dashed line and 100-direction as black dash-
dotted line. The k-resolved MAE, computed using the force theorem, is shown as a
red solid line. Calculations were performed with WIEN2k and the same settings as
were used for the results in Fig. 3.3.
Torque MethodAnother method for calculating the MAE, implemented in SPR-KKR and em-
ployed in various parts of this work, is the torque method [166]. If we consider
the first two θ -dependent terms in Eq. 2.22, the torque T on a magnetic mo-
ment in direction (sinθ cosφ ,sinθ sinφ ,cosθ) is
T (θ) =dFdθ
= K1 sin2θ +2K2 sin2 θ sin2θ . (3.21)
From Eq. 3.21 it is easy to evaluate the MAE as
EMAE = T (π4) = K1 +K2 = E(0)−E(
π2), (3.22)
if the torque can be calculated. In the multiple scattering formalism adopted
in SPR-KKR, the torque can be obtained through the formula [167]
T (θ) =− 1
πIm
∫ εF
dε ∑n
Tr
(∂ t−1
n
∂θτnn(ε)
). (3.23)
Fig. 3.2 contains a comparison between torque calculations and total energy
difference calculations for the MAE of Fe0.4Co0.6 as a function of c/a and the
agreement is excellent.
Computing the MAE(T )Soon after the introduction of zero temperature, ground state DFT, a general-
isation to finite temperatures was developed by Mermin [168]. In this case, a
56
grand potential
Ω =− 1
βlnTre−β (H −μN ) (3.24)
takes the place of the ground state energy. In Eq. 3.24, β = 1kBT , μ is the
chemical potential, H the Hamiltonian and N the number operator. Due to
difficulties in formulating useful temperature dependent exchange-correlation
functionals, the finite temperature DFT is not widely used [10]. Neverthe-
less, there are situations where DFT can be extended as a useful tool to cal-
culate finite temperature effects, in practice making use of the ground state
exchange-correlation functionals. One such case is found in the disordered
local moments (DLM) method [169], where the thermal disorder among the
spin magnetic moments is described using the CPA (see Sec. 3.1.3). In the
non-relativistic case, where the Hamiltonian is invariant under spin rotations,
the paramagnetic state can then be described by a two-component alloy with
equal spin up and spin down parts. In the relativistic case, there is additional
energy dependence of magnetisation directions, related to SOC, and the theory
must be extended accordingly. This has been done by Staunton et al. [79, 167]
in a way which allows for the calculation of temperature dependent MAE.
Let n denote the unit vector in the direction of magnetisation and mi the unit
vector in the direction of the atomic magnetic moment mi. In a classical mean
field description, the atomic magnetic moments should follow a probability
distribution
Pn(mi) =e−βh(n)n·mi∫
e−βh(n)n·mi dmi, (3.25)
where a Weiss field h(n) has been introduced. The Weiss field can be calcu-
lated as
h(n) =3
4π
∫(m · n)
⟨Ωn
⟩m
dm, (3.26)
where⟨
Ωn⟩
mdenotes averaging of the grand potential w.r.t. to moment direc-
tion, which can be performed with the CPA, as discussed in detail by Staunton
et al. [167]. In practice, the N magnetic moments are distributed in different
directions, e.g. on a rectangular grid in terms of the polar coordinates θ and φso that N = Nθ Nφ , and each moment direction is populated with a concentra-
tion proportional to the probability in Eq. 3.25 using the CPA.
To evaluate the MAE, a difference in free energy for different magnetisa-
tion directions should be calculated and formulas are available for this [167].
However, a computationally more efficient scheme is available by using a fi-
nite temperature version of the magnetic torque formula in Eq. 3.23, which
has been derived by Staunton et al. [167].
The above scheme for computing temperature dependence of the MAE, in-
cluding the torque formula, is available in the SPR-KKR package. Since it
relies on a mean field description of the magnetisation, one might expect this
57
to result in some error. Instead of self-consistently evaluating the Weiss field,
one can therefore take this as input if the magnetisation as a function of tem-
perature is known, e.g. from experiment or Monte Carlo simulations, similarly
as has been done in temperature dependent linear response calculations [162].
The methods discussed above have previously been applied to various ma-
terials, most notably FePt [79, 167] in the L10 ordered state, but also taking
into consideration disorder [170]. In FePt, it was able to explain the unusual
exponent in the anisotropy as function of magnetisation (see Sec. 2.1.2). It
has also been applied to describe surface and interface magnetism [171, 172],
where it has in some cases predicted spin reorientation transitions. As was
mentioned in Sec. 2.1.2, the (Fe1−xCox)2B alloy studied in Paper VI exhibits
an interesting MAE(T ) behaviour with spin reorientation transition for vari-
ous x and it appears motivated to investigate whether this can be reproduced
within the relativistic DLM (RDLM) approach.
Fig. 3.6 shows the results of RDLM calculations using SPR-KKR for Fe2B
and (Fe0.7Co0.3)2B. Calculations were performed with 10000 k-points in the
full Brillouin zone, 48 energy points on an arc in the complex energy plane
and angular momentum states up to and including l = 4. Magnetic moment
directions were uniformly distributed over a rectangular grid in terms of the
polar coordinates θ and φ with Nθ = 90 and Nφ = 20. Experimental data from
Paper VI is shown for comparison. In addition, computational results for the
temperature dependence of the MAE from Ref. [173] are also shown.
Fig. 3.6a) and Fig. 3.6c) show the temperature dependence of the exper-
imental magnetisation and calculated atom resolved average magnetic mo-
ments. A Langevin function is also shown for comparison. It is interesting
to note that the contribution from the induced moment on the B atoms van-
ishes already at low temperatures. The mean field description in the RDLM
underestimates the experimental magnetisation. According to the method of
Ebert et al. [162], the MAE(T ) behaviour has, therefore, also been calculated
with the experimental M(T ) curves as input, resulting in the curves marked
with MT in Fig. 3.6b) and Fig. 3.6d), which contain the temperature depen-
dence of the MAE. The MT method produces an MAE quantitatively some-
what different from RDLM, but in this material it does not appear to pro-
vide a significantly better comparison with experimental data. In both Fe2B
and (Fe0.7Co0.3)2B there is a significant quantitative difference between the
RDLM or MT calculations and the experiments, but order of magnitude as
well as the qualitative behaviour is well captured. In particular, the spin reori-
entation transition is reproduced for Fe2B, although at an underestimated tem-
perature. For (Fe0.7Co0.3)2B a monotonically decreasing MAE(T ) behaviour
is correctly observed. These results should be compared with the calculations
of Zhuravlev et al. [173]. Those calculations were performed with a similar
scheme as that used in the SPR-KKR calculations presented here, although
simplified in several aspects. Zhuravlev et al. used a linear muffin-tin or-
bital (LMTO) method, in the GGA, with a so called vector DLM model [174],
58
T/TC
0 0.2 0.4 0.6 0.8 1 1.2
m/m
0
0
0.2
0.4
0.6
0.8
1Exp.LangevinFe RDLMB RDLM
(a) Fe2B, magnetisation.
T/TC
0 0.2 0.4 0.6 0.8 1
MA
E(M
J/m
3)
-1
-0.5
0
0.5
1
RDLMMT
Zhuravlev et al.Exp.
(b) Fe2B, MAE.
T/TC
0 0.2 0.4 0.6 0.8 1 1.2
m/m
0
0
0.2
0.4
0.6
0.8
1
Exp.LangevinFe RDLMCo RDLMB RDLM
(c) (Fe0.7Co0.3)2B, magnetisation.
T/TC
0 0.2 0.4 0.6 0.8 1
MA
E(M
J/m
3)
-0.2
0
0.2
0.4
0.6
0.8 RDLMM
T
Zhuravlev et al.Exp.
(d) (Fe0.7Co0.3)2B, MAE.
Figure 3.6. Temperature dependence of magnetic moments and MAE from relativistic
DLM calculations in SPR-KKR. Experimental data is taken from Paper VI. The data
of Zhuravlev et al. is from Ref. [173]. MT denotes SPR-KKR calculations for the
MAE with M(T ) behaviour taken from experiments.
which by initially neglecting SOC only has to consider magnetic moments
distributed over θ and not φ . The SOC is then included as a perturbation in-
stead of considering a relativistic description as in SPR-KKR. Finally, instead
of self consistently evaluating the Weiss field, as in the RDLM method, it is
fitted to a Langevin function using the experimental Curie temperatures. With
the various simplifications, it appears somewhat surprising that the calcula-
tions of Zhuravlev et al. provide such a good agreement with experiment in
the case of Fe2B, as shown in Fig. 3.6b). For (Fe0.7Co0.3)2B, Zhuravlev et al.notably underestimate the MAE, in contrast to SPR-KKR which overestimates
it. This might partly be because they artificially introduce a scaling of the ex-
change field to reduce the magnetic moment of Co in hope of achieving better
agreement with experiment.
59
3.1.5 Exchange Coupling Parameters
The exchange coupling parameters Ji j of the Heisenberg Hamiltonian, dis-
cussed in Sec. 2.2, can be computed by considering the change in the energy
due to small variations in magnetic moments at sites i and j. Liechtenstein etal. [165, 175, 176] derived formulas for such calculations within KKR multi-
ple scattering and found that the exchange coupling parameters can be com-
puted as
Ji j =− 1
4πIm
∫ εF
dεTr[(t−1
i↑ − t−1i↓ )τ i j
↑ (t−1j↑ − t−1
j↓ )τji↓]. (3.27)
Eq. 3.27 is implemented in SPR-KKR and the result of its application on FeCo
alloys was shown in Fig. 2.7. With these Ji j as input, one can model finite
temperature effects in various ways including mean field theory, Monte Carlo
simulations (see Sec. 3.2) or atomistic spin dynamics based on the Landau-
Lifshitz-Gilbert equation [104]. Curie temperatures are evaluated with Monte
Carlo simulations and exchange coupling parameters obtained from Eq. 3.27
in Paper II and Paper X.
Eq. 3.27 is derived considering a collinear magnetic reference state. At
finite temperatures, non-collinear spin states will inevitably appear, whereby
Szilva et al. [177] derived a non-collinear version of Eq. 3.27, which in prin-
ciple allows for a more realistic description.
3.2 Monte Carlo Simulations
In statistical mechanics [178] one wishes to evaluate partition functions
Z = Tre−H/kBT (3.28)
and expectation values such as
〈A〉= 1
ZTrAe−H/kBT . (3.29)
Calculating these traces for complicated systems with many degrees of free-
doms, amounts to evaluating sums or integrals over a phase space with a large
number of dimensions. This soon becomes insurmountable with determin-
istic methods but it turns out that stochastic methods such as Monte Carlo
(MC) [111, 179] simulations, which calculate averages from large sets of ran-
dom numbers, are well suited to solve these problems. In general MC al-
lows for efficient evaluation of multidimensional integrals, since an integral
may be considered an expectation value of a probability distribution, and is
often more efficient than deterministic methods in more than three dimen-
sions [179]. When it comes to solving the problems of classical statistical
mechanics, the algorithm of Metropolis et al. [180], essentially a method for
60
importance sampling the Boltzmann distribution, provides a powerful method
of solution.
Here, we are mainly interested in the Heisenberg Hamiltonian in Eq. 2.41.
The state of such a system is described by the directions of all the N spins in
the system, i.e. the set {mi}. The average magnetic moment of a particular
configuration is
〈m〉= 1
N ∑i
mi (3.30)
and the energy E({mi}) is easily calculated from Eq. 2.41. By generating
many different states based on random numbers, one can evaluate thermo-
dynamic averages such as average energy per spin e = 〈H〉/N, specific heat
capacity
c =∂e∂T
=〈H2〉−〈H〉2
NT 2, (3.31)
or magnetic susceptibility
χ =∂m∂h
=〈m2〉−〈m〉2
NT, (3.32)
where h is an applied field.
The Metropolis algorithm applied on this type of system can be summarised
as follows:
1. Generate an appropriate initial configuration, e.g., random or all spins
aligned.
2. For each i, randomly generate a new trial state where mi is changed to m′i
and calculate the change in energy ΔE. Generate a uniformly distributed
random number r ∈ [0,1] and accept the new trial state if r < e−ΔE/kBT ,
otherwise keep the old state as the new state. To do this for each of the
N spins is known as one MC sweep.
3. Repeat the second step and after every other sweep measure wanted
quantities and evaluate thermodynamic averages. Repeat the procedure
for a large enough number of sweeps so that the averages are well con-
verged.
Before taking measurements one should run a number of sweeps to make the
system unbiased from the initial state and often it is also good to do a simu-
lated annealing where the temperature is slowly lowered to the measurement
temperature from a higher temperature in order to stop the system from being
trapped in a local energy minimum [111].
In practice, when performing simulations, one is limited to particle num-
bers which are very small compared to the sizes of real systems with ∼ 1023
particles. Recurrently, one might wish to analyse properties in the thermody-
namic limit, i.e., where the size of the system goes to infinity under constant
density, which can be done using the methods of finite size scaling. Criti-
cal points can be analysed using critical exponents and the Binder cumulant
61
method [111, 179]. One is then interested in the Binder cumulant,
U = 1− 〈m4〉〈m2〉2
, (3.33)
which is independent of system size at the critical point where a phase tran-
sition occurs. Hence, plotting this quantity as a function of a thermodynamic
variable, such as temperature, for various system sizes and finding the point
of intersection allows one to identify the critical point in the thermodynamic
limit.
Figure 3.7 shows the average moment and the magnetic susceptibility as
functions of temperature for the L10 alloy FeNi with various system sizes de-
scribed by L so that there is a total L3 unit cells included in the simulation. The
particular MC implementation used here and in the work behind Paper II and
Paper X is that of the UppASD code [104]. In Fig. 3.7a one can observe how the
average moment decreases with temperature and how it decreases particularly
fast close to the transition temperature of TC = 916 K. One can also see that
for larger L, the drop in the moment is steeper and the value above TC goes
closer to the value of zero which is expected in the thermodynamic limit. In
Fig. 3.7b) it is shown how the susceptibility diverges at the critical point and
the peak becomes sharper for larger L. A fast and easy way to identify the
point of the phase transition is to look for peaks in the susceptibility. Fig. 3.8
0 500 10000
0.5
1
1.5
T (K)
<m>
(μB)
(a) Average moment.
0 500 10000
0.01
0.02
0.03
0.04
T (K)
χ
(b) Susceptibility.
Figure 3.7. Average moment and magnetic susceptibility as functions of temperature
for L10 alloy FeNi with system sizes L.
shows the Binder cumulant as function of temperature for the FeNi systems
of size L. The inset shows a close-up of the region around the critical point
where one can see how the curves for different L intersect at TC.
62
0 500 10000.4
0.5
0.6
0.7
T (K)
Bin
der
cum
ula
nt
L=10L=14L=18L=22
880 900 920
0.6
0.62
0.64
0.66
Figure 3.8. Binder cumulant as a function of temperature for L10 alloy FeNi with
system sizes L.
3.3 TEM Simulations - The Multislice ApproachPapers XIV-XV, which are further discussed in Sec. 4.2, deal with magnetic
effects on the elastic scattering of electrons through magnetic materials, as
they might appear in transmission electron microscopy (TEM). Various com-
putational methods are available for computational studies of TEM, among
which the more common include Bloch waves and multislice methods [181].
Multislice methods have the advantage of not requiring periodicity and are
typically more efficient, unless perfect crystals with very small unit cells are
to be considered. In Papers XIV-XV the multislice approach is extended to
consider magnetism. As was described in Sec. 2.3, when dealing with mag-
netic vector potentials, periodicity is broken which makes multislice methods
more suitable. A brief introduction to the multislice methodology is provided
here while more comprehensive discussion is available in literature [181–186].
Since the methods usually rely on the so called paraxial approximation, this is
first introduced.
In a TEM, a highly energetic electron beam, typically with kinetic energy
in the range 100− 1000 keV, scatters through a sample specimen and infor-
mation about the sample is obtained from the scattered exit beam. Fig. 3.9
shows a schematic illustration of the relevant situation with a sample of thick-
ness t, which is described by a stationary electrostatic potential3 V (r). In
simulations for realistic materials V (r) is typically constructed from atomic
potentials [181], or possibly calculated from electronic structure calculations
3This description already introduces an approximation, since exchange and correlation effects
between the beam electron and electrons in the sample are neglected. This is expected to be a
good approximation because of the large kinetic energy and wavevector of the incoming elec-
tron beam.
63
Figure 3.9. Schematic image of the situation studied in a transmission electron mi-
croscope. An incoming wave ψin(r) enters a sample of thickness t, represented by
an electrostatic potential V (r). This results in a reflected wave ψr(r) and a transmit-
ted wave ψout(r). The diffraction pattern, i.e. the modulus squared of the Fourier
transform of ψout(r), is the quantity observed in experiment.
which also allows the study of effects from chemical bonding [187]. The
incoming beam created in the microscope is contained in the wavefunction
ψin(r) while the reflected part is ψr(r). Let z be the propagation direction and
z = 0 be the entrance plane of the sample so that z = t is the exit plane. For
t < 0 the wavefunction is then ψ(r) = ψin(r)+ψr(r). The wanted quantity
is ψ(x,y,z ≥ t), denoted by ψout(r). This should be obtained by solving the
Schrödinger equation
− h2
2m∇2ψ(r)− eV (r)ψ(r) = Eψ(r), (3.34)
which requires suitable boundary conditions for ψ(r). In the xy-plane these
should be considered as known (for example periodic). In the z-direction, on
the other hand, the boundary conditions are currently insufficient as the only
known quantity is ψin(r), which leaves both ψ(x,y,0) (because ψr(r) is un-
known) and ψ(x,y, t) (this is the wanted quantity) unknown. The first step,
related to the paraxial approximation, is thus the forward scattering approx-
imation, which neglects back scattering so ψr(r) = 0, whereby ψ(x,y,0) is
a known quantity. Alternatively, the boundary condition at z = 0 could have
been formulated in terms of a continuity equation which would require dif-
ferent computational methods to be used. Various efforts have attempted to
64
extend multislice methods beyond the forward scattering approximation [188,
189], but the work in this thesis is restricted to forward scattering. Being a
second order equation in z, Eq. 3.34 still requires additional information (e.g.
knowledge of ∂∂ z ψ(x,y,0)) to be solved and, furthermore, is computationally
demanding. This leads us to the paraxial approximation. When the incoming
electron is moving fast along the z-direction in free space, its wavefunction
should approximately be a plane wave eikz with a large k and large energy
E = h2k2
2m . It should then continue to approximately be so also in the sample. It
is thus suitable to introduce the ansatz
ψ(r) = eikzφ(r) (3.35)
where φ(r) is a relatively slowly varying function of z and the paraxial ap-
proximation reads ∣∣∣∣∣∂ 2φ∂ z2
∣∣∣∣∣�∣∣∣∣k ∂φ
∂ z
∣∣∣∣ . (3.36)
By substituting Eq. 3.35 into Eq. 3.34 and neglecting the term with a second
derivative of φ(r) w.r.t. z according to Eq. 3.36, it is straight forward to derive
a paraxial Schrödinger equation for φ(r) as
∂∂ z
φ(r) =i
k
(1
2∇2
xy +meV (r)
h2
)︸ ︷︷ ︸
H(r)
φ(r), (3.37)
where ∇xy = (∂x,∂y) is the gradient in the xy-plane. Eq. 3.37 should be accu-
rate for small angle scattering but the approximation in Eq. 3.36 breaks down
for larger scattering angles. It is interesting to note that Eq. 3.37 takes the form
of a time-dependent Schrödinger equation in two spatial dimensions under the
substitution of z = hkτ/m. Since Eq. 3.37 is first order in z it can be solved
at any z from knowledge of φ(x,y,z = 0) and this is what is done with the
multislice approach.
Before discussing how Eq. 3.37 is solved using multislice methods, it is
important to discuss the effect of relativity on the fast electrons, since this
is relevant for the large kinetic energies typically used in the TEM. For ex-
ample, an electron with 200 keV kinetic energy will travel at approximately
v/c = 0.7 so that relativistic effects clearly cannot be neglected. The rele-
vant equation to solve would thus be the Dirac equation (Eq. 2.2) rather than
the Schrödinger equation in Eq. 3.34. From a description of multiple scat-
tering of electrons in an electrostatic potential based on the Dirac equation,
Fujiwara [190] argued that relativistic effects due to the large kinetic energy
of the electrons can largely be described using a Schrödinger equation with
relativistically corrected mass m = γm0, where m0 is the rest mass and γ the
Lorentz factor, while also applying a relativistic correction to the wavelength
65
according to
λ =2πk
=hc√
K(2m0c2 +K), (3.38)
where K is the kinetic energy of the electron, which is often written K = |eU |with U being the acceleration voltage. The standard treatment of relativis-
tic effects is hence to make these substitutions into Eq. 3.37. This is also
equivalent to starting from the Klein-Gordon equation but neglecting a term
proportional to the electrostatic potential squared [182, 186], i.e. it describes
relativistic but spinless particles in an electrostatic potential. Numerical cal-
culations based on the Dirac equation have indicated that further relativistic
effects are negligible for the typical situation studied in a TEM and that ne-
glecting the V 2 term yields errors of less than 0.1%, even at relatively large
scattering angles in the 1st order Laue zone, when considering heavy atoms
such as Au [186] .
The solution to Eq. 3.37 is
φ(x,y,z+Δz) = Z{e∫ z+Δz
z H(x,y,z′)dz′}φ(r), (3.39)
where Z is the path ordering operator required when[H(x,y,z1), H(x,y,z2)
]�=
0 for z1 �= z2. Knowing φ(x,y,z) for z = 0, it can now be calculated at any z if
a practically feasible way of evaluating the exponential operator in Eq. 3.39 is
available and there are different flavours of the multislice scheme to do this. In
the conventional multislice method the Baker-Campbell-Hausdorff formula is
applied to separate the exponential into a product of several exponentials, pe-
riodicity is assumed in the xy-plane and forward and backward Fourier trans-
forms are applied to efficiently deal with kinetic energy term and the resulting
convolution in real space [181]. Alternatively, there is a real space multislice
method which, in addition to providing good numerical precision [185], does
not require periodicity in any spatial dimension. This will prove advantageous
when considering magnetism whereby we choose to focus on the real space
version of the multislice method. By writing
h = Z1
Δz
∫ z+Δz
zH(x,y,z′)dz′ (3.40)
the exponential in Eq. 3.39 can be expanded as
φ(x,y,z+Δz) =∞
∑n=1
Δzn
n!hn(r)φ(r). (3.41)
For thin Δz the series will converge rapidly and can be truncated after a suit-
able number of terms. Furthermore, for thin enough Δz, h ≈ H(x,y,z). It is
now straightforward to compute φ(x,y,z+Δz) from φ(x,y,z) and this can be
repeated a suitable number of times until z = t is reached.
66
3.3.1 Multislice Solution to Paraxial Pauli Equation
The discussion above considered the scattering of electrons from an elec-
trostatic potential V (r) but neglected magnetic fields, i.e. the flux density
B(r) = ∇×A(r) and vector potential A(r). Therefore, in Papers XIV-XV
where magnetic effects in elastic electron scattering are studied, a generalisa-
tion of the equations above for the case including magnetic fields is derived
and used. This generalisation is briefly presented and discussed here. It leads
to a paraxial form of the Pauli equation with relativistic corrections, or it can
equivalently be considered as a two-component squared Dirac formalism, ne-
glecting a term related to the spin-orbit coupling. To perform simulations for
realistic materials one also needs a description of the magnetic fields which
can be obtained according to the scheme described in Sec. 2.3.
When dealing with highly relativistic electrons in electromagnetic fields,
the natural starting point is the Dirac equation (Eq. 2.1-2.2). By multiply-
ing Eq. 2.2 from the left with[βmc2− (E + eV )−ααα · (p+ eA)
]it is straight
forward4 to write down the squared form of the Dirac equation
[ (E + eV )2− c2(p+ eA)2−m20c4︸ ︷︷ ︸
Klein-Gordon
−ech(cΣΣΣ ·B+ iααα ·E )]
ψ = 0. (3.42)
As has been pointed out by others before [186], neglecting the final two terms,
which are the only non-diagonal ones, results in four decoupled Klein-Gordon
equations. As mentioned above, this yields the description given in the previ-
ous section if A = 0 and the V 2 term is ignored. If one is interested in mag-
netism, it is crucial to include also the next term which describes a Zeeman
interaction of the electron spin with the magnetic field. The final term is an
additional relativistic interaction for spin half fermions which should contain,
among other effects, the spin-orbit coupling. It is the only term coupling the
upper and lower halves of ψ , whereby neglecting this term allows one to work
with an equation for two rather than four components. Whether it is acceptable
to neglect that term seems questionable considering that E =−∇V diverges at
the atomic positions, which should make it important in describing large angle
scattering. Nevertheless, if one is interested in extending existing models to
describe magnetism, this term should not be the most crucial as it describes a
coupling to the electric, not the magnetic, field and it should be reasonable to
ignore that term, at least when considering small scattering angles. Doing so
and dividing Eq. 3.42 by twice the energy E = γm0c2 leads to a relativistically
corrected two-component Pauli equation[1
2m
(p+ eA(r)
)2+
em
S ·B(r)− eV (r)]
Ψ(r) = EΨ(r), (3.43)
4One mainly needs to apply the rules for the Dirac matrices, namely α2i = β 2 = I, αiβ =−βαi
and (ααα ·a)(ααα ·b) = a ·b+ iΣΣΣ · (a×b), as well as the definitions B = ∇×A and E =−∇V .
67
where a new energy E has been introduced as
E =E2−m2
0c4
2E. (3.44)
If one substitutes γ = 1√1− v2
c2
into this energy, it leads to
E =1
2γm0v2 =
1
2mv2, (3.45)
i.e., the classical expression for the kinetic energy of the electron with the
relativistic mass m = γm0. To Eq. 3.43 it is straight forward to apply the same
steps that brought us from Eq. 3.34 to Eq. 3.37, starting with an ansatz
Ψ(r) = eikz(
ψ↑(r)ψ↓(r)
)(3.46)
and then neglecting second derivatives of ψ↑,↓(r) w.r.t. z. Doing so, applying
Coulomb gauge ∇ ·A = 0 and neglecting5 the term proportional to (eA)2 leads
to the paraxial form of the Pauli equation
∂∂ z
(ψ↑(r)ψ↓(r)
)=
imh(hk+ eAz)
−1
{h2
2m∇2
xy +iehm
Axy ·∇xy− hkeAz
m
− em
S ·B+ eV}(
ψ↑(r)ψ↓(r)
). (3.47)
Since Eq. 3.47 is of the form ∂zφ = Hφ , Eq. 3.39-3.41 can be applied without
modification and this is how magnetic effects in elastic, paraxial electron scat-
tering are studied in Papers XIV-XV, as discussed further in Sec. 4.2. Eq. 3.47
is equivalent to Eq. 3.37 for each component if A = B = 0. All terms in
Eq. 3.47 are spin diagonal except the S ·B term which yields off-diagonal con-
tributions proportional to Bx± iBy. Thus, it is only the x- and y-components
of the B-field which scatters spin up electrons to spin down states and vice
versa. As pointed out before Az = 0 when considering collinear magnetism
(see Sec. 2.3).
5One reason for this is to treat A and V on equal footing, but also one can confirm that this term
is small. For example |hk⊥| ≈ 7 · 10−23 kgm/s for a beam with 300 keV kinetic energy, while
|eA| ≈ 10−28 kgm/s and√
eh|B| ≈ 10−26 kgm/s, based on the data in Fig. 2.8.
68
4. Results
This chapter presents a brief overview of the key results of this thesis, while
readers interested in more details are referred to the papers. Sec. 4.1 discusses
results regarding MAE and permanent magnet materials, based on the work
in Papers I-XIII. Sec. 4.2 presents an overview of the results regarding mag-
netic effects in elastic electron scattering, as they might appear in transmission
electron microscopy, which is the topic of Papers XIV-XV.
4.1 Magnetocrystalline Anisotropy and PermanentMagnet Materials
Various 3d-based materials in non-cubic crystal structures are studied as they
might have large MAE and be of technological importance. Fe1−xCox alloys
and the possibility to increase the MAE by introducing tetragonal strain via in-
terstitial B or C impurities is discussed in Sec. 4.1.1. Tetragonal (Fe1−xCox)2B
and the effect on its MAE from 5d impurities is discussed in Sec. 4.1.2. The
results regarding magnetic properties of several binary compounds in the L10
structure, tetragonal Fe5Si1−xPxB2 and hexagonal Laves phase Fe2Ta1−xWxare overviewed in Sec. 4.1.3-4.1.5. Finally, a number of other possibly inter-
esting materials are briefly discussed in Sec. 4.1.6.
4.1.1 Fe1−xCox Alloys
Fe1−xCox has a large magnetic moment of more than 2.3 μB/atom at the max-
imum of the Slater-Pauling curve, around x≈ 0.2 [28, 152, 191]. The material
also has a high Curie temperature, above 1000 K, so two of the important
requirements for a good permanent are fulfilled. Unfortunately, most of the
Fe-Co phase diagram below 1100 K is bcc [192], whereby the MAE is van-
ishingly small. However, Burkert et al. [70] showed that an enormous MAE
could be obtained in tetragonally strained Fe1−xCox, given specific conditions
of c/a ≈ 1.2 and x ≈ 0.65. The work of Burkert et al. was based on the
VCA and, as discussed in Sec. 3.1.3, it significantly overestimates the MAE.
However, more realistic CPA or supercell treatment of disorder, as well as ex-
periments, confirm that a notable enhancement of the MAE can be obtained
in tetragonally strained Fe1−xCox [150, 154, 156–158, 193]. Clearly, this is
interesting in the context of rare-earth free permanent magnet replacement
materials.
69
Experimental studies [157, 158, 193] following the computational work of
Burkert et al. [70], on the increased MAE in bct Fe1−xCox, relied on thin film
strain engineering. In this approach there is eventually a lattice relaxation
back to the cubic structure, limiting the thickness of tetragonal films to a few
nm. Paper I, Paper III, Paper V and Papers VIII-IX focus on the possibility
to stabilise a tetragonal strain in Fe1−xCox alloys beyond thin films, using
interstitial dopants, and these results are discussed further below.
Co is found in the uniaxial hcp structure at room temperature and pressure,
which allows it to have a significantly higher MAE compared to bcc Fe. An-
other interesting system to explore could therefore be Fe1−xCox in the hcp
structure. One problem with this is that hcp is only the stable phase for a
narrow range of x close to one [192]. Nevertheless, FP-LAPW VCA calcu-
lations were performed to explore the magnetic properties of the hypothetical
hcp Fe1−xCox system for a complete range of x ∈ [0,1], also varying the lat-
tice parameters around their equilibrium values. It was found that large values
of MAE > 400 μeV/atom could be obtained for x ≈ 0.3 or smaller, but the
calculations indicate that the alloy is antiferromagnetic for x < 0.5, so that
no magnetisation would be left, making the system unsuitable as a permanent
magnet even if one would manage to alloy large amounts of Fe into the hcp
crystal.
Tetragonal Fe1−xCox Via C or B ImpuritiesThe metastable Fe-C martensite phase is an old and well known system in
which C atoms go into octahedral interstitial positions of the bcc Fe crystal,
where they cause a tetragonal distortion [194–196]. In practice this metastable
phase is obtained by rapid cooling of the high temperature fcc phase. Due to
similarities in the phase diagrams of the Fe-C and Fe-Co-C systems [192, 197],
it does not appear unreasonable to imagine the same type of structure occur-
ring in an alloy of Fe1−xCox-C. This would allow one to produce a tetragonal
Fe1−xCox based compound, potentially possessing the desired intrinsic prop-
erties of a permanent magnet, including a large MAE, if the desired conditions
of c/a≈ 1.2 and x≈ 0.65 pointed out in Ref. [70] can be realised.
In Paper I, a computational study regarding the structural and magnetic
properties of (Fe1−xCox)yC is presented. Minima in the energy as function
of volume and c/a, with c/a > 1, are presented for a number of internally
relaxed (Fe1−xCox)yC systems with y = 8, 16 and 24 and C in octahedral in-
terstitial positions as illustrated in Fig. 4.1a)-4.1c) and a few different values
of x around 0.65, as suggested in Ref. [70]. This indicates that it could indeed
be possible to find the desired type of martensite structures described above.
For systems with y = 16, relatively large tetragonal strains up to c/a ≈ 1.17
are found so that potentially large values of the MAE can be obtained. For
systems with a lower C content, i.e. y = 24, the strain is significantly lower
and c/a≈ 1.035.
70
The magnetic properties of the materials, including MAE, were calculated
using both WIEN2k with VCA and the force theorem as well as SPR-KKR in
the ASA with CPA and total energy differences. As expected, the VCA calcu-
lations overestimate the MAE significantly, but even with the CPA significant
MAE’s of up to EMAE = 41.6 μeV/atom = 0.59 MJ/m3 are found. Supercell
calculations in the FP-LAPW method, using special quasirandom structures
(SQS), result in a slightly larger value of EMAE = 0.75 MJ/m3. This might
be interpreted as an indication that the ASA underestimated the MAE, which
would be consistent with the results in Fig. 3.2. These results indicate that car-
bon doped Fe1−xCox can have an MAE around twice that of ferrite magnets
(see Table. 1.1). These systems also exhibit large saturation magnetisations
of μ0MS ≈ 2 T and if one can suspect that a small amount of non-magnetic
C does not drastically affect the strong exchange interactions of Fe and Co
atoms, it is reasonable to also expect a high Curie temperature.
(a) (Fe1−xCox)8C (b) (Fe1−xCox)16C (c) (Fe1−xCox)24C (d) (Fe1−xCox)32C
Figure 4.1. Illustrations of the various (Fe1−xCox)yC structures with interstitial C
atoms studied in Papers I and III.
These results point towards a potential route to a new permanent magnet
but, to this point, all results are theoretical suggestions. The obvious next
step should be an experimental confirmation. Such a confirmation is pro-
vided in Paper III, where pulsed laser deposition is used for epitaxial growth
of the (Fe1−xCox)yC system described above. It is found that when the ternary
(Fe1−xCox)yC system is grown on a CuAu buffer the tetragonal strain satu-
rates towards c/a≈ 1.03 as the film is grown thicker, in contrast to the binary
Fe1−xCox system, which rapidly saturates towards c/a = 1, clearly indicat-
ing that the C atoms indeed induce a tetragonal strain. A magnetocrystalline
anisotropy as large as 0.44 MJ/m3 was measured for x = 0.6. However, only
a small C content of around 2 at.% enters the system, rendering direct com-
parison to data in Paper I difficult. Hence, calculations were performed for
a (Fe0.4Co0.6)32C system, as that illustrated in Fig. 4.1d), with the experi-
71
mentally measured lattice parameters a = 2.81 and c/a = 1.03. Calculations
using WIEN2k with VCA and the force theorem indicate EMAE = 0.51 MJ/m3
while SPR-KKR in the ASA with CPA and total energy differences yields
EMAE = 0.22 MJ/m3. Again the computational results are consistent in the
way that VCA overestimates the MAE, while the ASA calculations underesti-
mate it.
If it is possible for C to go into octahedral interstitial positions, as discussed
above, it is easy to imagine also other atoms with similar size and properties,
such as B or N, to do the same. For B impurities in Fe, calculations have
shown that the desired octahedral interstitial positions are energetically pre-
ferred over tetrahedral interstitial positions [198]. Consequently, work very
similar to that in Paper I and Paper III was repeated with B impurities instead
of C and presented in Paper V. Similar results as for C are obtained in both
theory and experiment. However, the calculations indicate that the tetragonal
strain, and thus also the MAE, is larger for systems with y = 24 in the B case
than what was found with C. This is an indication that low amounts of B can
cause larger tetragonal strains, compared to a similar C content, and thus lead
to higher MAE per impurity content. Furthermore, the experimental work was
successful in using a higher percentage of B, compared to C, while still in-
creasing the c/a. A maximum c/a = 1.045 was experimentally observed for
Fe0.38Co0.62 with 4.2 at.% B. An MAE above 0.5 MJ/m3 was measured, again
indicating that the B-doped system is possibly more interesting than the C-
doped one. Higher B content leads to amorphisation and a reduction of both
c/a and MAE.
Papers VIII-IX contain further analysis of the B or C-doped Fe1−xCox. Pa-
per VIII contains results from ferromagnetic resonance (FMR) and XMCD
experiments on approximately 20 nm thick Fe0.4Co0.6-B films with 0, 4 or 10
at.% B content, which have c/a = 1.013, c/a = 1.034 and c/a = 1.02, re-
spectively. That the c/a is greater than unity without B indicates an effect of
lattice strain induced from the buffer layer of the film, while the reduction in
c/a when increasing the B-content from 4 to 10 at.% is an indication of the
amorphisation previously mentioned regarding high B-content films. XMCD
experiments reveal that the ratio of orbital magnetic moment to spin mag-
netic moment mL/mS is larger in the film with 4 at.% B than in that without
B. Comparison to calculations for Fe0.4Co0.6 without B shows that tetragonal
strain has very limited effect on mL/mS. On the other hand, the calculations
in Paper V revealed that the magnetic moments of the Fe/Co atoms near the
B impurities are strongly affected in a manner which can explain the observed
change in mL/mS. The FMR measurements indicate that the magnetic damp-
ing parameter is smaller in the B-doped sample, which is explained in terms
of a change in the density of states at the Fermi energy.
In thin films the magnetic anisotropy is heavily influenced by surface and
interface effects. In Paper IX, the thickness dependence of magnetic anisotropy
is studied and the anisotropy contributions of the buffer interface and sur-
72
B
Fe/Co
Figure 4.2. One unit cell
of (Fe1−xCox)2B in the
tetragonal crystal struc-
ture with space group
140.
face are determined. A huge surface contribution of 1.4 mJ/m2 is measured,
while FP-LAPW calculations for a slab geometry indicate 0.08 mJ/m2 sur-
face anisotropy. The very large surface anisotropy seen in the experiment is
thought to be due to oxidation effects.
4.1.2 (Fe1−xCox)2B
(Fe1−xCox)2B crystallises in the tetragonal structure illustrated in Fig. 4.2 for
all x. The MAE as a function of alloy concentration x and temperature in
the material was experimentally studied by Iga [94] and is interesting as the
MAE shows non-monotonic variations as function of both alloy concentra-
tion and temperature. The end compounds, Fe2B and Co2B, have negative
(in-plane) magnetocrystalline anisotropy, while for x ≈ 0.3, the MAE is pos-
itive for all temperatures [94, 199], potentially interesting in applications. At
low temperatures and x ≈ 0.3, the MAE is approximately 0.5 MJ/m3. In Pa-
per VI both computational and experimental results regarding the magnetism
of (Fe1−xCox)2B, with focus on the magnetocrystalline anisotropy, are pre-
sented.
The low temperature magnetisation and MAE of (Fe1−xCox)2B, as func-
tions of x, were computed using the SPR-KKR with ASA and CPA, WIEN2kwith the VCA, as well as the full potential local-orbital (FPLO) minimum-
basis [200] method with the VCA, where the latter two yield nearly identical
results. The SPR-KKR-ASA-CPA calculations provide a satisfactory descrip-
tion of the magnetic properties for x ≤ 0.6, while for larger x both magneti-
sation and MAE are in disagreement with experimental data. The full poten-
tial VCA calculations yield qualitatively similar results for x ≤ 0.6, while the
MAE results differ significantly with those from SPR-KKR-ASA-CPA calcu-
lations and are in better agreement with experiments for larger x, including
x = 1. This implies that full potential effects are particularly important for
Co2B. This conclusion was also supported by comparison between FP and
ASA calculations in SPR-KKR for the end compounds. All of the calcula-
tions, however, underestimate the magnetisation on the Co-rich side of the al-
loy, indicating that the GGA is insufficient to properly describe the magnetism
in this material. DMFT calculations with the full-potential linear muffin-tin
orbital [110] (FP-LMTO) method were therefore also used to calculate the
73
magnetic moments of the end compounds. DMFT was found to improve the
magnetic moment of Co2B, while for Fe2B the agreement with experiment
became a bit worse. The general conclusion is that a complete description of
magnetism in the material is challenging to obtain and requires an accurate
description of the electrostatic potential and charge density as well as a more
accurate description of electron correlations than what is accessible within the
GGA. Nonetheless, SPR-KKR with the ASA and CPA provided good agree-
ment with experiment for x≤ 0.6, whereby it is applied for further calculations
in this interval.
One interesting topic to address regarding this material is the temperature
dependence of the MAE, which was also discussed in Sec. 2.1.2 and Sec. 3.1.4.
In particular, one would like to explain the non-monotonic variations and spin
reorientation transitions which are observed for various x. In Paper VI this
is done by performing fixed spin moment calculations, which allow for the
MAE to be calculated as a function of the size of the magnetic moment when
the exchange splitting is varied. This is then mapped to temperature in a sim-
plified picture which separates contributions from longitudinal and transversal
thermal fluctuations of the spins. The longitudinal part is then described with
the fixed spin moment calculations and mapped to temperature via a simple
function m = m0
√1−T 2/T 2
C , relevant for weak ferromagnets [1] (there are
unoccupied majority spin states). The transversal fluctuations are separately
taken into account in a single-ion anisotropy manner, by enforcing a reduction
in MAE with the third order of the average magnetic moment, according to
the discussion in Sec. 2.1.2. The reduction in the average magnetic moment
due to transversal spin fluctuations is assumed to be described by a Langevin
function. In this model, transversal spin fluctuations result in a monotonic re-
duction of the magnitude of the MAE, while longitudinal fluctuations are the
source of the non-monotonic variations of the MAE(T ), caused by changes
in the band structure around the Fermi energy as the exchange splitting is re-
duced. This simple model is found to be sufficient to qualitatively describe the
variations in the MAE as function of temperature in Fe2B and Co2B. A more
sophisticated approach to this problem was considered in Sec. 3.1.4, where
calculations based on relativistic disordered local moments were used.
For being a 3d-based transition metal magnet, the low temperature MAE
of 0.5 MJ/m3 is relatively large. However, with the saturation magnetisa-
tion above 1.5 T, many applications, including permanent magnets, might re-
quire a larger MAE than that. An important question is therefore whether the
MAE can be increased further. To investigate this we consider the possibility,
brought up already in Sec. 2.1.2, of increasing the MAE by adding 5d ele-
ments with strong SOC. The MAE was calculated for the alloy with x ≈ 0.3,
but 2.5 or 5 at.% of different 5d elements substituting the Fe and Co. The
resulting MAEs for the case of 5 at.% 5d impurities are shown in Fig. 4.3a),
with the horizontal dashed line indicating the MAE of (Fe0.7Co0.3)2B. Some
74
5d elements, such as W or Re, cause a significant increase in the MAE, while
others, including all in the latter half of the series, yield a small decrease in the
MAE. In Paper VI, results for calculations without SOC on the 5d atoms were
also presented. These results illustrate that there is a trend in the variation of
the MAE which is similar without the SOC on the 5d atoms, but the variation
is weaker. This indicates that the large increase in MAE seen with Re or W is
due to the strong SOC of the 5d atoms, while other changes in the electronic
structure also play a role in determining the change in the MAE. Fig. 4.3a)
also shows the anisotropy in the induced orbital magnetic moment on the 5d
impurities. There is a drastic change in this anisotropy in going from Re to
Os, crossing from the first to the second half of the series, which appears to
correlate with a similar change in the MAE.
Lu Hf Ta W Re Os Ir Pt Au Hg
MA
E (
me
V/f
.u.)
0
0.1
0.2
0.3
Δ m
L o
f 5d (
10
-2μ
B)
-2
-1
0
1
(a)
Lu Hf Ta W Re Os Ir Pt Au Hg
5d m
om
ent (μ
B)
-0.4
-0.2
0
0.2
0.4
mS
10mL
(b)
Figure 4.3. a) MAE of (Fe0.675Co0.275Z0.05)2B with Z being various 5d elements, as
well as the anisotropy of the induced orbital magnetic moment on the 5d atoms. b)
Induced spin and orbital magnetic moments on the 5d atoms for spin quantisation axis
along the c-axis.
A crucial next question is of course whether these results can also be corrob-
orated experimentally. Therefore, single crystals of the (Fe0.675Co0.3Z0.05)2B
compound, alloyed with a few atomic percent of either Re or Ir, were synthe-
sised. MAE measurements on these crystals confirm a large increase in MAE
with Re doping and a small decrease in MAE with Ir doping, in excellent
agreement with the results in Fig. 4.3a) and Fig. 14 of Paper VI.
Fig. 4.3b) shows the calculated spin and orbital magnetic moments on the
5d impurities, for a magnetisation along the 001-direction. The induced mo-
ments show clear trends as one traverses the 5d series. In particular, the spin
moments are anti-parallel to the Fe/Co moments for the early 5d elements,
while they are parallel to the Fe/Co moments for the latter 5d elements. Such
a trend in the induced spin magnetic moments on 5d impurities embedded in a
magnetic 3d host has been found both computationally [201, 202] and experi-
mentally [203] before. As a first approximation one could expect the induced
orbital magnetic moments to obey Hund’s third rule, in which case mL and mSwould be parallel for the first half of the series and anti-parallel for the latter
75
half. This agrees quite well with the data in Fig. 4.3, although it does not hold
for W and Os. The magnetic moments on W has been discussed in relation to
Fe-W interfaces, where it was also found that the induced W moments violate
Hund’s third rule [204]. Nevertheless, Hund’s rules describe free atoms rather
than solids and can strictly speaking not be applied here. The small amounts
of 5d impurities have a limited effect on the density of states and spectral
functions at the Fermi energy, which should be decisive for the MAE. The
variations in the induced 5d moments shown in Fig. 4.3b) are instead expected
to be important to understand how the different dopants affect the MAE of the
alloy.
4.1.3 L10 Binary Compounds
Fig. 4.4 illustrates two different unit cells of the L10 crystal structure, one
fct-like cell with volume V1 = a2c and one bct-like, rotated by π4 about the
c-axies, with volume V2 = a′2c = V12 . The structure is often described in terms
of the fct-like structure, but for computational work it is beneficial to use the
smaller bct-like structure to reduce the system size and hence also computa-
tional effort. Certain binary alloys, such as FePt [78, 82, 205, 206], can have
an enormous MAE in this ordered structure. In the case of FePt, the 3d-5d
hybridisation and strong SOC of the Pt is important for the large MAE. Inter-
estingly, it is also possible to obtain large MAE without heavy elements such
as Pt, and materials of this kind have received attention for permanent magnet
applications. Two of the compounds which have recently received most atten-
tion in the context of replacement materials for rare-earth permanent magnets
are L10 structured FeNi [74, 75, 205, 207–212] and MnAl [93, 213–219]. Also
MnGa [220–222] and CoNi [223] have been studied in the L10 structure and
have a positive uniaxial MAE. In Paper II, a comprehensive computational
study for all four of these compounds is presented. A combination of WIEN2kand SPR-KKR, as well as MC calculations, is used to evaluate all three of the
important intrinsic magnetic properties of MAE, saturation magnetisation and
Curie temperature. The effect of disorder is also addressed using the CPA.
Figure 4.4. Two different perspectives on the L10 structure.
76
MnAl in the perfectly ordered stoichiometric case is found to not order
magnetically according to MC simulations, while MnGa does but with a very
low TC around 80 K. This is in strong contrast to experimental measurements
which report Curie temperatures above 600 K for these materials [213, 224].
However, these compounds are normally studied with some excess Mn con-
tent. Hence, the CPA is used to consider Mn-rich compounds with some extra
Mn substitution on the Al/Ga-site. This is found to result in an antiferromag-
netic coupling between the magnetic moments of Mn atoms on the different
sites, which stabilises a ferrimagnetic ordering with significant increase in TC,
reaching values of similar size as those experimentally reported. The ferrimag-
netic ordering, illustrated in Fig. 4.5, naturally also causes the saturation mag-
netisation to decrease, while calculations indicate that the MAE is increased.
Figure 4.5. Ferrimagnetic Mn1+xGa1−x.
One of the experimental challenges in obtaining good magnetic proper-
ties in the L10 compounds is to achieve a high degree of chemical ordering.
Substitutional disorder has been shown to significantly decrease the MAE of
FeNi [225] and similar observations have been made for FePt in the same
structure [77, 206]. In Paper II, the effect of substitutional chemical disorder
is studied for both FeNi and CoNi. In addition to a confirmation that the MAE
decreases with disorder, it is found that also the TC is slightly decreased in
both cases.
In Paper II, the MAE of MnGa is found to be bigger than that of MnAl. This
is also illustrated in Fig. 4.6, which shows the magnetic moments and MAE
of Mn1.1Al0.9−xGax, calculated with SPR-KKR and the CPA to treat disorder.
The MAE appears to increase monotonically and nearly linearly with the Ga
concentration x. One possible origin of the increase in MAE could be the
stronger SOC of Ga, which is below Al in the periodic table. However, as was
seen in Paper II, Al and Ga contributions to the DOS at the Fermi energy are
negligible, whereby the SOC of these atoms are not expected to be important
for the MAE. Calculations with the SOC of the Al and Ga atoms suppressed
confirm this conclusion, as it has a negligible (around 0.06%) effect on the
77
MAE. Instead the change in the MAE must be the result of other changes in
the electronic structure. It is interesting to note that also the magnetic moments
increase notably as one removes Al and substitutes it with Ga.
x
0 0.2 0.4 0.6 0.8
mo
me
nt
(μB)
2
2.5
3
3.5
Mn1
-Mn2
mtot
/f.u.
(a) m(x)
x
0 0.2 0.4 0.6 0.8
MA
E (
me
V/f
.u.)
0.3
0.35
0.4
0.45
0.5
(b) MAE(x)
Figure 4.6. Magnetic moments in a) and MAE in b) of Mn1.1Al0.9−xGax
In, for example FeNi, the saturation magnetisation of 1.33 MA/m is rather
large, while the MAE, even in the perfectly ordered case, is rather modest
0.77 MJ/m3, according to FP-LAPW calculations. This results in a magnetic
hardness parameter κ =√
K/μ0M2S well below unity, indicating that a further
increase in MAE (or decrease in MS) would be required for the material to
be useful as a permanent magnet. A potentially interesting path towards this
could be that discussed for (Fe1−xCox)2B in the previous section and Paper VI,
adding a few atomic percent of 5d dopants into the system.
Calculations using SPR-KKR in the ASA and with disorder treated in the
CPA have been performed for the two L10 structure compounds FeNi and
MnAl with one atomic percent of 5d impurities added into the system. In the
FeNi case, one atomic percent was substituted on each of the FeNi sites, while
in the MnAl case, an Mn rich alloy MnAl0.95Mn0.05 was considered and two
atomic percent of the excess Mn on the Al site was substituted for 5d elements.
The results are shown in Fig. 4.7. The first thing to note is that the induced
5d magnetic moments shown in Fig. 4.7b) and Fig. 4.7d) are similar to those
which were shown in Fig. 4.3b), indicating the generality of the trend in the
induced magnetic moments of 5d dopants in a magnetic 3d host. In Fig. 4.7b),
it is seen that the induced moments vary slightly depending on whether the
impurity is on the Fe or Ni site, while the general trend is very similar. Also in
the changes in the MAE seen in Fig. 4.7a) and Fig. 4.7c) there are similarities
with the results in Fig. 4.3. In all cases there is a strong increase in MAE as one
approaches the middle of the series from the left and then a rapid decrease as
one continues across to the right half of the series, e.g. in going from Re to Os
in Fig. 4.7c). There also appears to be a correlation between the anisotropy of
the induced orbital magnetic moments of the 5d dopants and the change in the
MAE. The changes in MAE of maximum around 20 percent seen in Fig. 4.7
78
are relatively small compared to the increase of almost 70 percent seen in
Fig. 4.3. However, in Fig. 4.7 low dopant concentrations are considered and
in Paper VI it was found that increasing the amount of dopants also notably
increased the effect on the MAE.
Lu Hf Ta W Re Os Ir Pt Au Hg
MA
E(m
eV/f.
u.)
0.1
0.15
mL
of5d
(B
)
0
0.01
(a)
Lu Hf Ta W Re Os Ir Pt Au Hg5d
mom
ent(
B)
-1.5
-1
-0.5
0
0.5
1
mS
at Fe
mS
at Ni
10mL
at Fe
10mL
at Ni
(b)
Lu Hf Ta W Re Os Ir Pt Au Hg
MA
E(m
eV/f.
u.)
0.25
0.3
0.35
0.4
mL
of5d
(B
)
-0.05
0
0.05
(c)
Lu Hf Ta W Re Os Ir Pt Au Hg
5dm
omen
t(B
)
-1.5
-1
-0.5
0
0.5
mS
10mL
(d)
Figure 4.7. MAE and orbital magnetic moment anisotropy of a) FeNi or c) MnAl with
1 at.% of various 5d dopants. The induced magnetic moments on the 5d atoms are
shown for FeNi in b) and MnAl in d).
4.1.4 Fe5Si1−xPxB2
Fe5PB2 and Fe5SiB2 crystallise in a tetragonal I4/mcm crystal structure and
Mössbauer measurements have shown that they are ferromagnetic [96, 226]
with Curie temperatures of approximately 628 K and 784 K for the P and Si
cases, respectively. Being tetragonal, Fe-rich ferromagnets with Curie temper-
atures well above room temperature, these compounds could be interesting as
permanent magnet materials
Paper VII contains experimental measurements of magnetisation as func-
tion of temperature as well as MAE for Fe5SiB2. The measured MAE is
0.25 MJ/m3 at 300 K. Additionally, Paper VII contains a comprehensive com-
putational study of the magnetic properties of Fe5Si1−xPxB2 for all x, with the
79
alloy treated within the VCA. The calculations indicate negative (in-plane)
magnetisation for x = 0 (Si) and positive (uniaxial) magnetisation for x = 1
(P), with a spin reorientation transition at x≈ 0.75. This is consistent with the
experimental data from Ref. [227], which reported a positive MAE for Fe5PB2
and Ref. [97], which reported in-plane magnetisation for Fe5SiB2 at low tem-
peratures. Paper VII also considers hypothetical S substitutions in Fe5PB2
and Co substitutions in Fe5SiB2. However, since neither Fe5SB2 nor Co5SiB2
seem to appear in the relevant phase diagrams, one might anticipate difficulties
in experimentally achieving such substitutions in the correct crystal structure.
On the other hand, Co5PB2 has been reported to exist [228], isostructural to
Fe5PB2, whereby Co substitutions in Fe5PB2 could be more interesting to con-
sider for future work. Calculations using SPR-KKR led to the conclusion that
the ASA is insufficient to describe the magnetism of these compounds and
yields incorrect magnetic moments in comparison to FP calculations.
In Paper X, the magnetic properties MS, MAE and TC, are experimentally
studied for all x in Fe5Si1−xPxB2. Additionally, exchange interactions are
computed for x = 0 and x = 1, using the scalar relativistic full potential mode
in the SPR-KKR method. In the exchange interactions one can see a decrease
in the important short range Fe-Fe interactions when going from x= 0 to x= 1,
which explains the higher TC of the compound containing Si. Using MC calcu-
lations, the Curie temperatures were also computed in satisfactory agreement
with experimental measurements.
In Paper VII, as well as Ref. [229], a peculiar behaviour is observed in
the magnetisation as function of temperature for Fe5SiB2, with a peak in the
magnetisation at approximately 172 K. So far this has mainly been discussed
in terms of a spin reorientation transition (SRT). Such a transition was sug-
gested to occur already by Ericsson et al. [96] and it was recently supported
by neutron diffraction experiments [97]. Interestingly, other materials which
are known to exhibit spin reorientation transitions, e.g. Fe2B discussed in
Sec. 4.1.1 and Paper VI, do not show similar anomalous M(T ) behaviour,
casting doubt on whether the magnetism of Fe5SiB2 is completely under-
stood. Furthermore, Ericsson et al. mentioned that their Mössbauer data in-
dicated that the low temperature magnetic structure of Fe5SiB2 might not be
collinear, whereby the idea that there is more happening than a SRT between
two collinear magnetic states around 172 K, seems reasonable.
Fig. 4.8 shows the results of MC simulations for Fe5SiB2, with the ex-
change interactions presented in Paper X. The calculations were performed for
a system of 10× 10× 10 unit cells (containing a total of 104 Fe atoms), con-
sidering interactions between atoms separated by distances up to 3a ≈ 17 Å,
where a is the in-plane lattice parameter. The calculations show a decrease in
magnetisation at low temperatures, as well as a peak in the susceptibility at
approximately 100 K, not far from the experimentally observed magnetisation
anomaly. Analysis of the static spin correlation function in reciprocal space
80
T (K)
0 200 400 600 800 1000
m(T
) (μ
B)
0
1
2
χ (
arb
. u.)
0
2
4
Figure 4.8. Average magnetisation and susceptibility as functions of temperature in
Fe5SiB2, from MC calculations.
Sk(q) =1√
2πN ∑r,r′
eiq·(r−r′)(⟨
mk(r)mk(r′)⟩−⟨
mk(r)⟩⟨
mk(r′)⟩)
, (4.1)
where q is a point in the Brillouin zone and k denotes a component of the three
dimensional vectors, reveals clear peaks for q = ± 110a(1,1,− 1
c/a). This indi-
cates the appearance of a spin spiral in the crystallographic (1,1,-1) direction,
with a wavelength of 10|(a,a,c)|. A possible explanation for the magneti-
sation anomaly in this material could therefore be a change in the magnetic
structure, from a collinear ferromagnetic state at high temperatures to a non-
collinear spin spiral at low temperatures.
4.1.5 Fe2Ta1−xWx
FePt in the L10 structure has received much attention for its large uniaxial
MAE [77, 78, 82, 205, 206]. The first crucial property allowing for this large
MAE is the tetragonal crystal structure. Next, the strong SOC of Pt and a sig-
nificant Fe-Pt hybridisation at the Fermi energy [82] is essential for the large
MAE. Unfortunately, Pt is a precious metal and FePt would not be viable in
large scale permanent magnet applications. Clearly, it should be of interest
to consider other 3d-5d compounds in uniaxial crystal structures to find simi-
lar properties. Preferably they should contain a large amount of magnetic 3d
elements in order to have a large saturation magnetisation and Curie temper-
ature. Additionally, considering the price and availability of the various 5d
elements [20], W should be of particular interest.
Based on the above reasoning, Fe2W, which crystallises in the hexagonal
Laves phase [230], should be of particular interest. Moreover, Fe2Ta crys-
tallises in the same structure [231] and it thus appears realistic to also con-
sider the alloy Fe2WxTa1−x. Until now, limited magnetic characterisation of
the compounds Fe2W and Fe2Ta is available, except recent studies on Fe2W
81
nanoparticles [232]. It is therefore motivated to investigate these compounds
further.
Paper XI contains a detailed computational study of the magnetic proper-
ties of Fe2WxTa1−x, using the FP-LAPW method in the GGA. Some focus is
put on the MAE, which is carefully analysed in terms of the electronic struc-
ture. Both end compounds are found to have a positive MAE, with values of
0.87 MJ/m3 and 1.25 MJ/m3 for Fe2W and Fe2Ta respectively. VCA calcu-
lations show that the MAE of the alloy is smaller than that for the end com-
pounds for all x. It should be noted that the MAEs of the end compounds are
much smaller than that of 6.6 MJ/m3 found in FePt [26]. Partly this might be
explained by a disadvantageous band structure around the Fermi energy, but
it is also found that there is a limited Fe-5d hybridisation at the Fermi energy,
whereby the MAE cannot benefit strongly from the strong 5d SOC. Never-
theless, a reciprocal space analysis of the MAE reveals that regions in the
Brillouin zone with stronger Fe-5d hybridisation provide the by far strongest
MAE contributions.
Fe2W was found to be a ferrimagnet, in contrast to a recent computational
study using pseudopotential DFT calculations in the GGA [233], which also
predicted a negative MAE in Fe2Ta. In the case of Fe2W the most proba-
ble reason for the discrepancy is thought to be that the authors of Ref. [233]
assumed a ferromagnetic ordering and that their calculation converged into a
metastable local energy minimum. For Fe2Ta, the discrepancy might be due to
insufficiencies of their computational methods in accurately describing effects
of the SOC. This is particularly important for the MAE which is a delicate
property requiring the most accurate methods to be well described in compu-
tational work.
4.1.6 Other Interesting Hard Magnetic Materials
Various other materials have been discussed as potential candidates for re-
placement permanent magnets without rare earths or other critical elements.
One candidate which has received interest because of its huge MAE is Fe2P [71,
234], but unfortunately it has rather low TC, which might, however, be possible
to raise by alloying with various other elements [235–239]. Another material
which has been studied as a potential permanent magnet is MnBi [91, 92, 240]
with the anomalous MAE which increases strongly with temperature [89], re-
sulting in a strongly uniaxial MAE at room temperature. There are plenty
of other materials which could also be potentially interesting in this context.
Any compound which contains a lot of magnetic 3d elements (i.e. Mn, Fe,
Co or Ni), combined with other non-critical elements, in a uniaxial crystal
should be of interest. If one considers ternary or even quaternary such phases,
a huge number of potential materials should be available, many of which one
will find have not been properly characterised in terms of magnetic properties.
82
Furthermore, the properties should also be possible to tune by considering, for
example, alloying. If the 3d elements could be combined with 5d elements,
such as W, this should be of particular interest. A few examples of potentially
interesting materials are discussed below.
The Heusler compounds make up a huge class of materials with varying
and tunable properties, due to which they have been discussed related to var-
ious different applications [241, 242]. Many of these materials are cubic but
some are tetragonal [243, 244] and could therefore have a large MAE, whereby
they have been discussed as possible hard magnetic materials. For example,
Mn3Ga has a strong magnetic anisotropy [245, 246]. Unfortunately, it has a
ferrimagnetic ordering with very small saturation magnetisation, which makes
it unsuitable for permanent magnet applications. So far it seems to be a general
trend that most tetragonal Heuslers are Mn-based compounds with low mag-
netisation, which has hindered the usefulness of these materials as permanent
magnets.
Fe3P is a tetragonal ferromagnet [247], but in comparison to Fe2P it has
a high Cure temperature of 700 K. The MAE is, however, strongly nega-
tive [248]. It could be interesting to alloy this compound with other elements
to modify the MAE. Indeed, alloying with Co or Mn has been attempted [248]
and in both cases the TC and Ms are found to decrease while a positive MAE is
not obtained. Another element which could potentially be interesting to con-
sider alloying with would be Ni, since Ni3P forms the same tetragonal crystal
structure [249].
Mn3B4 is an orthorhombic ferrimagnet with low magnetisation and order-
ing temperature of 392 K [250], seemingly uninteresting as permanent mag-
net. However, the alloy (Mn1−xWx)3B4. has been studied [251, 252] and it has
been shown that W alloying induces a ferromagnetic ordering, which is likely
to be connected to a lattice expansion. This also results in an increase in the
Curie temperature to values well above 500 K and, with Ta alloying instead of
W, Curie temperatures above 700 K are observed [251]. Ferromagnetic order-
ing leads to a larger magnetisation and with the increased TC, this could make
this material appealing as a permanent magnet. Additionally, W doping could
be beneficial in achieving a large MAE and this material should be of interest
for further study.
4.2 Magnetic Effects in Elastic Electron Scattering
In Papers XIV-XV, the computational methods developed in Sec. 3.3 together
with a description of a magnetic solid according to the discussion in Sec. 2.3,
are used to study the effect of magnetism on the elastic scattering of fast elec-
trons in a magnetic solid. Three types of probes, relevant in the context of
transmission electron microscopy, are considered. These are electron vortex
83
beams, spin polarised beams and phase aberrated electron beams, which are
each discussed in the three coming sections, respectively.
4.2.1 Electron Vortex Beams
The concept of an electron vortex beam was already introduced in Sec. 2.1,
where Eq. 2.4 essentially describes a relativistic electron vortex beam as the
cylindrically symmetric solution to the free electron Dirac equation. In the
non-relativistic theory, the equivalence is obtained by solving the cylindrical
free electron Schrödinger equation
−h2
2m∇2ψ(r,φ ,z) = Eψ(r,φ ,z), (4.2)
which has energy eigenstates
ψl,k⊥,kz(r,φ ,z) = Nl,kJl(k⊥r)eikzzeilφ (4.3)
with energy
E =h2(k2
z + k2⊥)
2m=
h2k2
2m, (4.4)
a normalisation constant Nl,k and Jl(x) denoting a Bessel function. The inte-
ger l is an orbital angular momentum quantum number while k⊥ and kz are
continuous quantum numbers related to the linear momentum in radial and z-
directions. Clearly ψl,k⊥,kz has a well-defined orbital angular momentum of lhalong the z-axis since
Lzψl,k⊥,kz =−ih∂
∂φψl,k⊥,kz = lhψl,k⊥,kz . (4.5)
An EVB with a general radial shape can be constructed as a superposition of
beams with different k⊥ to obtain
ψl,kz(r,φ ,z) = Neikzzeilφ f (r) (4.6)
with a radial shape function f (r). Such a state is still an eigenstate of Lzwith OAM of lh. In the computational work presented in Papers XIV-XV the
EVBs are created as uniform disks in reciprocal space with an additional phase
winding eilφk so
ψl(k⊥,φk) ∝ eilφk Θ(qmax− k⊥), (4.7)
where Θ(x) is a step function and (k⊥,φk) are reciprocal space polar coordi-
nates. It is straightforward to show that an inverse Fourier transform of Eq. 4.7
leads to a real space wavefunction of the form in Eq. 4.6 1. qmax is the max-
imum value of k⊥ and defines the convergence (semi) angle α of the beam
1If one wishes to do so it is convenient to use k · r = k⊥r(cosφk cosφ + sinφk sinφ) =k⊥r cos(φk − φ) and
∫ a+2πa ei(xcosθ+mθ) dθ = 2πimJm(x) [253] when computing the Fourier
transforms in polar coordinates.
84
by α ≈ tanα = qmax/kz. Example EVBs are shown in Fig. 4.9. With other
parameters fixed, the spatial dimensions of the beams increase with l.
1 nm1 nm1 nm1 nm1 nm1 nm1 nm1 nm1 nm1 nm
l=3 l=5
1 nm1 nm1 nm1 nm1 nm1 nm1 nm1 nm1 nm1 nm
l=10
1 nm1 nm1 nm1 nm1 nm1 nm1 nm1 nm1 nm
- 1 nm
1 nm1 nm1 nm1 nm1 nm
l=-1 l=0 l=1 l=2
Figure 4.9. Electron vortex beams with various l, convergence angle of 10 mrad and
kinetic energy of 200 keV. The phase of the wavefunction is indicated by the hue and
the logarithm of the modulus by the brightness. Note that there is one scale bar for the
top row and one for the bottom row.
As was discussed by Bliokh et al. [56] and in Sec. 2.1, the non-relativistic
case in Eq. 4.3 differs from the relativistic state in Eq. 2.4 mainly by an intrin-
sic spin-orbit coupling affecting the lower half of Eq. 2.4. This spin-orbit in-
teraction is neglected in the two-component formalism developed in Sec. 3.3.1
and used in this work. It should be of order√1− m0c2
Erelsinθ (4.8)
with relativistic energy Erel and the angle θ determined by the radial and z-
components of the wave vector according to Eq. 2.6. In a TEM the electron
energy is typically smaller than 2m0c2 and the maximum value of θ is deter-
mined by the convergence angle which is often around 30 mrad or smaller. The
intrinsic spin-orbit interaction of the EVBs is thus of order 2% and increases
with energy and convergence angle. In a first approximation it appears reason-
able to neglect this. However, in future work it would certainly be interesting
to extend the methods used here and perform fully relativistic calculations to
investigate possibilities for such relativistic phenomena to be observed in ex-
periments.
The question to address now is how a magnetic signal can be acquired from
the elastic scattering of electron vortex beams. A mirror operation in a plane
containing the propagation axis maps a +l EVB to a −l EVB. If the beam has
85
Figure 4.10. The intensity of the diffrac-
tion pattern is integrated over an annular
region in the diffraction plane I(q1,q2) =∫D |ψout(k)|2 dkx dky, where D is the an-
nular region D = {kx,ky;q1 <√
k2x + k2
y <
q2}. A magnetic signal M is ob-
tained by taking a difference for opposite
OAM beams, Ml(q1,q2) = I+l(q1,q2) −I−l(q1,q2).
scattered through an electrostatic potential which is symmetric under such a
mirror symmetry, the diffraction pattern for the opposite OAM beams should
differ by the mirror operation only. The intensity in the diffraction pattern in-
tegrated over an annular region with inner radius q1 and outer radius q2 (see
Fig. 4.10), or disk shaped region (q1 = 0) should thus be identical for the op-
posite sign OAM vortex beams. This symmetry is broken by the inclusion of
magnetic fields parallel to the propagation (z) direction, since these change
sign under mirror operations in mirror planes parallel to the z-axis. Therefore,
a magnetic signal is obtained by taking the difference in intensity integrated
over a disk shaped or annular region in the diffraction plane for the oppo-
site OAM beams. An example of signals obtained in such manner is shown in
Fig. 4.11. Various test calculations were performed with all magnetic fields set
to zero and, as expected, the magnetic signal was reduced to numerical noise
many orders of magnitude smaller than that seen in Fig. 4.11. For atomic res-
olution STEM imaging it becomes necessary to further consider the crystal
symmetries and compare the intensity collected for one beam position with
that collected at a beam position obtained by applying a mirror symmetry op-
eration of the crystal lattice to the first position. Some of the symmetry aspects
of using EVBs to measure magnetic signals in atomic resolution STEM have
previously been discussed in the context of EMCD [254].
In Paper XIV the elastic scattering of EVBs through bcc Fe is studied with
input attained according to the scheme described in Sec. 2.3 and using the
computational methods from Sec. 3.3.1. As was seen in Sec. 2.3, the magnetic
fields in a solid can be considered as a homogeneous field corresponding to
the saturation magnetisation and another microscopically varying field which
averages to zero over a unit cell. This results in different forms of magnetic
interactions for atomic sized beams or larger beams. For larger sized beams
the magnetic signal obtained as an intensity difference for opposite sign OAM
beams is found to increase approximately linearly with the magnitude of the
86
θ (mrad)
0 50 100
Inte
nsity
-4
-3
-2
-1
0
1
I1(θ)
I-1
(θ)
M1(θ)×10
6
I20
(θ)
I-20
(θ)
M20
(θ)×106/20
(a)
(mrad)0 25 50
t(u
.c.)
10
20
30
40 Mag
netic
sign
al10
5
-10
-5
0
5
10
(b)
Figure 4.11. a) Shows the intensity I±l(θ)= I±l(q1 = 0,q2), over a disk shaped region,
where θ ≈ q2/kz and the magnetic signal is Ml(θ) = Il(θ)− I−l(θ). The data shown is
for beams with OAM ±1, kinetic energy 300 keV and convergence angle of 40 mrad
and beams with OAM±20, kinetic energy 200 keV and convergence angle of 10 mrad,
after passing through 40 unit cells (11.48 nm) of bcc Fe. Such signals are strongly
dependent on thickness and b) shows the magnetic signal for the l = ±20 case as
function of both thickness and collection angle θ , which is how much of the data in
Paper XV is presented.
OAM, in accordance with Eq. 1.4. For large sized beams with initial OAM
of around 30h a relative magnetic signal close to 10−3 is predicted. Since
the sizes of the vortex beams increase with l (see Fig. 4.9), atomic resolution
beams can mainly be achieved with small OAM due to technological restric-
tions. Nevertheless, since the microscopic magnetic fields are of greater mag-
nitude than that corresponding to the saturation magnetisation (more than 40 T
compared to 2.2 T in bcc Fe) one might still expect a significant magnetic sig-
nal in the atomic resolution. However, atomic resolution STEM simulations in
Paper XIV indicate a rather weak magnetic signal in the order of 10−5, which
is expected to be difficult to measure in experiments.
In Paper XV, a comprehensive computational study on L10 FePt is per-
formed to investigate the effect of different beam parameters, including l, ac-
celeration voltage and convergence angle, on the magnetic signal. Results
indicate that large l, small acceleration voltages and small convergence an-
gles are beneficial to obtain stronger magnetic signal. This is, unfortunately,
discouraging for the possibility of atomic resolution imaging since the require-
ments on all three beam parameters also lead to larger sized electron beams.
Nevertheless, realistic parameter values are presented which increase the mag-
netic signal strengths with about an order of magnitude in comparison to the
results presented in Paper XIV. The antiferromagnet LaMnAsO is also con-
sidered and, in agreement with expectations, it is found that a magnetic signal
87
is only seen in atomic resolution imaging for antiferromagnets, since the sat-
uration magnetisation is zero. The effect of statistical and mechanical noise
is also considered and further support the result that a magnetic signal should
be obtainable according to the suggested methods by using nanometer sized
beams with large OAM, while atomic resolution measurements are deemed
challenging in experiments with current technology.
4.2.2 Spin Polarised Scattering
It was pointed out already by Bohr that an electron beam can not be split into
its spin up and spin down part using a Stern-Gerlach setup. The arguments are
based on the uncertainty principle and reviewed by Mott [255]. Such possibili-
ties have been revisited and discussed more recently [256, 257]. Nevertheless,
it remains challenging to produce electron beams with a significant degree of
spin polarisation. However, there has been recent progress in producing spin
polarised electron beams using spin polarised electron sources [46]. Other
schemes have also been discussed [258]. This makes it highly relevant to in-
vestigate the scattering of spin polarised electron beams in magnetic matter as
is done in Papers XIV-XV.
From Eq. 1.4 it is evident that, at least for large sized beams (larger than
atomic distances), a similar magnetic interaction is expected for spin polarised
beams as with electron vortex beams, that is as an intensity difference for op-
posite angular momentum beams. Indeed in Paper XIV it is found that a mag-
netic signal is obtained in the same manner with spin polarised beams as with
electron vortex beams and that the magnitude is similar as that from small
OAM vortex beams. For atomic resolution STEM imaging there is the sim-
plification that the magnetic signal can be obtained by comparing intensities
collected for the same beam position, without considering mirror points.
In Eq. 3.47 it was seen that only the components of the magnetic flux den-
sity which is in the xy-plane causes spin-flip scattering. One therefore expects
stronger spin flip effects for magnetisations perpendicular to the spin quanti-
sation axis. Hence, in Paper XV simulations were performed for FePt with
magnetisation perpendicular to the propagation direction while the spin polar-
isation is parallel to the propagation direction. It is found that this increases
spin scattering effects by several orders of magnitude, but still only about one
in 108 electrons from an initially spin up polarised beams are found in a spin
down state after passing through 80 nm of FePt. This corresponds to a rotation
of the expectation value of the spin operator by about 0.1 mrad.
4.2.3 Aberrated Probes
As mentioned previously, a new path towards measuring magnetism in the
(scanning) transmission electron microscopes was recently introduced with
88
the idea that electron beams with phase aberrations can interact with mag-
netism [47, 48]. This was recently experimentally corroborated in EELS ex-
periments [49], but has not yet been discussed in the context of elastic scatter-
ing. Hence, this is done here and in Paper XV.
As in the case of electromagnetic optics, electron optics suffer from aberra-
tion effects which restrict the resolution achievable to a level below that of the
diffraction limit [259], which is less than 0.1 Å for electrons with the energies
that are relevant in a (S)TEM. In attempts to reach an ever increasing reso-
lution, aberration correction techniques have been developed and eventually
allowed for Å-resolution imaging [260, 261] and beyond [36]. The develop-
ments in aberration correction technology also mean that the aberration of a
beam can be tuned towards other configurations than to vanish [49].
Aberrations introduce an additional phase factor with an aberration function
χ(k) to the wavefunction
ψ(k)→ ψ(k)eiχ(k). (4.9)
The aberration function may be written [50]
χ(k⊥,φk) =2πλ ∑
n,m
θ n+1
n+1
[Ca
n,m cos(mφk)+Cbn,m sin(mφk)
], (4.10)
where Can,m and Cb
n,m are aberration coefficients, related to each other by π2m
rotations. The non-negative integer n denotes the order of the aberration while
the non-negative integer m = n+ 1,n− 1,n− 3, ... denotes the order of the
rotational symmetry of the aberration. k⊥ = kz tanθ is the radial distance and
φk = arctan(ky/kx) the azimuthal angle in the diffraction plane.
Although modifying the phase distribution, the aberrations do not result in
providing the electron beam with OAM. Therefore, it is not expected that the
aberrated electron beams are useful to image magnetism at the nanoscale based
on an interaction such as that seen in Eq. 1.4. However, in the atomic resolu-
tion limit, the phase gradients can still couple locally to the magnetic fields,
e.g. via the Axy ·∇xy term in Eq. 3.47. As discussed in Paper XV, one then
expects that a magnetic signal can be obtained in the same way as it was for
the EVBs, by comparing the intensity integrated over an annular disk for op-
posite sign Cin,m (i = a, b), if certain symmetry requirements are fulfilled. The
first symmetry requirement is, similarly as for EVBs, that a mirror symmetry
operation of the crystal maps the +Cin,m term to a −Ci
n,m term. This was au-
tomatically fulfilled for EVBs as long as a mirror symmetry operation existed
for the crystal. In the case of aberrations, the aberration term must be chosen
to fit a symmetry depending on the crystal. The next requirement is that there
is not a rotational symmetry operation of the crystal which maps the +Cin,m
term to−Cin,m. Rotations about the z-axis leave magnetic fields along this axis
invariant, so if such symmetry exists, the diffraction patterns for±Cin,m should
89
only differ by such a rotation also with magnetic effects included. The inten-
sity over an annular region is then identical. These symmetry requirements for
the case of tetragonal FePt, which is studied in Paper XV, are summarised in
Fig. 4.12.
C12b -C12
b
C34b -C34
b
Figure 4.12. The tetragonal FePt crystal possesses a C4v symmetry, which includes
multiples of π2 rotations and four mirror symmetries, including mirroring in the y = x
axis. ±Cb1,2 aberrations differ by a π
2 rotation which leaves both electrostatic potential
and magnetic fields invariant. The intensity integrated over an annular disk is thus
identical, since the diffraction pattern should only be rotated for the two aberrations.
In the case of Cb3,4, the aberration is invariant under π
2 rotation but changes sign un-
der the mirroring in the y = x axis. The electrostatic potential is invariant under the
mirror operation but magnetism changes sign, whereby a non-zero magnetic signal is
obtained by comparing the intensity over an annular region in the diffraction plane for
±Cb3,4.
In Paper XV, simulations are performed for a beam with aberration Cb3,4 =
±14 μm, kinetic energy of 100 keV and convergence angle of 30 mrad. Indeed
a non-zero magnetic signal is observed and it is of similar size as that found
for small OAM vortex beams. Hence, it is most likely challenging to observe
in experimental setups using current technology, but might become realistic
with further developments.
90
5. Summary and Conclusions
The focus of this thesis has been on theoretical and computational modelling
of the physics of magnetism and magnetic materials. In particular, two top-
ics have been considered; first principles electronic structure theory has been
used to study the intrinsic magnetic properties of saturation magnetisation,
magnetocrystalline anisotropy and Curie temperature in transition metal based
magnets, with focus on the magnetocrystalline anisotropy, since it is critical
for novel permanent magnet materials. Additionally, magnetic effects on the
elastic scattering of high energy electrons have been studied in the context of
transmission electron microscopy. Relevant theory was introduced in Sec. 2
and the computational methods used were discussed in Sec. 3. Finally, an
overview of the results was provided in Sec 4.
As was introduced in Sec. 1.1, the challenge in obtaining permanent mag-
net replacement materials, without rare-earth or other critical elements, is to
obtain a large MAE in transition metal based magnets. Furthermore, the MAE
is one of the most important intrinsic magnetic properties, which is crucial in a
variety of applications [262], making the results interesting in a wider context.
The theory of MAE in transition metal magnets was reviewed in Sec. 2.1.2.
Based on this theory, it was concluded that a large MAE in the relevant mate-
rials, with weak SOC, first of all requires a non-cubic crystal structure. Next it
was found that the MAE depends sensitively on the electronic band structure
around the Fermi energy. The MAE can, therefore, be tuned by modifying the
band structure at the Fermi energy, which can in practice can be done, for ex-
ample, by alloying or strain engineering. These ideas have been explored by
using electronic structure calculations based on DFT (discussed in Sec. 3.1)
for a number of non-cubic 3d-based magnets, as overviewed in Sec. 4.1. A
pictorial overview is given in Fig. 5.1, showing the MAE and MS of various
materials studied in this thesis, together with some of the well known perma-
nent magnet materials, such as Nd2Fe14B and the ferrite BaFe12O19. The line
corresponding to the hardness parameter κ = 1 is also shown since it has been
suggested that κ > 1 is required for a useful permanent magnet [25, 26]. Out
of the materials studied in this thesis (see Sec. 4.1), Fe2W, Fe2Ta, MnAl and
MnGa are on the useful side of the κ > 1 line. From these, Fe2W has a very
small saturation magnetisation, smaller than that of BaFe12O19, due to the fer-
rimagnetic ordering. This would limit the energy product of a magnet from
this material and restrict its usefulness.
Another path towards new, high MAE, permanent magnets from transition
metals, which was investigated here, is to combine 3d-based compounds with
91
0M
s(T)
0 1 2
MA
E(M
J/m
3)
0
2
4
6
8
Nd2Fe
14B
FePt
BaFe12
O19 (Fe
0.4Co
0.6)0.96
B0.04
FeNiMnAl
MnGa
(Fe0.7
Co0.3
)2B
Fe2TaFe
2W
=1
Re
Figure 5.1. Overview of saturation magnetisation and MAE for various materials
interesting in the context of permanent magnet applications. A similar illustration
including a different set of materials was given by Hirosawa [25].
small amounts of 5d elements (Paper VI and Sec. 4.1.2-4.1.3). As can be
seen in Fig. 5.1, Re doping brings (Fe0.3Co0.7)2B closer to the κ = 1 line, but
not quite over to the desired side. However, the result that calculations and
experiments agree in that the MAE can be increased significantly by a few
atomic percent of 5d impurities is still an important one. As seen in Sec 4.1.3,
calculations indicate that the same result can be achieved in other systems.
This allows for the possibility of modifying a uniaxial 3d-based magnet with
an MAE which is somewhat too small, relative to its saturation magnetisation,
to obtain κ > 1.
To conclude this part of the work, it is perfectly realistic to have transition
metal based magnets, without any heavy elements, with MAE above 2 MJ/m3
(e.g. Fe2P, MnAl and MnGa). There is no apparent physical reason that this
cannot be combined with a large magnetisation and Curie temperature. If one
requires κ =√
Kμ0M2
S
> 1, an MAE of 2 MJ/m3 allows for a saturation mag-
netisation above 1.5 T, nearly has high as that of Nd2Fe14B. Hence, there
is no physical limitation hindering us from finding a 3d-based magnet with
properties comparable to Nd2Fe14B. Furthermore, by combining it with small
amounts of heavier elements, e.g. W, an even larger MAE, allowing for a
larger MS, should be realistic. It is then a question of finding the right mate-
rial. For this task, computational methods such as those used in this thesis,
should be of great value, since they allow one to evaluate each of the relevant
intrinsic material properties with relative ease compared to experiments. One
92
way of doing this is to calculate properties of carefully selected, known phases
and investigate how these can be tuned and modified by, for example, alloy-
ing. This is essentially the path which has been explored in this thesis. Another
way is to perform data mining [263, 264], possibly even combined with crys-
tal structure prediction [265]. Such methods have recently been applied to the
search for permanent magnet materials [266, 267]. However, these methods
must be used with care, especially when considering delicate properties, such
as the MAE, which require careful calculations with the most accurate compu-
tational methods. Otherwise there is significant risk of producing misleading
results.
In the other part of the work in this thesis, a realistic model to describe the
scattering of fast electrons in a magnetic material has been described and im-
plemented in computational methods. This model has been used to study the
effects of magnetism on electron vortex beams, spin polarised electron beams
and electron beams with phase aberrations. In each case, a magnetic signal
was found by considering an intensity difference for beams with opposite an-
gular momentum polarisation or aberration coefficients. Only in the case of
large orbital angular momentum beams, with spatial dimensions of one or a
few nanometers, the signal was considered strong enough to be feasible to
be observed in experiments with current technology, after taking into account
various noise sources. These results potentially allow for a new technique
of imaging magnetic materials in scanning transmission electron microscopy
with nanometer spatial resolution and call for experimental confirmation. As
a next step in this work, it should be interesting to computationally explore
how various magnetic structures in the nano regime, such as domain walls,
skyrmions or magnetic interfaces, can be probed by this method.
Electron beams in transmission electron microscopy typically have a ki-
netic energy of hundreds of kiloelectronvolt, resulting in highly relativistic
electrons, often travelling at more than half the speed of light. An interesting
development of the work in Papers XIV-XV would therefore be to extend the
methods discussed in Sec. 3.3 to the fully relativistic case of solving the four
component Dirac equation. It has been suggested that the relativistic mass cor-
rection, commonly applied in TEM simulations, is sufficient for an accurate
description of the relevant scattering processes [186]. Nevertheless, a descrip-
tion based on the Dirac equation would allow one to explore novel phenomena,
which only appear in a more complete treatment of relativity, for example, the
intrinsic spin-orbit interaction of electron vortex beams mentioned in Sec. 2.1.
93
6. Sammanfattning
Studier av magnetism och magnetiska material har resulterat i teknologiska
upptäckter som ett flertal gånger omvälvat människans samhälle och sätt att
leva. Först genom upptäckten av kompassen för cirka ett millennium sedan,
vilken möjliggjorde resor över planeten. Senare genom utvecklingen av mag-
neter som tillåter konvertering mellan elektrisk och mekanisk energi, samt ef-
fektiv energiöverföring med högspänningsteknik. Även mer nyligen genom
att tillåta lagring av enorma informationsmängder i magnetiska hårddiskar1.
Det fasta tillståndets magnetism har sitt ursprung i elektronernas spinn (ett in-
trinsiskt magnetiskt moment) och banrörelsemängdsmoment (ett magnetisk
moment relaterat till elektronernas rörelse). Magnetisk ordning och andra
fenomen uppstår genom intra- och interatomär elektronväxelverkan. Förståelse
av magnetismen kräver därmed en kvantmekanisk beskrivning av komplicer-
ade system med stora antal elektroner. Kraftfulla verktyg för att studera den
typen av system har gjorts tillgänglig med utvecklingen av avancerad kvant-
mekanisk beräkningsmetodik, till exempel täthetsfunktionalteori [8]. Med
hjälp av kraftulla moderna datorer tillåter dessa metoder noggranna beräkningar
av materialegenskaper, även i komplexa material, såsom legeringar, heterostruk-
turer och liknande. Syftet med denna avhandling är att tillämpa teoretisk mod-
ellering och storskaliga datorberäkningar, ofta baserade på täthetsfunktional-
teori, för att studera två utvalda ämnen inom magnetism, relevanta för modern
teknologi.
Det första ämnet som behandlas är den magnetiska anisotropin, det vill säga
den fria energins beroende av magnetiseringsriktningen, i övergångsmetalls-
baserade magneter. Magnetisk anisotropi har sitt ursprung i den relativistiska
spinn-ban-kopplingen, vilken sammanbinder spinn och banrörelsemängdsmo-
ment hos elektroner. Den magnetiska anisotropin är en av de viktigaste intrin-
siska magnetiska egenskaperna, nödvändig i ett flertal teknologiska tillämp-
ningar, bland annat i permanentmagneter. Permanentmagneter används till
exempel för att konvertera mellan elektrisk och mekanisk energi i motorer och
generatorer. En bra permanentmagnet kräver en stark magnetisk anisotropi.
De permanentmagneter som idag kan lagra störst energitäthet innehåller säll-
synta jordartsmetaller, vilka bidrar med stark spinn-ban-koppling. Det är ön-
skvärt att uppnå samma egenskaper som dessa material har utan de sällsynta
jortartsmetallerna [18, 268], en uppgift som visat sig vara utmanande. Ett av
1Uppskattningsvis översteg världens totala informationslagring 1020 byte redan för ett dece-
nium sedan och mängden fördubblas på lite mer än ett år [2]. Under senare år har huvuddelen
av denna information lagrats på magnetiska hårddiskar.
94
målen i denna avhandling är därför att tillämpa elektronstrukturteori och tä-
thetsfunktionalteoretiska beräkningar för att finna sätt att öka den magnetiska
anisotropin i övergångsmetallsbaserade magneter. Tre sätt att uppnå detta
poängteras, nämligen att endast betrakta icke-kubiska kristaller, att påverka
elektronstrukturen kring Fermienergin, samt att kombinera magnetiska 3d-
metaller med grundämnen från 5d-serien i det periodiska systemet, vilka har
en starkare spinn-ban-koppling. Att kontrollera elektronstrukturen kring Fer-
mienergin, och därmed den magnetiska anisotropin, är möjligt, till exem-
pel, genom legering eller genom att påverka kristallstrukturens gitterparame-
trar. I praktiken studeras därför flera Fe-Co-baserade legeringar, till exem-
pel (Fe1−xCox)2B, där den magnetiska anisotropin kan kontrolleras genom att
styra legeringskoncentrationen. Vidare påvisas att anisotropin kan ökas av-
sevärt genom att även legera med några (ca 2.5-5) atomprocent av 5d-atomer.
I rymdcentrerade kubiska legeringar av järn och kobolt uppnås, enligt både
beräkningar och experimentell data som presenteras i denna avhandling, en
starkt ökad magnetisk anisotropi genom att tillsätta små mängder interstitiella
B eller C-atomer. Dessa inducerar en tetragonalitet i den annars kubiska
kristallen, vilket leder till den ökade magnetiska anisotropin. Magnetiska
egenskaper hos kemiskt ordnade binära sammansättninger i den tetragonala
L10-strukturen studeras även, då dessa kan uppvisa stark magnetisk anisotropi.
Även hos den tetragonala legeringen Fe5Si1−xPxB2 och den hexagonala leg-
eringen Fe2Ta1−xWx genomförs studier av de magnetiska egenskaperna.
Då magnetiska fenomen normalt har ett atomärt eller subatomärt ursprung,
samt då ny teknik genomgår en kontinuerlig förminskning i storlek, är ex-
perimentalla metoder för att studera magnetism och magnetiska material med
högsta möjliga upplösning av avgörande betydelse. Detta leder oss till det
andra ämnet som behandlas i denna avhandling, nämligen magnetiska effek-
ter i elastisk elektronspridningsteori. Den nyligen utvecklade möjligheten
att producera elektronvirvelstrålar [38–40], det vill säga elektronstrålar med
ett banrörelsemängdsmoment vilket kan växelverka med magnetfält, gör det
synnerligen intressant att studera sådana effekter. Effekterna studeras un-
der förutsättningar som är releventa inom transmissionselektronmikroskopi,
där elektronvirvelstålarna potentiellt kan möjliggöra experimentella analysme-
toder av magnetiska material med mycket hög, möjligen atomär, upplösning.
En modell för att beskriva den relevanta typen av processer från Pauliekva-
tionen härleds, tillsammans med en realistisk beskrivning av magnetismen i
fasta tillståndet som utgår från täthetsfunktionalteoretiska beräkningar. Mod-
ellering görs sedan för elektronvirvelstrålar vilka passerar genom magnetiska
material och resultaten indikerar en möjlighet att uppmäta en magnetisk sig-
nal som en intensitetsskillnad i diffraktionsmönstret för virvelstrålar med mot-
satt banrörelsemängdsmoment, integrerat över en cirkulär skiva i diffraktion-
splanet. För virvelstrålar med en utbreddning på några nanometer och med
ett banrörelsemängdsmoment på ca 20-40 gånger Plancks konstant h uppnås
en relativ magnetisk signal av storleksordning 10−3, vilket bedöms möjligt att
95
uppmäta i realistiska experimentalla uppställningar. Med de mindre virvel-
strålar som krävs för att nå atomär upplösning är den magnetiska signalen en
storleksordning svagare och bedöms i nuläget vara bortom vad som kan upp-
mätas i experiment.
Nyligen har vissa framsteg uppnåtts kring att producera spinnpolariserade
elektronstålar [46], vilket öppnar för en ny väg till att observera magnetism i
transmissionselektronmikroskopi. En annan möjlighet som nyligen påvisats är
att utnyttja aberrationskorrektionsteknologi för att styra elektronstålarnas fas-
fördelningar och på så sätt uppnå en magnetisk växelverkan [49]. Dessa möj-
ligheter till att åstadkomma en magnetisk signal genom elastisk elektronsprid-
ning i transmissionselektronmikroskopet studerats också med de utvecklade
beräkningsmodellerna. Resultaten visar på möjligheten att uppmäta en mag-
netisk signal på liknande sätt som med elektronvirvelstrålarna, men i samtliga
fall är signalen runt en storleksordning svagare än den starkaste som uppnås
med virvelstrålar och förväntas därmed vara svår att uppmäta i experiment
med nu tillgänglig teknologi. Den mest intressanta tillämpningen av elastisk
elektronspridning i magnetiska material som studerats bedöms därmed vara i
studier av nanomagnetism, till exempel magnetiska domänväggar, gränsskikt
eller skyrmioner, vilket kräver ytterligare beräkningsstudier.
96
7. Acknowledgements
First of all, I thank Ján Rusz and Olle Eriksson for their outstanding super-
vision throughout my PhD studies and for giving me with the opportunity to
work with good people in an excellent research environment. I am grateful
for everything you have taught me and for always being willing to guide or
support me when needed, while also trusting me to work more independently
and explore my own ideas.
I’m thankful to all my colleagues at the division for materials theory for
the good work environment, discussions and all the fika we had together. In
particular, I want to thank Pablo Maldonado for four enjoyable years as office
mates, friends and colleagues, and for always being willing to discuss any-
thing from physics to the latest stereotypes regarding Swedish people. I also
thank Iulia Brumboiu for spreading chocolate, kindness and happiness in the
work place. I’m grateful to Jonathan Chico for discussions and for helping me
use the UppASD code. I also want to express my appreciation to Konstantinos
Koumpouras for all the good times in Uppsala, Honolulu, Palermo and Coven-
try. Additionally, over the years a large number of people in this big division
have in some way contributed to making my time here more enjoyable and
rewarding, via scientific collaborations or discussions, lunchtime discussions
or good company during other times. I will not attempt to write an exhaustive
list, but I hope that those on it will know that I appreciated our time together.
I want to thank Mirosław Werwinski for good scientific collaborations,
for the enjoyable times spent together in Uppsala, Poznan, Rhodes, Dresden,
Madrid, Vienna and Annapolis, and for the help with reading my thesis and
finding seemingly invisible errors.
In addition to my colleagues at the division for materials theory, I have had
the opportunity to interact with many people at other parts of the Ångström
laboratory. I have had the pleasure to have many fruitful discussions and work
with others sharing the common interest of permanent magnet materials, in-
cluding Klas Gunnarsson, Peter Svedlindh, Johan Cedervall, Martin Sahlberg,
Ocean Fang, as well as Sofia Kontos, who also deserves a special thanks for
making the years in Uppsala more fun.
In the early days of my PhD studies, my work was part of the EU project
REFREEPERMAG. I would like to thank all members of this project for the
good collaborations and discussions. In particular, I want to thank Ludwig
Reichel for the fruitful collaborations and Prof. Dimitrios Niarchos for his
hard work in coordinating the project.
I thank Prof. Shunsuke Muto, as well as all members of his research group
at Nagoya University, for the kind hospitality during my stay in Japan in the
97
summer of 2015. I am also grateful to the Japan Society for the Promotion
of Science (JSPS) for funding my travel to Japan and for organising a very
enjoyable and memorable summer program.
During my PhD studies I also enjoyed the opportunity to visit many other
nice places around the world for scientific conferences, collaborations, meet-
ings, workshops or summer schools. I thank Anna Maria Lundins stipendie-
fond at Smålands nation, Liljevalchs stipendiefond and the Graduate School in
Advanced Materials for the 21st century (GradSAM21) for financial support.
Jag vill också tacka Anna Gál för allt hennes tålamod medan jag arbetat
med denna avhandling. Slutligen vill jag tacka min snälla farmor för att hon
brytt sig, oroat sig och hjälpt mig på olika sätt under min studietid.
98
References
[1] P. Mohn, Magnetism in the Solid State An Introduction (Springer-Verlag,
2002).
[2] M. Hilbert and P. López, Science 332, 60 (2011).
[3] J. H. Van Vleck, Rev. Mod. Phys. 50, 181 (1978).
[4] A. Fert, Rev. Mod. Phys. 80, 1517 (2008).
[5] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, Journal of Physics D:
Applied Physics 43, 264001 (2010).
[6] S. Krause and R. Wiesendanger, Nature Materials 15, 493 (2016).
[7] O. Gutfleisch, M. A. Willard, E. Brück, C. H. Chen, S. G. Sankar, and J. P.
Liu, Advanced materials 23, 821 (2011).
[8] W. Kohn, Rev. Mod. Phys. 71, 1253 (1999).
[9] J. Stöhr and H. C. Siegmann, Magnetism - From Fundamentals to NanoscaleDynamics (Springer, Berlin, 2006).
[10] R. M. Martin, Electronic Structure (Cambridge University Press, 2004).
[11] Top 500, Performance Development , (2015).
[12] Y. Yang, A. Walton, R. Sheridan, K. Güth, R. Gauß, O. Gutfleisch, M. Buchert,
B.-M. Steenari, T. Van Gerven, P. T. Jones, and K. Binnemans, Journal of
Sustainable Metallurgy , 1 (2016).
[13] R. A. Griffiths, A. Williams, C. Oakland, J. Roberts, A. Vijayaraghavan, and
T. Thomson, Journal of Physics D: Applied Physics 46, 503001 (2013).
[14] R. Yamamoto, A. Yuzawa, T. Shimada, Y. Ootera, Y. Kamata, N. Kihara, and
A. Kikitsu, Japanese Journal of Applied Physics 51, 046503 (2012).
[15] R. Skomski and J. M. D. Coey, Permanent Magnetism (Institute of Physics
Publishing, 1999).
[16] M. Sagawa, S. Fujimura, N. Togawa, H. Yamamoto, and Y. Matsuura, Journal
of Applied Physics 55, 2083 (1984).
[17] R. Skomski and J. M. D. Coey, Phys. Rev. B 48, 15812 (1993).
[18] D. Kramer, Physics Today 63, 22 (2010).
[19] N. Jones, Nature 472, 22 (2011).
[20] J. Coey, Scripta Materialia 67, 524 (2012).
[21] European Comission, Report on Critical Raw Materials for the EU (2014).
[22] U.S. Department of Energy, Critical Materials Strategy (2011).
[23] K. J. Strnat, in Ferromagnetic Materials, Vol. 4, edited by E. P. Wohlfarth and
K. H. J. Buschow (Amsterdam, North-Holland, 1988) p. 135.
[24] Sveriges Geologiska Undersökning (SGU), Metallprisutveckling 11 , (2015).
[25] S. Hirosawa, Journal of the Magnetics Society of Japan 39, 85 (2015).
[26] J. M. D. Coey, IEEE Transactions on Magnetics 47, 4671 (2011).
[27] R. Skomski, Simple Models of Magnetism (Oxford University Press, 2008).
[28] S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, 1997).
[29] B. T. Thole, P. Carra, F. Sette, and G. V. D. Laan, Phys. Rev. Lett. 68, 1943
(1992).
99
[30] P. Carra, B. T. Thole, M. Altarelli, and X. Wang, Phys. Rev. Lett. 70, 694
(1993).
[31] P. Bruno, Phys. Rev. B 39, 865 (1989).
[32] J. Stöhr, Journal of Magnetism and Magnetic Materials 200, 470 (1999).
[33] C. Hébert and P. Schattschneider, Ultramicroscopy 96, 463 (2003).
[34] P. Schattschneider, S. Rubino, C. Hébert, J. Rusz, J. Kunes, P. Novák,
E. Carlino, M. Fabrizioli, G. Panaccione, and G. Rossi, Nature 441, 486
(2006).
[35] J. Rusz, O. Eriksson, P. Novák, and P. M. Oppeneer, Phys. Rev. B 76, 060408
(2007).
[36] R. Erni, M. D. Rossell, C. Kisielowski, and U. Dahmen, Phys. Rev. Lett. 102,
096101 (2009).
[37] P. Schattschneider, M. Stöger-Pollach, S. Rubino, M. Sperl, C. Hurm,
J. Zweck, and J. Rusz, Phys. Rev. B 78, 104413 (2008).
[38] M. Uchida and A. Tonomura, Nature 464, 737 (2010).
[39] J. Verbeeck, H. Tian, and P. Schattschneider, Nature 467, 301 (2010).
[40] B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J.
McClelland, and J. Unguris, Science 331, 192 (2011).
[41] J. Rusz and S. Bhowmick, Phys. Rev. Lett. 111, 105504 (2013).
[42] J. Verbeeck, P. Schattschneider, S. Lazar, M. Stöger-Pollach, S. Löffler,
A. Steiger-Thirsfeld, and G. Van Tendeloo, Applied Physics Letters 99, 10
(2011).
[43] V. Grillo, G. C. Gazzadi, E. Mafakheri, S. Frabboni, E. Karimi, and R. W.
Boyd, Phys. Rev. Lett. 114, 1 (2015).
[44] K. Saitoh, Y. Hasegawa, N. Tanaka, and M. Uchida, Journal of electron
microscopy 61, 171 (2012).
[45] M. Babiker, J. Yuan, and V. E. Lembessis, Phys. Rev. A 91, 013806 (2015).
[46] M. Kuwahara, F. Ichihashi, S. Kusunoki, Y. Takeda, K. Saitoh, T. Ujihara,
H. Asano, T. Nakanishi, and N. Tanaka, Journal of Physics: Conference Series
371, 012004 (2012).
[47] J. Rusz, J.-C. Idrobo, and S. Bhowmick, Phys. Rev. Lett. 113, 145501 (2014).
[48] J. Rusz and J. C. Idrobo, Phys. Rev. B 93, 104420 (2016).
[49] J. C. Idrobo, J. Rusz, J. Spiegelberg, M. A. McGuire, C. T. Symons, R. R.
Vatsavai, C. Cantoni, and A. R. Lupini, Advanced Structural and Chemical
Imaging 2, 1 (2016).
[50] O. L. Krivanek, N. Dellby, and A. R. Lupini, Ultramicroscopy 78, 1 (1999).
[51] G. E. Uhlenbeck and S. Goudsmit, Die Naturwissenschaften 13, 953 (1925).
[52] P. A. M. Dirac, Proceedings of the Royal Society A 117, 610 (1928).
[53] P. A. M. Dirac, Proceedings of the Royal Society A 118, 351 (1928).
[54] F. Gross, Relativistic Quantum Mechanics and Field Theory (John Wiley &
Sons, Inc., 1993).
[55] P. Strange, Relativistic Quantum Mechanics (Cambride University Press,
1998).
[56] K. Y. Bliokh, M. R. Dennis, and F. Nori, Phys. Rev. Lett. 107, 174802 (2011).
[57] S. Gasiorowicz, Quantum Physics (John Wiley & Sons Inc., 2003).
[58] A. R. Mackintoch and O. K. Andersen, in Electrons at the Fermi surface,
edited by M. Springford (Cambridge University Press, 1980).
100
[59] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics ofContinuous Media (Pergamon Press, Oxford, England, 1984).
[60] J. H. V. Vleck, Phys. Rev. 52, 1178 (1937).
[61] I. G. Rau, S. Baumann, S. Rusponi, F. Donati, S. Stepanow, L. Gragnaniello,
J. Dreiser, C. Piamonteze, F. Nolting, S. Gangopadhyay, O. R. Albertini, R. M.
Macfarlane, C. P. Lutz, B. A. Jones, P. Gambardella, A. J. Heinrich, and
H. Brune, Science 344, 988 (2014).
[62] H. Brooks, Phys. Rev. 58 (1940).
[63] E. I. Kondorskii and E. Straube, Journal of Experimental and Theoretical
Physics 63, 188 (1973).
[64] dos Santos Dias, M., “Topological orbital magnetic moments,” (2016),
EU-Japan Workshop on Computational Materials Design.
[65] G. Tatara and H. Kohno, Phys. Rev. B 67, 113316 (2003).
[66] M. Hoffmann, J. Weischenberg, B. Dupé, F. Freimuth, P. Ferriani,
Y. Mokrousov, and S. Heinze, Phys. Rev. B 92, 020401 (2015).
[67] E. Abate and M. Asdente, Phys. Rev. 140, A1303 (1965).
[68] H. Takayama, K.-P. Bohnen, and P. Fulde, Phys. Rev. B 14, 2287 (1976).
[69] G. H. Lander, M. S. S. Brooks, B. Lebech, P. J. Brown, O. Vogt, and
K. Mattenberger, Applied Physics Letters 57, 989 (1990).
[70] T. Burkert, L. Nordström, O. Eriksson, and O. Heinonen, Phys. Rev. Lett. 93,
027203 (2004).
[71] M. Costa, O. Grånäs, A. Bergman, P. Venezuela, P. Nordblad, M. Klintenberg,
and O. Eriksson, Phys. Rev. B 86, 085125 (2012).
[72] J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 2nd ed.
(Addison-Wesley, San Francisco, 1994).
[73] C. Andersson, B. Sanyal, O. Eriksson, L. Nordström, O. Karis, D. Arvanitis,
T. Konishi, E. Holub-Krappe, and J. Dunn, Phys. Rev. Lett. 99, 177207
(2007).
[74] Y. Miura, S. Ozaki, Y. Kuwahara, M. Tsujikawa, K. Abe, and M. Shirai,
Journal of physics: Condensed matter 25, 106005 (2013).
[75] M. Kotsugi, M. Mizuguchi, S. Sekiya, M. Mizumaki, T. Kojima, T. Nakamura,
H. Osawa, K. Kodama, T. Ohtsuki, T. Ohkochi, K. Takanashi, and
Y. Watanabe, Journal of Magnetism and Magnetic Materials 326, 235 (2013).
[76] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
[77] S. Okamoto, N. Kikuchi, O. Kitakami, T. Miyazaki, Y. Shimada, and
K. Fukamichi, Phys. Rev. B 66, 024413 (2002).
[78] T. Burkert, O. Eriksson, S. Simak, A. Ruban, B. Sanyal, L. Nordström, and
J. Wills, Phys. Rev. B 71, 134411 (2005).
[79] J. Staunton, S. Ostanin, S. Razee, B. Gyorffy, L. Szunyogh, B. Ginatempo,
and E. Bruno, Phys. Rev. Lett. 93, 257204 (2004).
[80] O. N. Mryasov, U. Nowak, K. Y. Guslienko, and R. W. Chantrell, Europhysics
Letters (EPL) 69, 805 (2005).
[81] A. B. Shick and O. N. Mryasov, Phys. Rev. B 67, 172407 (2003).
[82] P. Ravindran, a. Kjekshus, H. Fjellvåg, P. James, L. Nordström, B. Johansson,
and O. Eriksson, Phys. Rev. B 63, 144409 (2001).
[83] H. Callen and E. Callen, Journal of Physics and Chemistry of Solids 27, 1271
(1966).
101
[84] W. P. Wolf, Phys. Rev. 108, 1152 (1957).
[85] G. P. Rodrigue, H. Meyer, and R. V. Jones, Journal of Applied Physics 31,
S376 (1960).
[86] W. J. Carr, Phys. Rev. B 109, 1971 (1958).
[87] E. E. Anderson, Phys. Rev. 134, A1581 (1964).
[88] W. Sucksmith and J. E. Thompson, Proceedings of the Royal Society of
London A 225, 362 (1954).
[89] P. A. Albert and W. J. Carr, Journal of Applied Physics 32, S201 (1961).
[90] V. P. Antropov, V. N. Antonov, L. V. Bekenov, A. Kutepov, and G. Kotliar,
Phys. Rev. B 90, 054404 (2014).
[91] N. V. R. Rao, A. M. Gabay, and G. C. Hadjipanayis, Journal of Physics D:
Applied Physics 46, 062001 (2013).
[92] N. V. R. Rao, A. M. Gabay, W. F. Li, and G. C. Hadjipanayis, Journal of
Physics D: Applied Physics 46, 265001 (2013).
[93] J. M. D. Coey, Journal of physics. Condensed matter : an Institute of Physics
journal 26, 064211 (2014).
[94] A. Iga, Japanese Journal of Applied Physics 9, 415 (1970).
[95] M. Däne, S. Kyung Kim, M. P. Surh, D. Åberg, and L. X. Benedict, Journal of
Physics: Condensed Matter 27, 266002 (2015).
[96] T. Ericsson, L. Häggström, and R. Wäppling, Physica Scripta 17, 83 (1978).
[97] J. Cedervall, S. Kontos, T. C. Hansen, O. Balmes, F. J. Martinez-Casado,
Z. Matej, P. Beran, P. Svedlindh, K. Gunnarsson, and M. Sahlberg, Journal of
Solid State Chemistry 235, 113 (2016).
[98] N. W. Ashcroft and D. N. Mermin, Solid State Physics (Brooks/Cole, Belmont,
USA, 1976).
[99] W. Nolting and A. Ramakanth, Quantum Theory of Magnetism(Springer-Verlag, 2009).
[100] E. Sasıoglu, L. M. Sandratskii, and P. Bruno, Phys. Rev. B 70, 024427 (2004).
[101] J. Rusz, L. Bergqvist, J. Kudrnovský, and I. Turek, Phys. Rev. B 73, 214412
(2006).
[102] S. V. Halilov, H. Eschrig, A. Y. Perlov, and P. M. Oppeneer, Phys. Rev. B 58,
293 (1998).
[103] L. Bergqvist, A. Taroni, A. Bergman, C. Etz, and O. Eriksson, Phys. Rev. B
87, 144401 (2013).
[104] B. Skubic, J. Hellsvik, L. Nordström, and O. Eriksson, Journal of Physics:
Condensed Matter 20, 315203 (2008).
[105] Pajda, M. and Kudrnovský, J. and Turek, I. and Drchal, V. and Bruno, P., Phys.
Rev. B 64, 174402 (2001).
[106] P. Blaha, G. Madsen, K. Schwarz, D. Kvasnicka, and J. Luitz, “WIEN2k, An
Augmented Plane Wave + Local Orbitals Program for Calculating Crystal
Properties,” (2001).
[107] D. Bonnenberg, K. A. Hempel, and H. P. J. Wijn, “Figs. 94 - 114, tables 18 -
19,” (Springer, 1986) Chap. 3d, 4d and 5d Elements, Alloys and Compounds.
[108] J. D. Jackson, Classical electrodynamics, 3rd ed. (Wiley, New York, 1999).
[109] H. Eschrig, The Fundamentals of Density Functional Theory (Dresden, 2003).
[110] J. M. Wills, M. Alouani, P. Andersson, A. Delin, O. Eriksson, and
O. Grechnyev, Full-Potential Electronic Structure Method (Springer, 2010).
102
[111] D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in StatisticalPhysics (Cambridge University Press, 2000).
[112] P. Hohenberg and W. Kohn, Phys. Rev. 136, 864 (1964).
[113] W. Kohn and L. J. Sham, Phys. Rev. 140, 1133 (1965).
[114] L. O. Wagner, T. E. Baker, E. M. Stoudenmire, K. Burke, and S. R. White,
Phys. Rev. B 90, 045109 (2014).
[115] L. Hedin and B. I. Lundqvist, Journal of Physics C: Solid State Physics 4,
2064 (1971).
[116] U. von Barth and L. Hedin, Journal of Physics C: Solid State Physics 5, 1629
(1972).
[117] A. Zunger and J. P. Perdew, Phys. Rev. B 23, 5048 (1981).
[118] J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992).
[119] V. L. Moruzzi, A. R. Williams, and J. F. Janak, Phys. Rev. B 15, 2854 (1977).
[120] D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).
[121] S. H. Vosko, L. Wilk, and M. Nusair, Canadian Journal of Physics 58, 1200
(1980).
[122] J. P. Perdew, A. Ruzsinszky, J. Tao, V. N. Staroverov, G. E. Scuseria, and G. I.
Csonka, The Journal of Chemical Physics 123, 062201 (2005).
[123] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A.
Constantin, X. Zhou, and K. Burke, Phys. Rev. Lett. 100, 136406 (2008).
[124] Z. Wu and R. E. Cohen, Phys. Rev. B 73, 235116 (2006).
[125] F. Tran, R. Laskowski, P. Blaha, and K. Schwarz, Phys. Rev. B 75, 115131
(2007).
[126] V. Ozolinš and M. Körling, Phys. Rev. B 48, 18304 (1993).
[127] P. Bagno, O. Jepsen, and O. Gunnarsson, Phys. Rev. B 40, 1997 (1989).
[128] P. Haas, F. Tran, and P. Blaha, Phys. Rev. B 79, 085104 (2009).
[129] V. I. Anisimov, F. Aryasetiawan, and A. I. Lichtenstein, Journal of Physics:
Condensed Matter 9, 767 (1997).
[130] G. Kotliar and D. Vollhardt, Phys. Today 57, 53 (2004).
[131] G. Kotliar, S. Savrasov, K. Haule, V. Oudovenko, O. Parcollet, and
C. Marianetti, Reviews of Modern Physics 78, 865 (2006).
[132] O. Eriksson, B. Johansson, R. C. Albers, A. M. Boring, and M. S. S. Brooks,
Phys. Rev. B 42, 2707 (1990).
[133] K. Lejaeghere, G. Bihlmayer, T. Björkman, P. Blaha, S. Blügel, V. Blum,
D. Caliste, I. E. Castelli, S. J. Clark, A. Dal Corso, S. de Gironcoli, T. Deutsch,
J. K. Dewhurst, I. Di Marco, C. Draxl, M. Dułak, O. Eriksson, J. A.
Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi,
S. Goedecker, X. Gonze, O. Grånäs, E. K. U. Gross, A. Gulans, F. Gygi, D. R.
Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iusan, D. B. Jochym, F. Jollet,
D. Jones, G. Kresse, K. Koepernik, E. Küçükbenli, Y. O. Kvashnin, I. L. M.
Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordström,
T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson,
M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz,
S. Sharma, F. Tavazza, P. Thunström, A. Tkatchenko, M. Torrent,
D. Vanderbilt, M. J. van Setten, V. Van Speybroeck, J. M. Wills, J. R. Yates,
G.-X. Zhang, and S. Cottenier, Science 351, 1415 (2016).
[134] R. V. Noorden, B. Maher, and R. Nuzzo, Nature 514, 530 (2014).
103
[135] S. Cottenier, Density Functional Theory and the Family of (L)APW-methods: astep-by-step introduction (Instituut voor Kern- en Stralingsfysica, K.U.Leuven,
Belgium, 2002).
[136] O. K. Andersen, Phys. Rev. B 12, 3060 (1975).
[137] J. C. Slater, Phys. Rev. 51, 846 (1937).
[138] P. E. Blöchl, O. Jepsen, and O. K. Andersen, Phys. Rev. B 49 (1994).
[139] D. D. Koelling and H. N. Harmon, Journal of Physics C: Solid State Physics
10 (1977).
[140] A. H. MacDonald, W. E. Picket, and D. D. Koelling, Journal of Physics C:
Solid State Physics 13, 2675 (1980).
[141] H. Ebert, “The Munich SPR-KKR package, version 6.3,” (2012).
[142] H. Ebert, D. Ködderitzsch, and J. Minár, Reports on Progress in Physics 74,
096501 (2011).
[143] J. Korringa, Physica 13, 392 (1947).
[144] W. Kohn and N. Rostoker, Phys. Rev. 94, 1111 (1954).
[145] E. N. Economou, Green’s Functions in Quantum Physics (Springer Berlin
Heidelberg, 2006).
[146] J. Zabloudil, R. Hammerling, P. Weinberger, and L. Szunyogh, ElectronScattering in Solid Matter (Springer, 2005).
[147] D. J. Chadi and M. L. Cohen, Phys. Rev. B 8, 5747 (1973).
[148] J. S. Faulkner, Progress in Materials Science 27, 1 (1982).
[149] A. Zunger, S.-H. Wei, L. G. Ferreira, and J. E. Bernard, Phys. Rev. Lett. 65,
353 (1990).
[150] S. Steiner, S. Khmelevskyi, M. Marsmann, and G. Kresse, Phys. Rev. B 93,
224425 (2016).
[151] P. Söderlind, B. Johansson, and O. Eriksson, Journal of Magnetism and
Magnetic Materials 45, 2037 (1992).
[152] R. H. Victora and L. M. Falicov, Phys. Rev. B 30, 259 (1984).
[153] P. James, Calculation of magnetism and its crystal structure dependence,
Ph.D. thesis, Uppsala University (1999).
[154] I. Turek, J. Kudrnovský, and K. Carva, Phys. Rev. B 86, 174430 (2012).
[155] Y. Kota and A. Sakuma, Applied Physics Express 5, 113002 (2012).
[156] C. Neise, S. Schönecker, M. Richter, K. Koepernik, and H. Eschrig, Physica
Status Solidi (B) 248, 2398 (2011).
[157] G. Andersson, T. Burkert, P. Warnicke, M. Björck, B. Sanyal, C. Chacon,
C. Zlotea, L. Nordström, P. Nordblad, and O. Eriksson, Phys. Rev. Lett. 96,
037205 (2006).
[158] P. Warnicke, G. Andersson, M. Björck, J. Ferré, and P. Nordblad, Journal of
Physics: Condensed Matter 19, 226218 (2007).
[159] P. Soven, Phys. Rev. 156 (1967).
[160] B. L. Gyorffy, Phys. Rev. B 5, 2382 (1972).
[161] H. Ebert, S. Mankovsky, D. Ködderitzsch, and P. J. Kelly, Phys. Rev. Lett.
107, 066603 (2011).
[162] H. Ebert, S. Mankovsky, K. Chadova, S. Polesya, J. Minár, and
D. Ködderitzsch, Phys. Rev. B 91, 165132 (2015).
[163] G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuurmans, Phys. Rev. B 41,
11919 (1990).
104
[164] J. Trygg, B. Johansson, O. Eriksson, and J. M. Wills, Phys. Rev. Lett. 75, 2871
(1995).
[165] A. Liechtenstein, M. Katsnelson, V. Antropov, and V. Gubanov, Journal of
Magnetism and Magnetic Materials 67, 65 (1987).
[166] X. Wang, R. Wu, D. Wang, and A. Freeman, Phys. Rev. B 54, 61 (1996).
[167] J. Staunton, L. Szunyogh, A. Buruzs, B. Gyorffy, S. Ostanin, and L. Udvardi,
Phys. Rev. B 74, 144411 (2006).
[168] N. D. Mermin, Phys. Rev. 137, A1441 (1965).
[169] B. L. Györffy, A. J. Pindor, J. Staunton, G. M. Stocks, and H. Winter, Journal
of Physics F: Metal Physics 15, 1337 (1985).
[170] A. Deák, E. Simon, L. Balogh, L. Szunyogh, M. dos Santos Dias, and J. B.
Staunton, Phys. Rev. B 89, 224401 (2014).
[171] A. Buruzs, P. Weinberger, L. Szunyogh, L. Udvardi, P. I. Chleboun, A. M.
Fischer, and J. B. Staunton, Phys. Rev. B 76, 064417 (2007).
[172] A. Buruzs, L. Szunyogh, L. Udvardi, P. Weinberger, and J. Staunton, Journal
of Magnetism and Magnetic Materials 316, e371 (2007), proceedings of the
Joint European Magnetic Symposia.
[173] I. A. Zhuravlev, V. P. Antropov, and K. D. Belashchenko, Phys. Rev. Lett. 115,
217201 (2015).
[174] B. S. Pujari, P. Larson, V. P. Antropov, and K. D. Belashchenko, Phys. Rev.
Lett. 115, 057203 (2015).
[175] A. I. Liechtenstein, M. I. Katsnelson, and V. A. Gubanov, Journal of Physics
F: Metal Physics 14, L125 (1984).
[176] A. Liechtenstein, M. Katsnelson, V. Antropov, and V. Gubanov, Journal of
Magnetism and Magnetic Materials 54, 965 (1986).
[177] A. Szilva, M. Costa, A. Bergman, L. Szunyogh, L. Nordström, and
O. Eriksson, Phys. Rev. Lett. 111, 127204 (2013).
[178] R. K. Pathria, Statistical Mechanics, 2nd ed. (Elsevier, 1996).
[179] M. Wallin, Monte Carlo Simulations in Statistical Physics (KTH, Stockholm,
Sweden, 2005).
[180] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and
E. Teller, The Journal of Chemical Physics 21, 1087 (1953).
[181] E. J. Kirkland, Advanced Computing in Electron Microscopy, 2nd ed.
(Springer, 2009).
[182] A. Lubk and J. Rusz, Phys. Rev. B 92, 235114 (2015).
[183] J. M. Cowley and a. F. Moodie, Acta Crystallographica 10, 609 (1957).
[184] A. F. Moodie and P. Goodman, Acta Crystallographica 30, 280 (1970).
[185] C. Y. Cai, S. J. Zeng, H. R. Liu, and Q. B. Yang, Micron 40, 313 (2009).
[186] A. Rother and K. Scheerschmidt, Ultramicroscopy 109, 154 (2009).
[187] J. C. Meyer, S. Kurasch, H. J. Park, V. Skakalova, D. Künzel, A. Groß,
A. Chuvilin, G. Algara-Siller, S. Roth, T. Iwasaki, U. Starke, J. H. Smet, and
U. Kaiser, Nature Materials 10, 209 (2011).
[188] J. H. Chen and D. V. Dyck, Ultramicroscopy 70 (1997).
[189] J. Spiegelberg and J. Rusz, Ultramicroscopy 159, 11 (2015).
[190] K. Fujiwara, Journal of the Physical Society of Japan 16, 2226 (1961).
[191] A. R. Williams, V. L. Moruzzi, A. P. Malozemoff, and K. Terakura, IEEE
Transactions on Magnetics 19, 1983 (1983).
105
[192] D. Bonnenberg, K. A. Hempel, and H. P. J. Wijn, “1.2.1 Alloys between Fe,
Co or Ni, 1.2.1.1 Phase diagrams, lattice parameters,” (Springer, 1986) Chap.
3d, 4d and 5d Elements, Alloys and Compounds.
[193] F. Yildiz, M. Przybylski, X.-D. Ma, and J. Kirschner, Phys. Rev. B 80, 064415
(2009).
[194] G. Kurdjumov, Metallurgical Transactions A 7, 999 (1976).
[195] R. Naraghi, M. Selleby, and J. A. gren, Calphad 46, 148 (2014).
[196] G. Kurdjumov and A. Khachaturyan, Acta Metallurgica 23, 1077 (1975).
[197] L. C. Materials Science International Team MSIT, Andy Watson, “Carbon -
cobalt - iron, iron systems: phase diagrams, crystallographic and
thermodynamic data, 1.2.1.1 phase diagrams, lattice parameters,” (Springer,
2008) Chap. 3d, 4d and 5d Elements, Alloys and Compounds.
[198] S. S. Baik, B. I. Min, S. K. Kwon, and Y. M. Koo, Phys. Rev. B 81, 144101
(2010).
[199] M. D. Kuz’min, K. P. Skokov, H. Jian, I. Radulov, and O. Gutfleisch, Journal
of Physics: Condensed Matter 26, 064205 (2014).
[200] K. Koepernik and H. Eschrig, Phys. Rev. B 59, 1743 (1999).
[201] H. Akai, Hyperfine Interactions 43, 253 (1988).
[202] P. Dederichs, R. Zeller, H. Akai, and H. Ebert, Journal of Magnetism and
Magnetic Materials 100, 241 (1991).
[203] R. Wienke, G. Schütz, and H. Ebert, Journal of Applied Physics 69, 6147
(1991).
[204] F. Wilhelm, P. Poulopoulos, H. Wende, A. Scherz, K. Baberschke,
M. Angelakeris, N. K. Flevaris, and A. Rogalev, Phys. Rev. Lett. 87, 207202
(2001).
[205] Y. Kota and A. Sakuma, Journal of the Physical Society of Japan 81, 084705
(2012).
[206] J. B. Staunton, S. Ostanin, S. S. A. Razee, B. Gyorffy, L. Szunyogh,
B. Ginatempo, and E. Bruno, Journal of Physics: Condensed Matter 16,
S5623 (2004).
[207] T. Kojima, M. Mizuguchi, and K. Takanashi, in Journal of Physics:Conference Series, Vol. 266 (2011) p. 012119.
[208] T. Kojima, M. Mizuguchi, T. Koganezawa, K. Osaka, M. Kotsugi, and
K. Takanashi, Japanese Journal of Applied Physics 51, 010204 (2012).
[209] L. H. Lewis, F. E. Pinkerton, N. Bordeaux, A. Mubarok, E. Poirier, J. I.
Goldstein, R. Skomski, and K. Barmak, IEEE Magnetics Letters 5, 1 (2014).
[210] E. Poirier, F. E. Pinkerton, R. Kubic, R. K. Mishra, N. Bordeaux, A. Mubarok,
L. H. Lewis, J. I. Goldstein, R. Skomski, and K. Barmak, Journal of Applied
Physics 117 (2015).
[211] N. Bordeaux, A. Montes-Arango, J. Liu, K. Barmak, and L. Lewis, Acta
Materialia 103, 608 (2016).
[212] A. Frisk, B. Lindgren, S. D. Pappas, E. Johansson, and G. Andersson, Journal
of Physics: Condensed Matter 28, 406002 (2016).
[213] A. J. J. Koch, P. Hokkeling, M. G. v. d. Steeg, and K. J. de Vos, Journal of
Applied Physics 31, S75 (1960).
106
[214] J. H. Park, Y. K. Hong, S. Bae, J. J. Lee, J. Jalli, G. S. Abo, N. Neveu, S. G.
Kim, C. J. Choi, and J. G. Lee, Journal of Applied Physics 107, 09A731
(2010).
[215] S. H. Nie, L. J. Zhu, J. Lu, D. Pan, H. L. Wang, X. Z. Yu, J. X. Xiao, and J. H.
Zhao, Applied Physics Letters 102, 152405 (2013).
[216] L. G. Marshall, I. J. McDonald, and L. Lewis, Journal of Magnetism and
Magnetic Materials 404, 215 (2016).
[217] H. Fang, S. Kontos, J. Ångström, J. Cedervall, P. Svedlindh, K. Gunnarsson,
and M. Sahlberg, Journal of Solid State Chemistry 237, 300 (2016).
[218] H. Fang, J. Cedervall, F. J. M. Casado, Z. Matej, J. Bednarcik, J. Ångström,
P. Berastegui, and M. Sahlberg, Journal of Alloys and Compounds 692, 198
(2017).
[219] Y. Kinemuchi, A. Fujita, and K. Ozaki, Dalton Trans. 45, 10936 (2016).
[220] A. Sakuma, Journal of Magnetism and Magnetic Materials 187, 105 (1998).
[221] K. Wang, E. Lu, J. W. Knepper, F. Yang, and A. R. Smith, Applied Physics
Letters 98, 162507 (2011).
[222] E. Lu, D. Ingram, A. Smith, J. Knepper, and F. Yang, Phys. Rev. Lett. 97,
146101 (2006).
[223] S. Fukami, H. Sato, M. Yamanouchi, S. Ikeda, and H. Ohno, Applied Physics
Express 6, 073010 (2013).
[224] M. Tanaka, J. P. Harbison, J. DeBoeck, T. Sands, B. Philips, T. L. Cheeks, and
V. G. Keramidas, Applied Physics Letters 62, 1565 (1993).
[225] T. Shima, M. Okamura, S. Mitani, and K. Takanashi, Journal of Magnetism
and Magnetic Materials 310, 2213 (2007).
[226] L. Häggström, R. Wäppling, T. Ericsson, Y. Andersson, and S. Rundqvist,
Journal of Solid State Chemistry 13, 84 (1975).
[227] T. N. Lamichhane, V. Taufour, S. Thimmaiah, D. S. Parker, S. L. Bud’ko, and
P. C. Canfield, Journal of Magnetism and Magnetic Materials 401, 525 (2016).
[228] Rundqvist, Acta Chemica Scandinavica 16, 1 (1962).
[229] M. A. McGuire and D. S. Parker, Journal of Applied Physics 118 (2015).
[230] H. Arnfelt and A. Westgren, Jernkontorets Annaler , 185 (1935).
[231] Ž. Blažina and S. Pavkovic, Journal of the Less Common Metals 155, 247
(1989).
[232] M. A. Koten, P. Manchanda, B. Balamurugan, R. Skomski, D. J. Sellmyer,
and J. E. Shield, APL Materials 3, 076101 (2015).
[233] P. Kumar, A. Kashyap, B. Balamurugan, J. E. Shield, D. J. Sellmyer, and
R. Skomski, Journal of physics: Condensed matter 26, 064209 (2014).
[234] H. Fujii, T. Hokabe, T. Kamigaichi, and T. Okamoto, Journal of the Physical
Society of Japan 43, 41 (1977).
[235] R. Fruchart, Journal of Applied Physics 40, 1250 (1969).
[236] Catalano, A. and Arnott, R. J. and Wold, A, Journal of Solid State Chemistry
7, 262 (1973).
[237] C. Jian-wang, L. He-lie, and Z. Qing-Qi, Journal of Physics: Condensed
Matter 5, 9307 (1993).
[238] S. N. Doli, A. Krishnamurthy, V. Ghose, and B. K. Srivastava, Journal of
Physics: Condensed Matter 5, 451 (1993).
107
[239] J. Leitão, M. van der Haar, a. Lefering, and E. Brück, Journal of Magnetism
and Magnetic Materials 344, 49 (2013).
[240] A. Sakuma, Y. Manabe, and Y. Kota, Journal of the Physical Society of Japan
82, 73704 (2013).
[241] T. Graf, C. Felser, and S. S. Parkin, Progress in Solid State Chemistry 39, 1
(2011).
[242] C. Felser, L. Wollmann, S. Chadov, G. H. Fecher, and S. S. P. Parkin, APL
Materials 3, 041518 (2015).
[243] C. Felser, V. Alijani, J. Winterlik, S. Chadov, and A. K. Nayak, IEEE
Transactions on Magnetics 49, 682 (2013).
[244] L. Wollmann, S. Chadov, J. Kübler, and C. Felser, Phys. Rev. B 92, 064417
(2015).
[245] J. Winterlik, B. Balke, G. H. Fecher, C. Felser, M. C. M. Alves, F. Bernardi,
and J. Morais, Phys. Rev. B 77, 054406 (2008).
[246] A. Edström, Theoretical Magnet Design, Licentiate thesis, Uppsala University
(2014).
[247] E. J. Lisher, C. Wilkinson, T. Ericsson, L. Häggstrom, L. Lundgren, and
R. Wappling, Journal of Physics C: Solid State Physics 7, 1344 (1974).
[248] A. Broddefalk, P. Granberg, P. Nordblad, H.-P. Liu, and Y. Andersson, Journal
of Applied Physics 83, 6980 (1998).
[249] S. Rundqvist, E. Hassler, and L. Lundvik, Acta Chemica Scandinavica 16, 242
(1962).
[250] S. Neov and E. Legrand, Physica status solidi (b) 49, 589 (1972).
[251] T. Ishii, M. Shimada, and M. Koizumi, Journal of Magnetism and Magnetic
Materials 31, 151 (1983).
[252] A. Iga and Y. Tawara, Journal of the Physical Society of Japan 24, 28 (1968).
[253] A. M. Cormack, Acta Crystallographica 10, 354 (1957).
[254] J. Rusz, S. Bhowmick, M. Eriksson, and N. Karlsson, Phys. Rev. B 89,
134428 (2014).
[255] N. F. Mott, Proceedings of the Royal Society A 124, 425 (1929).
[256] H. Batelaan, T. J. Gay, and J. J. Schwendiman, Phys. Rev. Lett. 79, 4517
(1997).
[257] G. H. Rutherford and R. Grobe, Phys. Rev. Lett. 81, 4772 (1998).
[258] V. Grillo, L. Marrucci, E. Karimi, R. Zanella, and E. Santamato, New Journal
of Physics 15, 093026 (2013).
[259] S. J. Pennycook and P. D. Nellist, Scanning Transmission Electron Microscopy- Imaging and Analysis (Springer, New York, 2006).
[260] M. Haider, S. Uhlemann, E. Schwan, H. Rose, B. Kabius, and K. Urban,
Nature 392, 768 (1998).
[261] M. Haider, H. Rose, S. Uhlemann, B. Kabius, and K. Urban, Journal of
Electron Microscopy 47, 395 (1998).
[262] J. M. D. Coey, Magnetism and magnetic materials, 1st ed. (Cambridge
University Press, Cambridge, 2010).
[263] C. Ortiz, O. Eriksson, and M. Klintenberg, Computational Materials Science
44, 1042 (2009).
[264] S. Curtarolo, G. L. W. Hart, M. B. Nardelli, N. Mingo, S. Sanvito, and
O. Levy, Nature Materials 12, 191 (2013).
108
[265] C. C. Fischer, K. J. Tibbetts, D. Morgan, and G. Ceder, Nature Materials 5,
641 (2006).
[266] N. Drebov, A. Martinez-Limia, L. Kunz, A. Gola, T. Shigematsu, T. Eckl,
P. Gumbsch, and C. Elsässer, New Journal of Physics 15, 125023 (2013).
[267] X. Zhao, M. C. Nguyen, W. Y. Zhang, C. Z. Wang, M. J. Kramer, D. J.
Sellmyer, X. Z. Li, F. Zhang, L. Q. Ke, V. P. Antropov, and K. M. Ho, Phys.
Rev. Lett. 112, 045502 (2014).
[268] E. Jonsson, K. Högdahl, and V. Troll, Forskning och Framsteg 7 (2012).
109
Acta Universitatis UpsaliensisDigital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1440
Editor: The Dean of the Faculty of Science and Technology
A doctoral dissertation from the Faculty of Science andTechnology, Uppsala University, is usually a summary of anumber of papers. A few copies of the complete dissertationare kept at major Swedish research libraries, while thesummary alone is distributed internationally throughthe series Digital Comprehensive Summaries of UppsalaDissertations from the Faculty of Science and Technology.(Prior to January, 2005, the series was published under thetitle “Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology”.)
Distribution: publications.uu.seurn:nbn:se:uu:diva-304666
ACTAUNIVERSITATIS
UPSALIENSISUPPSALA
2016