Institut für Chemie

Masterstudiengang Chemie

Masterarbeit

Ab Initio Magnetic Properties of Rare-Earth Lean Nd-based Hard Magnets

vorgelegt von: Stephan Erdmann

Matrikelnummer: 4546697

Betreuender Gutachter: Herr Prof. Dr. Thorsten Klüner

Zweitgutachter: Herr Dr.-Ing. Halil İbrahim Sözen

Oldenburg, 20.12.2022

###### Contents

  1. 1 Introduction to Permanent Magnets and Magnetism
    1. 1.1 Basic Properties of Magnets and Fundamentals
    2. 1.2 Historical Development of Magnets
    3. 1.3 World Market and Usage of Magnets
  2. 2 Case Description and Motivation of the Work
  3. 3 Theoretical Background
    1. 3.1 Density Functional Theory
    2. 3.2 Schrödinger’s Equation
    3. 3.3 Born-Oppenheimer Approximation
    4. 3.4 The Hohenberg-Kohn Theorem
    5. 3.5 The Kohn-Sham Equations
    6. 3.6 Exchange-Correlation Energy
    7. 3.7 Local Density Approximation
    8. 3.8 Local Spin Density Approximation
    9. 3.9 Generalized Gradient Approximation
    10. 3.10 Ultra-soft Pseudopotentials and the Projector- Augmented Wave Method
    11. 3.11 Magnetism in Density Functional Theory
    12. 3.12 Treatment of f-electrons
  4. 4 Computational Details
    1. 4.1 Vienna ab initio Simulation Package Calculations
    2. 4.2 Korringa-Kohn-Rostoker Calculations
  5. 5 Results and Analysis
    1. 5.1 Determination of the Hubbard U Parameter by Density of States Calculations
    2. 5.2 Calculation of Lattice Constants and Cell Volume
    3. 5.3 Total Magnetic Moment and Magnetization
    4. 5.4 Maximum Energy Product
    5. 5.5 Curie Temperature
    6. 5.6 Finite Temperature Magnetization
    7. 5.7 Magnetocrystalline Anisotropy
    8. 5.8 Hardness Factor
  6. 6 Conclusion and Outlook

Bibliography

List of Abbreviations

Erklärung

###### List of Figures

  1. 1.1 Hysteresis loop for soft (left) and hard (hard) magnetic materials. Important magnetic properties like the magnetization saturation $M_{S}$ are marked and the maximum energy product |BH|max is is depicted as a blue square in the second quadrant.
  2. 1.2 Different magnetic domain configurations for the different types of magnetism with and without an external field applied. The image is taken from [7].
  3. 1.3 a) Change of the magnetic ordering from ferro- to paramagnetism with increasing temperature up to the Curie temperature $T_{C}$; b) Finite temperature magnetization curves for the elements Fe, Co and Ni up to their $T_{C}$’s of 1044, 1388 and 628 K. Image b) is based on data from [6].
  4. 1.4 Unequal energies for the different magnetization directions. Image adapted from [8].
  5. 1.5 Change of the magnetic energy in relationship to the angle between the easy axis and the magnetization direction with a peak at an angle of 90°.
  6. 1.6 Development of the different types of magnets in the 20${}^{\text{th}}$ century and the beginning of the 21${}^{\text{st}}$ century. The volume needed of each magnetic material to achieve the same magnetic strength is shown under the diagram. Image taken from [12].
  7. 1.7 Application of permanent magnets by market share in 2019. The image is based on data from [21].
  8. 1.8 Worldwide production of sintered Nd-Fe-B magnets from 1984 to 2017. The image is based on data from [22].
  9. 1.9 Percentage sales of major permanent magnet alloy types in 2020 in the global market. The image is based on data from [17]
  10. 2.1 Calculated competition energies of NdFe12 alloys with increasing concentrations of Ti and Ce or Y with three different temperatures 0, 750 and 1500 K. The calculations for the (Nd,X)Fe12-yTiy alloys (X = Y and Ce) were performed in 2 formula units with 26 atoms supercells in the limits of $0<x<0.5$ and $0<y<1$. Linear interpolation has been performed between the calculation points that correspond to intermetallic compositions. Images are taken from [35].
  11. 3.1 Increasing number of publications using DFT. The Pople endorsement and the Nobel prize in chemistry are marked according to the appropriate year. The image is based on data from [44]
  12. 3.2 Jacob’s ladder showing the different functionals (LDA, GGA, Meta GGA, Hybrid GGA, Hybrid Meta GGA and Fully Non-Local) leading to increases of the accuracy of calculations. The image is adapted from [47].
  13. 3.3 Radial dependence of typical pseudopotentials in comparison with the atomic Coulomb potential (blue). In the lower part a soft-core pseudopotential and a hard-core pseudopotential are shown with the solid and dashed red curve respectively. In the top part are the wave functions and the pseudo wave function resulting from the atomic (blue) and the pseudo-potential (red) depicted. It has to be noted, that both wave functions have the same values above a certain point r. The image is taken from [57]
  14. 5.1 Calculated density of states (DOS) for the NdFe11Ti compound. The Dudarev approach is used with the U parameter ranging from 1 to 8 eV. Vertical dotted line represents for the Fermi level. The 4f peaks (red lines) change according to the U parameter and are located for small U near the Fermi energy.
  15. 5.2 Schematic representation of the 2 formula unit (f.u.) (26 atoms) body-center-tetragonal ThMn12 (I4/mmm, space group number 139) 1:12 phases. (a) Unstable RFe12, (b) 1 Ti atom substituted (3.8 at.%) R2Fe23Ti (RFe11.5Ti0.5) and (c) 2 Ti atom substituted (7.7 at.%) and nitrogenated supercell RFe11TiN (R2Fe22Ti2N2). Ti atoms substituted in energetically favorable 8i sites and 2 N atom considered to fully occupy interstitial 2b site.
  16. 5.3 Schematic representation of the cell volume trends for two different substitutions, which are Ti substitution on Fe sublattice and Ce and Y substitutions in Nd sublattice, via GGA calculations. Purely Nd-containing compounds are depicted in black while compounds only including Y and Ce are shown as blue and red, respectively. The quarternary compounds containing Nd,Y and Nd,Ce are illustrated as cyan and orange, respectively. The resulting trend for each substitution is further clarified by arrows.
  17. 5.4 Total magnetic moments of the clean (RFe24-yTiy), N doped (RFe24-yTiyN2) and N removed but volume kept fixed, and atoms are allowed to relax cases (RFe24-yTiyE2). Since DFT+U calculations do not change the results significantly, only DFT results are given. The light(dark) grey background coloring shows the magnetovolume (chemical) effect. The blue circles show theoretical data [97, 88] for the clean alloys from others, the red circles show experimental data [37, 96, 81, 92] for the clean alloys, the blue squares show theoretical nitrogenated cases [97, 77, 88], and the red squares show the experimental nitrogenated cases [92].
  18. 5.5 Theoretical maximum energy product |BH|max values for considered compounds and nitrogenated cases. In addition to conventional DFT, DFT+U with U=6 eV has been considered for stable Nd-contained alloys. Experimental values [10, 101] of most common hard magnets are given as horizontal dotted lines.
  19. 5.6 Calculated Curie temperatures $T_{\text{C}}$ for the clean and nitrogenated 1:12 phases. Exchange interaction energies have been calculated with both ordered ferromagnetic (FM) and disordered local moment disorder (LMD) states, and the input lattice information is taken from our theoretically relaxed calculations (check Tab. 5.1). Experimental data is taken from [86, 104].
  20. 5.7 Finite temperature magnetization calculated on basis of the Körmann and Kuzmin approaches. For comparison the experimental data by Herper et al. [75], Bozukov et al. [108], Akayama et al. [81] and Tereshina et al. [80] are shown. For the NdFe11Ti and YFe11Ti compounds the Kuzmin and the Körmann approaches are used. For the CeFe11Ti compound the Kuzmin approach with $p$=8 (red) and $p$=6 (blue) are depicted.
  21. 5.8 Calculated energy differences of the magnetocrystalline energy for the shift of the crystallographic direction from [001] to [110](black) and [100] to [010] (red) for the clean (circles) and nitrogenated (squares) NdFe11Ti compound. On the left side are DFT and on the right side DFT+U results shown. For the change from [001] to [110] the angles 0, 22.5, 45, 67.5 and 90° are considered. For the change from [001] to [010] only the 0 and 90 ° angles are considered.
  22. 5.9 Influence of interstitial nitrogenation on the anisotropy constant $K_{1}$ (black) and anisotropy field $H_{a}$ (red). Treatment with the GGA (GGA+U) functional is depicted as squares (circles). Clean compounds are shown as empty and nitrogenated ones as filled symbols.
  23. 5.10 Calculated hardness factors $\kappa$ for GGA (black), GGA+U (red) and LSDA (blue) functionals. The clean compounds are represented by circles and the nitrogenated compounds considered in the GGA and GGA+U approaches are depicted as squares. The region colored as dark grey represents soft magnets ($\kappa$ < 0.1), the light grey one semi-hard magnets (0.1 < $\kappa$ < 1.0) and the white background hard magnets ($\kappa$>1). The magnetic hardness factor of the most well-known magnets are also shown by horizontal dotted lines.

###### List of Tables

  1. 1.1 Curie temperature $T_{C}$, magnetization $\mu_{0}$MS, anisotropy constant $K_{1}$ and hardness factor $\kappa$ for common hard magnets. Data is taken from [11].
  2. 1.2 Production of RE-oxides and available reserves for different countries. Data is taken from [23].
  3. 5.1 Calculated lattice constant, cell volume and bulk modulus of the considered alloys and nitrogenated cases using DFT. In the case of Nd containing stable compounds DFT+U has been considered only for Nd 4f-electrons with U=6 eV and is given in parenthesis. N is assumed to occupy all available N(2b) sites in the 2 f.u. supercell with 2 atoms (see Fig. 5.2 c)). For instance, Y-based 1:12 phases have the following compositions after nitrogenation: YFe12N, Y2Fe23Ti2N2 and YFe11TiN.
  4. 5.2 Calculated magnetic moments at rare-earth site $m^{2a}_{tot}$, total magnetic moments $m_{tot}$ ($\mu_{\text{B}}/f.u$) and saturation magnetization $\mu_{0}M_{S}$ (T) with comparison against available experimental data. In the case of Nd contained compounds DFT+U with U= 6 eV is applied to Nd 4f-electrons only and results are given in parenthesis. 2 N atoms have been considered in the N(2b) site in the 2 f.u. supercell for the nitrogenated cases.
  5. 5.3 Calculated theoretical magnetocrystalline anisotropy constant $K_{1}$ according to Eq. 5.5, anisotropy field $H_{a}$ according to Eq. 5.6 and magnetic hardness factor $\kappa$ from Eq. 5.7 for stable compounds with available experimental data. Values are given for both GGA and LSDA treatments for the exchange-correlation potential and DFT+U (U= 6 eV) results are reported in parenthesis Nd contained compounds. The experimental value of $K_{1}$ of well known hard magnets is also given for a comparison.

## Acknowledgements/Danksagung

After writing this work i want to thank Prof. Dr. Thorsten Klüner who enabled
me to write this thesis in his working group and for supervising this work. I
also want to thank Dr. Ing. Halil İbrahim Sözen for taking the position as the
second supervisor. I also want to thank him for helping me with the program
package VASP, his tutelage with the different calculations and his general
support in this work. Furthermore, I want to thank my friends and especially
my family for the support I got in the last years of my studies. Special
thanks also go to the working group Theoretische Chemie of the Carl-von-
Ossietzky university in Oldenburg who were all ready to help if it was needed.
Lastly, I want to thank the team of the HPC-Cluster CARL of the university on
which the calculations in this work were done.

Mein besonderer Dank gilt Herrn Prof. Dr. Thorsten Klüner dafür, dass ich
diese Abschlussarbeit in seinem Arbeitskreis anfertigen konnte. Weiterhin
möchte ich besonders Dr. Ing. Halil İbrahim Sözen für seine Hilfe im Umgang
mit dem Programmpaket VASP, seiner Unterstützung bei den Berechnungen, der
generellen Betreuung durch ihn sowie der Kontrolle als Zweitprüfer in dieser
Arbeit. Ebenfalls danke ich meinen Freunden und besonders meiner Familie
welche mich während des Studiums stets unterstützt und mir geholfen haben.
Auch möchte ich generell der Arbeitsgruppe der Theoretischen Chemie in
Oldenburg für die Bereitschaft zur Hilfe bei Fragen und für die gute Aufnahme
im Arbeitskreis danken. Zuletzt möchte ich mich beim Team des HPC-Clusters
CARL der Universität Oldenburg bedanken, auf dem die Berechnungen in dieser
Arbeit durchgeführt wurden.

##  Chapter 1 Introduction to Permanent Magnets and Magnetism

Magnetic materials have played a critical role in the development of electric
motors and renewable energy technologies in the last years and are an
essential part of many other applications like generators or actuators. Since
the need and demand for these technologies are steadily increasing, more and
more effort is put into the development of these materials [1]. A big share of
this interest is focused on the improvement of rare-earth-based hard magnetic
materials that are mostly composed of rare-earth (RE) and transition metal
(TM) elements. Due to resource criticality, a big focus is devoted to the
reduction of the rare-earth elements in these compounds and on the development
of new rare-earth lean or rare-earth free hard magnets [2, 3].

In the first chapter some fundamental and basic magnetic properties such as
magnetic moment $m_{tot}$, magnetization $M_{S}$, the maximum energy product
$|BH|_{max}$, the Curie temperature $T_{C}$ and the magnetocrystalline
anisotropy energy (MAE) are explained to provide a robust understanding for
the readers, who are unfamiliar with the topic. In the second part, the
historical development of magnetic materials is reported to demonstrate the
historical impact of magnets on world development. In the third part, the
situation of the current global market and the main usage fields of hard
magnetic materials in the last decade are described. At the end of this
chapter, the motivation for this thesis is explained.

###  1.1 Basic Properties of Magnets and Fundamentals

Since the discovery of the first magnetic material, which were naturally
magnetized pieces of iron called lodestone, its properties to attract ferrous
materials has drawn the attention of many scientists and researchers to it.
These properties had a great impact on the development of the world. For
instance, the invention of the compass, one of the earliest magnetic devices,
made the navigation on sea much easier and also enabled much larger
explorations. These explorations included the discovery of America by
Christopher Columbus in 1492 or the voyage around the world by Ferdinand
Magellan from 1519 to 1522 [4].

The origin of magnetism can be attributed to the magnetic moments of the
different atoms in a compound. The classical Bohr definition explains their
origin in the motion of an electrical charge, in other words of electrons, and
the resulting magnetic field that is generated with their orbital motion. This
orbital motion generates an orbital magnetic moment and a spin magnetic moment
associated with it, both of which are measured in Bohr magneton $\mu_{B}$ [4,
5, 6]. The intrinsic spin angular momentum of electrons is according to the
quantum mechanical definition $s=\frac{1}{2}$. The total magnetic moment is
therefore made out of the orbital (L) and the spin (S) magnetic moment that
has two ways to be put together in the forms of LS and JJ couplings with J =
L+S. When the electron shell of an element is filled completely the electrons
pair themselves with those electrons that have an opposing intrinsic magnetic
moment. This leads to an overall total magnetic moment of zero.

Until the 19${}^{\text{th}}$ century, the shape of magnets was strongly
restricted due to the shape-dependent demagnetization caused by the internal
magnetic field $H_{d}$ that occurred in these materials ($H_{d}$ = - NM, with
N as the shape-dependent demagnetization factor and M as the magnetization).
This limited the shape of magnets to be bar and needle-like. A small
improvement was the development of magnets in the shape of a horseshoe that
avoided the self-demagnetization problem but still restricted the shape to a
U-form. Nevertheless, these magnets at least led in combination with copper
coils and iron to the first magnetic motors and generators in the
19${}^{\text{th}}$ century [4, 6].

When the first ferrimagnetic hexagonal ferrites were discovered in 1951, a
better understanding of the coercivity led to further progress and enabled the
manufacturing of magnets with a coercivity $H_{c}$ higher than their
spontaneous magnetization $M_{S}$ and a break of the shape restrictions. At
that time, it was possible to design magnets in every way possible, which
caused a significant increase in usability. Ideally, a high coercivity and a
high magnetization are desired for hard magnetic materials. This can best be
explained with the hysteresis loop (Fig. 1.1), a plot of the magnetization $M$
against the magnetic field strength $H$ [4, 5, 6].

Figure 1.1: Hysteresis loop for soft (left) and hard (hard) magnetic
materials. Important magnetic properties like the magnetization saturation
$M_{S}$ are marked and the maximum energy product |BH|max is is depicted as a
blue square in the second quadrant.

In the hysteresis loop, the magnetization starts at zero at point 1 and
increases with the magnetic field strength until the maximum at point 2 is
reached. This maximum is the magnetization saturation $M_{S}$ and describes
the maximum possible magnetization for a compound. In this process, the
magnetic domains in the material, small regions with the same direction of the
magnetic moments, are at first oriented in different directions to minimize
the magnetostatic energy. When the material reaches $M_{S}$, all the domain
walls move in the direction of the external field and align parallel to this
direction. Once the magnetic field $H$ is reduced to zero, it consequently
leads to a slight decrease in magnetization. At this point 3, the remaining
magnetization is called remanence magnetization $M_{r}$ with most of the
magnetic domains still lined up in the same direction and only a few having
changed back to their former state. The remanence magnetization $M_{r}$ for
hard magnetic materials is needed to be as large as possible as it describes
how much a magnet stays magnetized without an external field applied. In the
next step of the hysteresis, the magnetic field $H$ is further decreased until
the magnetization reaches a value of zero. This point 4 defines the coercivity
field strength $H_{c}$ and relates to the field strength needed to completely
demagnetize the material. At this point, an equal amount of magnetic domains
are aligned in two opposing directions and therefore cancel each other out. A
further reduction of the magnetic field strength followed by an increase back
to zero results in a mirrored version of the first two quadrants of this loop.
With the hysteresis loop the maximum energy product $|BH|_{max}$, a
quantitative measurement for the performance of a magnet, can be derived by
the area of a square in the second quadrant of the loop. With these
properties, the shape of the hysteresis loop for hard and soft magnetic
materials can be defined as it is shown in Fig. 1.1. For soft magnetic
materials, a small $|BH|_{max}$ is resulting from a narrow hysteresis loop due
to a low coercivity. This means that soft magnetic materials are very easy to
demagnetize. Hard magnetic materials on the other hand have a wide hysteresis
loop, therefore a high coercivity and a high $|BH|_{max}$ which results in a
high resistance to demagnetization.

Since the magnetic domains can react in a variety of ways to the appliance of
the external field $H$, different types of magnetism are possible. This
response can be used to classify materials into one of five different classes
of magnetism, namely diamagnetism, paramagnetism, ferromagnetism,
ferrimagnetism and antiferromagnetism as visible in Fig. 1.2. Note that
ferrimagnetism and antiferromagnetism are considered as subclasses of
ferromagnetism [5, 6].

Figure 1.2: Different magnetic domain configurations for the different types
of magnetism with and without an external field applied. The image is taken
from [7].

The first two types, dia- and paramagnetism, are the most common classes of
magnetism and only noticeable when an external field is applied. Diamagnetism
is a very weak form of magnetism and is found in most materials. Without an
external field, no magnetic moments can be attributed to the material. When an
external field is applied the magnetic moments are induced by the field and
aligned in the opposite direction of the field. This causes a weakening of the
magnetic field due to the opposite direction of the induced magnetic moments.
Diamagnetism is generally attributed to materials in which all electrons are
paired and hence have a total magnetic moment of zero.

With paramagnetism, the local magnetic moments in a material are all finite,
due to unpaired electrons, and independently leading in different directions
which results in an overall magnetic moment of zero. When an external field is
applied, like before with diamagnetism, the magnetic moments are realigned
into the same direction as the external field [5, 6].

Ferromagnetism has two additional subclasses, which are ferrimagnetism and
antiferromagnetism. In a ferromagnetic material, the local magnetic moments
are all linked and oriented parallel to each other. With an external field,
the direction of these linked local magnetic moments changes to the same
direction as the field. When the field is removed the magnetic moments stay in
the induced direction which leads to the macroscopic effect of magnetism. The
best-known material for this type of magnetism is, as the name already
suggests, iron.

In the subclass of antiferromagnetism instead of one sublattice with linked
magnetic moments, there are two sublattices alternating with each other. The
magnetic moments of these lattices are identical in their order to each other
but are aligned in opposing directions. This means that the magnetic moments
cancel each other out and a total magnetic moment of zero is the result [5,
6].

The subclass of ferrimagnetism shows a similar case to antiferromagnetism as
the magnetic moments are linked to each other in two alternating magnetic
sublattices directed in opposing directions. Nevertheless, the difference from
antiferromagnetism is the magnetic moments of the sublattices are not equal to
each other and do not cancel each other out completely. This means that
ferrimagnetic materials still have a net magnetic moment similar to
ferromagnets but are considerably weaker. This subclass is often seen in
materials consisting of different atoms such as ferrites where each atom
species provides one of the sublattices [5, 6].

Figure 1.3: a) Change of the magnetic ordering from ferro- to paramagnetism
with increasing temperature up to the Curie temperature $T_{C}$; b) Finite
temperature magnetization curves for the elements Fe, Co and Ni up to their
$T_{C}$’s of 1044, 1388 and 628 K. Image b) is based on data from [6].

With the knowledge of the different types of magnetism and magnetization,
which is a result of the realignment of the magnetic domains, it is now
possible to explain the influence of temperature on the strength of
magnetization in ferromagnetic materials. At low temperatures, the magnetic
domains in a ferromagnet are strictly ordered and aligned in the same
direction as explained before. As magnetic ordering is a thermodynamic process
an increase in the temperature causes the magnetic order to get slowly
disrupted until the order is completely lost (see Fig. 1.3 a)) This results in
a transition from a ferromagnetic material at low temperatures to a
paramagnetic material above a certain temperature called the Curie temperature
$T_{C}$. Above $T_{C}$ the total magnetization is zero, which means that
$T_{C}$ is an important intrinsic magnetic property that has to be taken into
account for the possible field of application. A magnet with a low $T_{C}$ can
not be utilized in various industrial fields, which need high operation
temperatures. In Fig. 1.3 b) the finite temperature magnetization of the
metals Fe, Co and Ni are shown with their respective $T_{C}$’s of 1044, 1388
and 628 K [6].

Figure 1.4: Unequal energies for the different magnetization directions. Image
adapted from [8].

As mentioned above, a better understanding of coercivity was a big milestone
in the evolution of magnets as it enabled the manufacturing of magnets with a
higher coercivity $H_{c}$ than their magnetization $M_{S}$ and a break of the
shape restrictions. A key part of this coercivity is the magnetocrystalline
anisotropy energy (MAE). In a ferromagnet, the magnetization direction of the
magnetic domains usually aligns through one or more easy axes that represent
the energetically most favored directions. This implies that the magnetization
direction plays an important role as the energy needed to realign the domains
is not the same for each direction (see Fig. 1.4). With only one easy
magnetization direction in a compound, the anisotropy is called uniaxial [4,
6]. For the ThMn12-type compounds, considered in this work, only one easy axis
and therefore uniaxial anisotropy is reported [9].

Since the origin of the magnetism lies within the motion of the electrons and
therefore in the circulating electron currents, the energy for a certain
magnetization distribution in the direction r $M(r)$ is the same as for the
opposing direction along the same axis ($M(r)=-M(r)$). This implies that the
energy is different for deviations from the easy axis with an energy peak at
an angle of $\theta=90^{\circ}$ as illustrated in Fig. 1.5.

Figure 1.5: Change of the magnetic energy in relationship to the angle between
the easy axis and the magnetization direction with a peak at an angle of 90°.

In the case of uniaxial anisotropy the energy difference $\Delta E_{MAE}$
resulting for each angle differing from the easy magnetization axis can be
expressed with the following equation [10]

$\Delta E_{MAE}=K_{1}\cdot sin(\theta)^{2},$ (1.1)

with $\theta$ as the angle between the direction of the magnetization M and
the easy axis and $K_{1}$ as the anisotropy constant. Normally the anisotropy
energy is described by more than one constant ($K_{2}$, $K_{3}$) but with
uniaxial anisotropy $K_{1}$ alone is sufficient as all higher constants are at
least 2 orders smaller than $K_{1}$. Both $E_{MAE}$ and $K_{1}$ are measured
in J/m3, ranging from low values below 1 kJ/m3 up to more than 10 MJ/m3. Most
commonly both values are expressed in MJ/m3 [4].

Another property influenced by the anisotropy energy, and therefore by the
anisotropy constant, is the anisotropy field $H_{a}$. As seen in the
hysteresis loop, given in Fig. 1.1, the anisotropy field $H_{a}$ has an upper
limit represented by the coercivity field $H_{c}$ ($H_{a}\leq H_{c}$) as this
point marks the complete demagnetization of the material. The connection of
the anisotropy constant $K_{1}$ to the anisotropy field $H_{a}$ can be
expressed as [10]

$H_{a}=\frac{2K_{1}}{\mu_{0}M_{S}},$ (1.2)

with the magnetic constant $\mu_{0}=4\pi\times 10^{-7}$ J/(A2m).

The suitability of a magnetic material to serve as a hard magnet can be
determined with the anisotropy constant $K_{1}$ and the magnetization $M_{S}$
by considering the magnetic hardness factor $\kappa$ as

$\kappa=\sqrt{\frac{K_{1}}{\mu_{0}M_{S}^{2}}}.$ (1.3)

The magnetic hardness factor $\kappa$ defines whether a magnet is accepted as
a soft ($\kappa<0.1$), a semi-hard ($0.1<\kappa<1.0$) or a hard magnet
($\kappa>1.0$). The most prominent magnet Nd2Fe14B is attributed with a
hardness factor of 1.54 (see Tab. 1.1).

Table 1.1: Curie temperature $T_{C}$, magnetization $\mu_{0}$MS, anisotropy constant $K_{1}$ and hardness factor $\kappa$ for common hard magnets. Data is taken from [11].  Magnet | Tc (K) |  $\mu_{0}$MS (T) |  $K_{1}$ (MJ/m3) | $\kappa$
---|---|---|---|---
Nd2Fe14B | 588 | 1.61 | 4.9 | 1.54
Sm2Co17 | 1190 | 1.21 | 4.2 | 1.89
SmCo5 | 1020 | 1.05 | 17.0 | 4.40
Alnico 5 | 1210 | 1.40 | 0.32 | 0.45

Magnets composing of RE and TM elements, such as Nd2Fe14B, have in most cases
a high hardness factor $\kappa$ due to a high anisotropy and a high
magnetization. The high anisotropy can be attributed to the RE-elements with
their 4f-electrons. The high magnetization of these magnets on the other hand
mostly belongs to the TM elements, foremost to elements like Fe or Co. The
hardness factors $\kappa$ of a few common magnetic materials are listed in
Tab. 1.1 in addition to some intrinsic magnetic properties such as Curie
temperature $T_{C}$, saturation magnetization $M_{S}$ and magnetocrystalline
anisotropy constant $K_{1}$.

###  1.2 Historical Development of Magnets

The first discovered magnetic materials were naturally magnetized iron pieces
called lodestone. It is assumed that these iron pieces were magnetized by huge
electrical currents in lightning strikes and were already known by the people
in ancient Greece and China since the 4${}^{\text{th}}$ to 6${}^{\text{th}}$
century. First devices using these iron pieces were compasses and a tool named
a "South pointer". The South pointer was used in China for geomancy and
utilized a lodestone carved into a spoon that turned on a small base to align
its handle with the earth’s magnetic field. It was used to design the grid-
like street plans of old Chinese towns. The invention of the first
navigational compasses around 1088 by Shen Kua and a century later in Europe
(first European reference around 1157-1217 by Alexander Neckham [5]) made
greater explorations at sea possible and led to the discovery of America in
1492 by Christopher Columbus or the voyage around the world by Ferdinand
Magellan from 1519 to 1522 [4].

In the middle ages many superstitions were attributed to magnets like the
possibility of a perpetuum mobile and magnetic levitation, of which the latter
was actually achieved at the end of the 20${}^{\text{th}}$ century. In 1269
the french crusader Petrus Peregrinus de Maricourt was the first in Europe to
give a detailed description of a floating compass and also the first one to
discover the magnetic lines and poles of a magnet. Based on this work William
Gilbert extended in his monograph De Magnete in 1600 the idea of Peregrinus
and came to the conclusion that the alignment of free-rotating magnets into a
particular direction is due to the earth itself being one big magnet [5].

After this the next noteworthy development with magnetic materials appeared in
1743, when Daniel Bernoulli invented the first horseshoe magnet that did not
suffer from demagnetization due to its own internal field. Magnets in this
U-shape are even today one of the most commonly applied symbols for magnetism
[4, 5]. The next milestone in the history of magnets was the connection of
electricity and magnetism found by Hans-Christian Oerstedt in 1820. On basis
of his discovery Faraday found the phenomenon of magnetic induction in 1821
and also the connection of light and magnetism with the magneto-optic Faraday
effect in 1845. The results of both, Oerstedt and Faraday, motivated James
Clerk Maxwell in the middle of the 19${}^{\text{th}}$ century to the
formulation of the Maxwell equations which laid the theoretical foundation for
the further development of hard magnets in the future [4, 5].

However, until the early 20${}^{\text{th}}$ century almost no further
improvement on permanent magnets was made due to the aforementioned problem of
the demagnetization and the resulting shape restrictions. It is therefore no
surprise that the scientific focus of that time was mainly on electromagnetism
as it seemed far more effective and enabled inventions like the first
electrical motors. Nevertheless, the development of hard magnets was at a
breaking point and each developed type of magnet was further improved with a
better performing one in a short period of time. This evolution in the
beginning of the 20${}^{\text{th}}$ century up to the start of the
21${}^{\text{st}}$ century is shown in Fig. 1.6.

Figure 1.6: Development of the different types of magnets in the
20${}^{\text{th}}$ century and the beginning of the 21${}^{\text{st}}$
century. The volume needed of each magnetic material to achieve the same
magnetic strength is shown under the diagram. Image taken from [12].

The first developed magnets in the 20${}^{\text{th}}$ century were steel
magnets [13] coming up around 1917 with a very low maximum energy product
$|BH|_{max}$ of only 8 kJ/m³ (1 MGOe). These magnets were improved using W and
Cr together with Fe-C alloys to suppress the domain wall movement by treatment
under appropriate heat. Later the addition of Co led to an increase of the
energy product and the focus of attention was then shifted generally to Co
alloys and the resulting Alnico compounds [14] around 1940.

These compounds, as the name suggests, consist of aluminum (Al) nickel (Ni)
and cobalt (Co) and offered much better magnetic properties than the magnetic
steels with a high Curie temperature $T_{C}$, a high remanent force, a good
resistance against corrosion in addition with a great reduction of the
material volume needed to achieve the same strength of the magnetic
properties, as can be seen in Fig. 1.6. A downside of these compounds is a low
coercivity [15] and therefore an easy demagnetization of these magnets.

Another group of magnets developed around the 1930s and becoming popular in
the 1950s were the ferrite magnets [16], also called ceramic magnets,
consisting mainly of ferric oxide $\alpha$-Fe2O3. The reason they gained
popularity was the invention of hard hexagonal ferrites together with their
extremely low production costs, due to easy production steps, cheap raw
materials and a good corrosion resistance. Even today, they are still used as
the second highly preferred magnet in the market [17] because of their low
costs although they have a comparably low energy product of 25 kJ/m3, which is
less than the more expensive alnico magnets have. The main drawback of the
ferrite magnets are their brittleness, which makes it harder to process them
into certain shapes, and a low mechanical strength.

In the mid of the 1960s a breakthrough took place in the historical
development of permanent magnets by using RE-TM consisting magnets. The first
magnet of this group was the YCo5 compound [18], and it showed that the
combination of RE and TM provides superior magnetic properties such as high
$T_{C}$ and high magnetization, attributed to the TM elements, and high
anisotropy due to the RE elements. The soon after discovered SmCo5 compound
has an energy product of 200 kJ/m3 (22.5 MGOe), which is a significant
increase compared to the alnico 5 compound with 60 kJ/m3 (10 MGOe). This
improvement led to an further decrease by about a factor of 2 of the needed
material volume to achieve a specific magnetic strength compared to the alnico
magnets. These Sm-Co compounds were further refined with the Sm2Co17 phase (it
is also called 2:17 phase) that enhanced the $|BH|_{max}$ to a value of 294
kJ/m³. This group of magnets is still used today as high performance magnets
in motors or other complex applications due to their high $T_{C}$ and high
$M_{S}$. However, the usage of these magnets is rather limited due to the high
costs of the Sm and Co elements, which is also the greatest disadvantage of
these magnets.

After the Sm-based magnets, the strongest type of RE-TM-magnets with the
Nd2Fe14B compounds were invented. These compounds were independently reported
in 1984 for the first time by Sagawa et al. [19] and Croat et al. [20] with
the $|BH|_{max}$ of 286 kJ/m³ (36 MGOe). As a result of the high costs for the
elements Sm and Co the Nd2Fe14B compounds were preferred for many applications
that do not require a high $T_{C}$. Main advantage of these materials is a
high energy product that was improved to a value of more than 500 kJ/m3. The
greatest disadvantage of the Nd2Fe14B magnets is their low $T_{C}$ of only 585
K, which makes it impossible to use these magnets in applications with high
working temperatures. This drawback can be treated by the addition of
expensive and as critically labeled elements like Co or Dy.

The last depicted magnet type in Fig. 1.6 are the Sm-Fe-N magnets which are
still in focus of research today. They exhibit promising properties such as a
high $T_{C}$, a high magnetization $M_{S}$ and a high resistance to
demagnetization.

###  1.3 World Market and Usage of Magnets

In the last decade, the demand for hard magnets has steadily increased as
magnets play an essential role in an emerging number of industrial
applications [1]. In Fig. 1.7 the percentual application of magnets in
different technological fields is shown. The largest application field of
permanent magnets is appeared as motors with 45 %, which also yields the
importance of the motor technology as well. The mentioned trend of an increase
can also be seen in Fig. 1.8 by the world production of sintered Nd-Fe-B
magnets from 1984 to 2016 for selected countries. Since the start of the
21${}^{\text{st}}$ century the production of these hard magnets has increased
significantly from roughly 20000 tons in the year 2000 to around 120000 tons
in 2016, thus increasing to 6 times the amount than before. Further, it has to
be noted that Fig. 1.8 only shows the production of sintered Nd-Fe-B magnets
and does not include other types of magnets like hard ferrites or Sm-based
ones.

Figure 1.7: Application of permanent magnets by market share in 2019. The
image is based on data from [21].  Figure 1.8: Worldwide production of
sintered Nd-Fe-B magnets from 1984 to 2017. The image is based on data from
[22].

The percentage of sales of Nd-Fe-B magnets has to be included to get a broader
and complete picture of the increasing demand for hard magnets. In Fig. 1.9,
the percentage sales of the different types of magnets are shown. With a
majority of 62 % the Nd-Fe-B magnet takes the biggest share followed
surprisingly by the hard ferrite magnets. However, magnetic steels and Alnico
magnets do not have sufficiently enough magnetic properties to reach the
requirements for most industrial applications, and the strong Sm-based magnets
are too expensive for a broad utilisation. This results in only small shares
of these magnets on the general sales.

Figure 1.9: Percentage sales of major permanent magnet alloy types in 2020 in the global market. The image is based on data from [17] Table 1.2: Production of RE-oxides and available reserves for different countries. Data is taken from [23].  Country |  | Production
---
in 2016 [t]
| Production
---
in 2017 [t]
| Production
---
in 2018 [t]
| Production
---
in 2019 [t]
| Production
---
in 2020 [t]
| Reserves
---
in 2020 [t]
China | 105.000 | 105.000 | 120.000 | 132.000 | 140.000 | 44.000.000
United States | 0 | 0 | 18.000 | 26.000 | 39.000 | 1.400.000
India | 1.700 | 1.500 | 2.900 | 3.000 | 2.900 | 6.900.000
Australia | 14.000 | 20.000 | 21.000 | 21.000 | 21.000 | 4.000.000
Russia | 3.000 | 3.000 | 2.700 | 2.700 | 2.700 | 21.000.000
Malaysia | 300 | 300 | 0 | 0 | 0 | n.a.
Brazil | 1.100 | 2.000 | 1.100 | 1.000 | 600 | 21.000.000
Thailand | 800 | 1.600 | 1.000 | 1.800 | 3.600 | n.a.
Vietnam | 300 | 100 | 920 | 920 | 700 | 22.000.000
Myanmar | 0 | 0 | 19.000 | 25.000 | 31.000 | n.a

As demonstrated before the RE-TM magnets, in particular Nd-Fe-B magnets make
up the majority of hard magnets used in industrial processes, the importance
of a steady supply of the needed resources for these magnets has a fundamental
importance. This marks a crucial problem associated with RE elements as most
of them like Nd, Dy, Sm or Tb are labeled as critical elements for future
applications [24, 25]. Another downside with the RE elements lies within the
available global reserves as a majority of them are located in China, as
visible in Tab. 1.2. In addition to the high amount of reserves, China also
produced around 60% of the world RE elements (measured in RE-oxides) in 2020.
With this monopoly in the world market, China has a significant impact on the
market prices of RE elements. This was experienced in 2009, when the prices
changed by a factor of up to ten due to export quotas, export tariffs and
other restrictions implemented by the Chinese government. This monopoly was
also used as leverage against the United States in 2019, when tensions between
both countries arose due to trade restrictions by the US. As a response to
China’s position in the global market, some countries such as the US have
started to raise their production of RE elements in the last years, as given
in Tab. 1.2.

##  Chapter 2 Case Description and Motivation of the Work

As mentioned in Sec. 1.3 there is an increasing demand for affordable and
resource-efficient hard magnets in rising application fields such as renewable
energies and electrical transportation [1]. Most of the currently applied hard
magnets consist of RE elements together with 3d TM elements, with Nd2Fe14B as
the strongest magnet at the moment [20, 19].

However, as mentioned before, most of the RE elements used in hard magnetic
materials such as Nd, Dy, Sm and Tb are often considered to be critical
elements for future applications [24, 25]. In addition, given the increasing
demand in combination with high prices for RE-elements and the supply monopoly
in the global market (Tab. 1.2), there are considerable efforts to design new
and alternative RE-lean and RE-free hard magnetic materials [2, 3].

In order to overcome this challenge, several approaches have been proposed in
the literature. First, instead of using critical Dy, coercivity enhancement
has been achieved by grain size reduction [26, 27]. Second, the grain boundary
diffusion process has been performed, which eliminates Dy and Tb-like critical
materials and enhances coercivity substantially [28, 29]. Third, the critical
RE component has been partially substituted by a more abundant element that
offers a low-cost alternative product for low to moderate-temperature
applications. As a potential candidate Ce has been investigated intensively
[30, 31].

To this extent, this work is focused on the last strategy, developing RE-lean
permanent magnets based on the ThMn12 prototype structure (will be referred as
the 1:12 phase). Note that the composition ratio of RE:Fe=1:12 has a lower RE
content than Nd2Fe14B magnets and promising magnetic properties [32, 33, 34].
In a work by Sözen et al., ab initio-based thermodynamic phase stabilities of
such RE-lean magnets have been reported [35]. Abundant and inexpensive Ce and
Y have been chosen for a potential substitution of the critical Nd and their
impact on the stability of the 1:12 phase has been discussed for Nd-Fe-Ti
compounds.

In that study, state-of-the-art approaches for vibrational, electronic and
magnetic entropy contributions of the Helmholtz free energy, $F$($T,V$), have
been calculated for all hard magnetic and relevant competing phases.
Competition energy formalisms have been developed for each ternary and
quaternary system to understand the relative finite temperature phase
stabilities. It has been revealed that Ce is a promising candidate to
compensate Nd with a composition of (Nd,Ce)Fe11Ti, which have appeared as
stable quaternarys at all considered temperatures from 0 to 1500 K (Fig. 2.1).
For further details about competition energy formalisms it is advised to read
Ref. [35] as competition energies are not a focus of this thesis.

Figure 2.1: Calculated competition energies of NdFe12 alloys with increasing
concentrations of Ti and Ce or Y with three different temperatures 0, 750 and
1500 K. The calculations for the (Nd,X)Fe12-yTiy alloys (X = Y and Ce) were
performed in 2 formula units with 26 atoms supercells in the limits of
$0<x<0.5$ and $0<y<1$. Linear interpolation has been performed between the
calculation points that correspond to intermetallic compositions. Images are
taken from [35].

Nevertheless, the effect of such substitutions on the intrinsic magnetic
properties was not addressed before. In order to fill this gap and produce
sustainable and reliable hard magnetic materials with reduced Nd content,
detailed ab initio-based investigations on the intrinsic magnetic properties
of (Nd,X)-Fe-Ti-N alloys (X= Y and Ce) have been performed in this thesis. In
order to have a systematic understanding, the intrinsic magnetic properties
have been calculated starting from RFe12 (R: Y, Ce and Nd) binaries. Since,
the 1:12 phase is thermodynamically not stable for the considered RE-elements
[36, 37, 38], Ti is considered as a stabilizer in different concentrations for
ternary RFe12-yTiy ($0.5\leq y\leq 1$) compounds. Then as quaternary
(Nd,X)Fe12-yTiy compounds, half of the Nd atoms have been replaced with
alternative elements, which are Y and Ce, respectively. In total this makes 15
different ThMn12-type Nd-X-Fe-Ti compounds and in addition, their nitrogenated
cases have been examined as well.

##  Chapter 3 Theoretical Background

In order to investigate the intrinsic magnetic properties of the considered
compounds in this study, ab initio calculations have been performed. The main
goal is to achieve precise and accurate magnetic properties providing an
atomistic understanding. In each calculated elastic and magnetic material,
properties have been computed and compared with available experimental data.

In this chapter the theoretical fundamentals of the used methods are
introduced. First of all, density functional theory (DFT) is explained as it
is the core method in this thesis. Furthermore, fundamental points of quantum
mechanics are explained, and a focus is put on the treatment of magnetism and
f-electrons in DFT. Due to the limitations of DFT regarding localized
electrons, a DFT+U approach is also considered in this study, and its
technical aspects are summarised.

###  3.1 Density Functional Theory

Density functional theory (DFT) is one of the most used quantum mechanical
methods to describe the electronic structure of many-body systems in chemistry
and physics. It is based on the Hohenberg-Kohn theorems [39] postulated in
1964 by Hohenberg and Kohn. As Hartree-Fock (HF) and post-Hartee-Fock (post-
HF) methods, DFT describes the interactions of electrons based on quantum
mechanical approaches using universal principles such as the Schrödinger
equation. These kinds of calculations are called first-principle or ab initio
methods as they do not require additional experimental data with the selected
input.[40, 41]

Contrary to the Hartree-Fock and post-Hartree-Fock methods, which use a Slater
determinant, a determinant of single-electron wave functions to describe the
many-electron wave function, DFT relies on the electron density of the
investigated system to express the ground state properties of it. The many-
body wave functions with 3N variables are reduced to the electron density that
only depends on 3 variables and is therefore much easier to handle. This
approach by Hohenberg and Kohn was improved by Kohn and Sham with the Kohn-
Sham equations [42] only one year later. They partially reintroduced the
description via a wave function and mapped the many-body problem onto a system
of non-interacting quasiparticles, thus simplifying the multi-electron problem
into a problem of non-interacting electrons in an effective potential [7]. In
this potential, the external potential and the Coulomb interactions between
the electrons are included. This approach is the basis of current DFT methods
[43]. Since then, the popularity of DFT increased steadily as can be seen in
Fig. 3.1. This can mostly be attributed to the development of better and more
precise functionals. A major acknowledgment of DFT was the awarded Nobel Prize
in chemistry in 1998 by Walter Kohn and John Pople.

Figure 3.1: Increasing number of publications using DFT. The Pople endorsement
and the Nobel prize in chemistry are marked according to the appropriate year.
The image is based on data from [44]

###  3.2 Schrödinger’s Equation

As mentioned above, ab initio methods are independent of experimental data and
they only rely on the Schrödinger equation. This equation is the starting
point of quantum mechanical understanding of a given material system [7]. The
Schrödinger equation forms the basis of many ab initio approaches and its time
independent, non-relativistic state is defined as follows

$\hat{H}\Psi_{i}(x_{1},x_{2},...x_{N},R_{1},R_{2},...,R_{M})=E_{i}\Psi_{i}(x_{1},x_{2},...x_{N},R_{1},R_{2},...,R_{M}),$
(3.1)

with $\hat{H}$ as the Hamiltonian for a molecular system consisting of M
nuclei and N electrons in the absence of magnetic or electric fields. This
many-body Hamiltonian is a differential operator that represents the total
energy with the following expression

$\hat{H}=\hat{T}_{e}+\hat{T}_{n}+\hat{V}_{nn}+\hat{V}_{en}+\hat{V}_{ee}.$
(3.2)

In this equation, $\hat{T}_{e}$ is the kinetic energy of the electrons, and
$\hat{T}_{n}$ is the kinetic energy of the nuclei. $\hat{V}_{nn}$,
$\hat{V}_{en}$ and $\hat{V}_{ee}$ describe the electrostatic energy of
attraction and repulsion for the nuclei-nuclei, electron-nuclei and electron-
electron interactions. When using atomic units the Hamiltonian has the
following form

$\hat{H}=-\frac{1}{2}\sum_{i=1}^{N}\nabla_{i}^{2}-\frac{1}{2}\sum_{A=1}^{M}\frac{1}{M_{A}}\nabla_{A}^{2}-\sum_{i=1}^{N}\sum_{A=1}^{M}\frac{Z_{A}}{r_{iA}}+\sum_{i=1}^{N}\sum_{j>i}^{N}\frac{1}{r_{ij}}+\sum_{A=1}^{M}\sum_{B>A}^{M}\frac{Z_{A}Z_{B}}{R_{AB}},$
(3.3)

with $i$ and $j$ marking the $N$ electrons and $A$ and $B$ denoting the $M$
nuclei. $M_{A}$ represents the mass of nucleus A, and Z as the charge of the
respective nucleus [41].

###  3.3 Born-Oppenheimer Approximation

Due to the complexity of the Schrödinger equation, an exact solution for most
systems of interest is impossible. To reduce the complexity and make solutions
possible, some approximations and methods were developed. The first and most
prominent approximation is the Born-Oppenheimer approximation [45], which
separates the electronic and ionic degrees of freedom from each other. This
assumption is based on the much higher velocity of the electrons due to their
lower weight in comparison to the nucleus (as an example, in the case of H the
nucleus weighs $\sim$ 1800 times more than its electron). This leads to an
approximation in which the cores are seen as stationary. With this, the
kinetic energy of the nuclei is now zero and the potential energy of the
nucleus-nucleus repulsion is a constant. This reduces the complex Hamiltonian
to the electronic Hamiltonian $\hat{H}_{elec}$:

$\hat{H}_{elec}=\hat{T}_{e}+\hat{V}_{en}+\hat{V}_{ee}=-\frac{1}{2}\sum_{i=1}^{N}\nabla_{i}^{2}-\sum_{i=1}^{N}\sum_{A=1}^{M}\frac{Z_{A}}{r_{iA}}+\sum_{i=1}^{N}\sum_{j>i}^{N}\frac{1}{r_{ij}}.$
(3.4)

Solving the Schrödinger equation with $\hat{H}_{elec}$ leads consequently to
the electronic wave function $\Psi_{elec}$ and the electronic energy
$E_{elec}$. Since $\Psi_{elec}$ only depends on the electronic coordinates and
the coordinates of the nuclei are only entered parametrically and do not
appear explicitly, the total energy $E_{tot}$ is a sum of the electronic
energy $E_{elec}$ and the constant nuclear repulsion term
$E_{nuc}=\sum_{A=1}^{M}\sum_{B>A}^{M}\frac{Z_{A}Z_{B}}{R_{AB}}$ [41].

With the total energy of the ground state of an electronic system calculated
via Eq. 3.4, the ground state energy $E_{0}=E[N,V_{ext}]$ and the ground state
wave function $\psi_{0}$ can be determined with the variational principle for
a system with $N$ electrons and a given nuclear potential $V_{ext}$. A full
minimization of $E[\Psi]$ will then lead to the true ground state $\Psi_{0}$

$E_{0}=\underset{\Psi\rightarrow
N}{\text{min}}E[\Psi]=\underset{\Psi\rightarrow
N}{\text{min}}\langle\Psi|\hat{T}_{e}+\hat{V}_{Ne}+\hat{V}_{ee}|\Psi\rangle.$
(3.5)

###  3.4 The Hohenberg-Kohn Theorem

Even though the Born-Oppenheimer approximation simplifies the Hamiltonian to a
great amount, it is still too complicated to be solved. Especially the
$V_{ee}$ term with a 3N dimensionality turns out to be a challenging part. To
solve these problems Hohenberg and Kohn postulated the Hohenberg-Kohn theorems
[39] in 1964 and reduced the many-body wave function to the electron density
$\rho(r)$, which represents the fundamental base for DFT[40, 41].

In their first theorem, they proved that since the electron density $\rho$
determines the number of electrons in a system, it also defines with $\rho(r)$
the ground state wave function $\Psi$ and every other electronic property. In
a relatively simple way, they showed this by employing only the minimum-energy
principle for the ground state.

They considered the electron density $\rho(r)$ for the nondegenerate ground
state of N-electron systems and defined N by quadrature. With the electron
density, the external potential $\nu(r)$ and all other properties were also
defined. When two external potentials $\nu$ and $\nu^{\prime}$ are differing
by more than a constant and each one gives the same density $\rho$ for its
ground state then one could have two different Hamiltonians $\hat{H}$ and
$\hat{H}^{\prime}$ with the same ground state density but with different wave
functions $\Psi$ and $\Psi^{\prime}$. Trying to solve the $\hat{H}$ problem
with $\Psi^{\prime}$ as a trial function and also the opposite way with
$\hat{H}^{\prime}$ and $\Psi$ one could get the following expressions

$E_{0}<\langle\Psi^{\prime}|\hat{H}|\Psi^{\prime}\rangle=\langle\Psi^{\prime}|\hat{H}^{\prime}|\Psi^{\prime}\rangle+\langle\Psi^{\prime}|\hat{H}-\hat{H}^{\prime}|\Psi^{\prime}\rangle=E_{0}^{\prime}+\int\rho(r)[\nu(r)-\nu^{\prime}(r)]\,dr,$
(3.6)

and

$E_{0}^{\prime}<\langle\Psi|\hat{H}^{\prime}|\Psi\rangle=\langle\Psi|\hat{H}|\Psi\rangle+\langle\Psi|\hat{H}^{\prime}-\hat{H}|\Psi\rangle=E_{0}-\int\rho(r)[\nu(r)-\nu^{\prime}(r)]\,dr,$
(3.7)

where $E_{0}$ and $E_{0}^{\prime}$ are the ground state energies for $\hat{H}$
and $\hat{H}^{\prime}$. Adding both equations the contradiction
$E_{0}+E_{0}^{\prime}<E_{0}^{\prime}+E_{0}$ is obtained, which indicates that
there cannot be two different $\nu$ that give the same electron density for
their ground states. With this they proved that the electron density
determines $N$ and $\nu$ and all following ground state properties like the
kinetic energy $T(\rho)$, the potential energy $V(\rho)$ and the total energy
$E(\rho)$ [40, 41].

In their second theorem, Hohenberg and Kohn introduced the variational
principle in the following form

$E_{0}\leq
E(\tilde{\rho})=T(\tilde{\rho})+V_{Ne}(\tilde{\rho})+V_{ee}(\tilde{\rho}),$
(3.8)

with $\tilde{\rho}(r)$ as any trial density that satisfies the needed boundary
conditions such as $\tilde{\rho}(r)\geq 0$ and $\int\tilde{\rho}(r)\,dr=N$.
This means that the minimum energy for the system can only be attained if the
correct ground state density is used. Every other trial density
$\tilde{\rho}(r)$ results in higher energies than the ground state energy
which makes it possible to determine the ground state density by the variation
of $\tilde{\rho}(r)$ until a minimum is reached [40, 41]. This way we arrive
at the following equation

$\langle\tilde{\Psi}|\hat{H}|\tilde{\Psi}\rangle=T[\tilde{\rho}]+V_{ee}[\tilde{\rho}]+\int\tilde{\rho}(r)\nu_{ext}\,dr=E[\tilde{\rho}]\geq
E_{0}[\tilde{\rho}_{0}]=\langle\Psi_{0}|\hat{H}|\Psi_{0}\rangle.$ (3.9)

Additionally, the kinetic energy $T(\rho)$ and the Coulomb interaction energy
$V_{ee}(\rho)$ can be summarized in the Hohenberg-Kohn functional
$F_{HK}(\rho)$

$F_{HK}(\rho)=T(\rho)+V_{ee}(\rho).$ (3.10)

Furthermore, the Coulomb interaction energy $V_{ee}(\rho)$ in this assumption
includes the classical repulsion term $J(\rho)$ and a non-classical exchange
term $K_{ex}(\rho)$.

$V_{ee}(\rho)=J(\rho)+K_{ex}(\rho).$ (3.11)

###  3.5 The Kohn-Sham Equations

However, Eq. 3.9 is not providing an exact solution. Therefore, Kohn and Sham
reformulated the current approach in 1965 [42] and introduced a new scheme by
mapping the fully interacting electronic system onto a fictitious system of
non-interacting quasi particles moving in an effective potential. The Kohn-
Sham (KS) equations can be reformulated as

$\hat{H}_{KS}\psi_{i}=\epsilon_{i}\psi_{i},$ (3.12)

with $\epsilon_{i}$ as the orbital energy of the corresponding KS orbital
$\psi_{i}$ and the KS Hamiltonian defined as

$\hat{H}_{KS}=[-\frac{1}{2}\nabla^{2}+V_{eff}(r)],$ (3.13)

with $V_{eff}(r)$ as an effective local potential. Therefore, the problem of
finding a solution to the many-body Schrödinger equation is now replaced by
solving single particle equations. Since the KS Hamiltonian is a functional of
just one electron at point $r$, the electron density can be defined according
to the HK theorem.

$\rho(r)=\sum_{i=1}^{N}|\psi_{i}(r)|^{2}.$ (3.14)

Besides, the kinetic energy term and the classical Coulomb interaction energy
of the electrons can be defined as

$T_{e}=-\frac{1}{2}\sum_{i=1}^{N}\int\,d^{3}r|\nabla\psi_{i}(r)|^{2},$ (3.15)
$J(\rho)=\frac{1}{2}\int\,d^{3}rd^{3}r^{\prime}\frac{\rho(r)\rho(r^{\prime})}{|r-r^{\prime}|}.$
(3.16)

Then, the Hohenberg-Kohn ground state energy can be written according to the
Kohn-Sham approach

$E_{KS}=\sum_{i}^{N}\epsilon_{i}-J(\rho)+E_{xc}-\int\frac{\delta
E_{xc}}{\delta\rho(r)}.$ (3.17)

$\epsilon_{i}$ are the one electron energies and are a result of the KS
equations. However, they have limited physical meaning. The last term in Eq.
3.17 is the exchange-correlation $E_{xc}$ term, which contains all many-body
interactions of exchange and a substantial part of the electron interactions
(besides the Hartree term)[7]. The exchange-correlation $E_{xc}$ can therefore
be defined as

$E_{xc}=T_{exact}(\rho)-T_{S}(\rho)+V_{ee}(\rho)-J(\rho)=T_{exact}(\rho)-T_{S}(\rho)+K_{ex}(\rho).$
(3.18)

This means that $E_{xc}$ is the sum of the difference between the exact
kinetic energy of the interacting system $T_{exact}(\rho)$ and the kinetic
energy of the non-interacting system $T_{S}(\rho)$, achieved by the Kohn-Sham
approach, and the non-classical exchange part $K_{ex}(\rho)$ of the Coulomb
interaction energy between the electrons $V_{ee}(\rho)$. Since $E_{xc}$
includes the unknown terms of $T_{exact}(\rho)$ and $K(\rho)$ the exchange-
correlation energy can not be calculated exactly[40, 41].

###  3.6 Exchange-Correlation Energy

As explained with the Kohn-Sham equations, the exchange-correlation functional
can not be calculated exactly and has to be approximated. For this
approximation different methods with significantly differing quality depending
on the actual system and their level of complexity have been developed. The
accuracy of these functionals is often correlating with the computational cost
and thus a functional that treats the exchange part correctly is also
computationally very demanding. Thus, one has to find a reasonable balance
between the accuracy needed and the computational demand for the calculation.
In 2001 Perdew et al. [46] introduced the Jacob’s ladder, as seen in Fig. 3.2,
which is a useful way to categorize various exchange-correlation functionals.

Figure 3.2: Jacob’s ladder showing the different functionals (LDA, GGA, Meta
GGA, Hybrid GGA, Hybrid Meta GGA and Fully Non-Local) leading to increases of
the accuracy of calculations. The image is adapted from [47].

Each functional on this ladder is categorized according to its level of
accuracy on one of the steps. The "earth" of the ladder is representing the HF
theory and the "heaven" of the ladder is the exact exchange-correlation
functional. Therefore, the higher a functional is located on the ladder the
more precise and computationally demanding it is. The first two steps on the
ladder are the local density approximation (LDA) [48] and the generalized
gradient approximation (GGA) [49]. Both of these functionals have been used in
this work and are the most used functionals today.

###  3.7 Local Density Approximation

The first step on the Jacob’s ladder is the local density approximation (LDA).
Although it is considered as a functional with a low accuracy, it is virtually
the basis for all other exchange-correlation functionals. The core of this
approximation is the model of a uniform electron gas with the electrons moving
on a positive background charge distribution to achieve an electrically
neutral system. The number of electrons N as well as the volume of the
electron gas V are considered to approach infinity. The electron density N/V
on the other hand is assumed as finite with a constant value everywhere.

$E_{xc}^{LDA}=\int\rho(r)\epsilon_{xc}[\rho(r)]\,dr,$ (3.19)

with $\epsilon_{xc}[\rho(r)]$ as the exchange-correlation energy per particle
of a uniform electron gas of density $\rho(r)$. This energy per particle is
weighted with the probability $\rho(r)$ that there is in fact an electron at
this point in space.

This model physically resembles an idealized metal with a perfect crystal of
valence electrons and smeared out positive cores. For simple metals like
sodium or systems with only a small variation of the density this assumptions
is relatively good. However, for most situations in atoms and molecules, it is
far from reality as the density often varies in these cases rapidly. LDA is
the only functional in DFT for which the form of the exchange functional is
known to a very high accuracy [41, 40].

###  3.8 Local Spin Density Approximation

The local density approximation can furthermore be reformulated within an
unrestricted version. In this version, not the electron density $\rho(r)$ is
employed but the two spin densities $\rho_{\alpha}(r)$ and $\rho_{\beta}(r)$
(with $\rho_{\alpha}(r)+\rho_{\beta}(r)=\rho(r)$). For the functional, this
provides additional flexibility as it now holds two variables instead of one
thus enabling more accuracy in case of an unequal number of electrons for
$\alpha$ and $\beta$. This functional is known as the local-spin density
approximation (LSDA) [41, 40].

$E_{xc}^{LSDA}[\rho_{\alpha}\rho_{\beta}]=\int\rho(r)\epsilon_{xc}[\rho_{\alpha}(r),\rho_{\beta}(r)]\,dr.$
(3.20)

With LSDA there are expressions known for the exchange and correlation
energies for spin-compensated situations
($\rho_{\alpha}(r)=\rho_{\beta}(r)=\frac{1}{2}\rho(r)$ as well as spin-
polarized cases ($\rho_{\alpha}(r)\neq\rho_{\beta}(r)$.

Since LDA and LSDA are only the first steps on the Jacobs ladder, there are
several drawbacks to be expected from these functionals. LDA and LSDA tend to
under-predict the ground state energies as well as the ionization of systems
and to over-predict the binding energies. It has also to be noted that high
spin structures are often favored [41].

###  3.9 Generalized Gradient Approximation

Since the low accuracy of LDA and LSDA is insufficient for most applications
in chemistry, these functionals are rarely used in computational chemistry. To
account for the inhomogenity of the electron density, the gradient of it
$\nabla\rho(r)$ is considered instead of only the charge density at a point
$r$.

$E_{xc}^{GGA}[\rho]=\int\rho(r)\epsilon_{xc}(\rho(r),\nabla\rho)\,dr$ (3.21)

In the case of a generalized gradient approximation (GGA) functional [41, 40],
it takes a slowly varying electron density into account paired together with a
few restrictions to restore the hole constraints lost in comparison to LDA.
Generally, GGA delivers more accurate results than LDA, but in some cases, LDA
still works better. Commonly used GGA functionals are the functional by Perdew
and Wang (PW91) [50, 51] and the functional by Perdew, Burke and Ernzerhof
(PBE) [49].

The choice of the considered functional has a huge impact on the results,
therefore both the LDA (LSDA) and the GGA functional have been used in this
study to achieve the most accurate results and compare the performance of
functionals. As it was shown in works by Sözen et al. [7, 52, 53, 35] the
lattice parameters and magnetic properties like the total magnetic moment can
properly be calculated employing the PBE functional. However, in this work it
has been seen that the GGA approach fails for magnetic anisotropy
calculations, therefore, it has been additionally considered the LSDA
functional in these cases to compare both results.

###  3.10 Ultra-soft Pseudopotentials and the Projector- Augmented Wave Method

Apart from the calculation of the exchange-correlation energy $E_{xc}$ the
treatment of the nuclei-electron interactions plays an important role in DFT
as well. The most effective and best known methods are the projector-augmented
wave method (PAW) of Blöchl [54, 55] and the ultrasoft pseudopotentials (USPP)
method of Vanderbilt [56].

In these approaches, the effects of the motion of the core electrons is
replaced by an effective potential. In the USPP method a certain core radius
gets replaced by a soft nodeless pseudo wave function. This leads to a
reduction of the basis set size. It is important to note that the pseudo wave
function needs to have the same norm as the original wave function in the
determined core radius and has to be identical to it outside of this radius.
The drawback of this approach is that elements with strongly localized
electrons require a large basis set for these pseudopotentials and therefore
they are expensive to calculate. Another disadvantage is that the
pseudopotentials are hard to generate as many parameters have to be chosen
accordingly.

Figure 3.3: Radial dependence of typical pseudopotentials in comparison with
the atomic Coulomb potential (blue). In the lower part a soft-core
pseudopotential and a hard-core pseudopotential are shown with the solid and
dashed red curve respectively. In the top part are the wave functions and the
pseudo wave function resulting from the atomic (blue) and the pseudo-potential
(red) depicted. It has to be noted, that both wave functions have the same
values above a certain point r. The image is taken from [57]

In the PAW method, these short-comings were removed by Blöchl with the
introduction of a linear transformation from the pseudo wave function to the
many-electron wave function. He achieved the PAW total energy by applying this
transformation to the KS functional. Furthermore, Blöchl splitted the grid of
augmentation charges into a radial support grid and a regular grid. This makes
it possible to use the PAW method directly with the many-electron wave
function and the according many-electron potentials [7].

###  3.11 Magnetism in Density Functional Theory

Since the theorems postulated by Hohenberg and Kohn [42] are only developed
for spinless cases they are insufficient for the hard magnetic materials
discussed in this work. The formalism was extended by Barth and Hedin [58] for
spin polarized cases, and the generalization was done by a replacement of the
scalar density with a Hermitian 2 $\times$ 2 matrix. This density matrix
$n(r)$ can be defined as [59]

$n_{\alpha\beta}(r)=\sum_{i=1}^{N}\psi_{i}^{*\alpha}(r)\psi_{i}^{\beta}(r),$
(3.22)

with $\alpha,\beta=(+),(-)$. Decomposing this matrix further, a scalar and a
vectorial part corresponding to charge and magnetization densities with the
Pauli matrices can be written as

$n(r)=\frac{1}{2}(n(r)I+\sigma\cdot
m(r))=\frac{1}{2}\begin{pmatrix}n(r)+m_{z}(r)&M_{x}(r)-im_{y}(r)\\\
m_{x}(r)+im_{y}(r)&n(r)-m_{z}(r)\end{pmatrix}.$ (3.23)

Accordingly, the Schrödinger equation can be re-written as follows

$[(-\frac{\hbar^{2}}{2m}\nabla^{2}+\sum_{\alpha}\int\frac{n_{\alpha\alpha(r^{\prime})}}{|r-r^{\prime}|}\,dr^{\prime})I+\nu(r)+\frac{\delta
E_{xc}}{\delta n(r)}]\begin{pmatrix}\psi_{i}^{(+)}r\\\
\psi_{i}^{(-)}\end{pmatrix}=\epsilon_{i}\begin{pmatrix}\psi_{i}^{(+)}r\\\
\psi_{i}^{(-)}\end{pmatrix},$ (3.24)

with I as a 2 $\times$ 2 unit matrix and the exchange-correlation matrix as
well as a 2 $\times$ 2 matrix [59].

Another way to write the potential matrices are the following expressions

$\nu(r)=\nu(r)I+\mu_{B}\sigma\cdot B(r),$ (3.25)

and

$\nu_{XC}(r)=\nu_{XC}(r)I+\mu_{B}\sigma\cdot B_{xc}(r),$ (3.26)

with $B(r)$ as a magnetic field and $\mu_{B}=\frac{e\hbar}{2mc}$. It is
assumed that the magnetic structure is collinear. The potential matrices in
this case are considered as diagonal which means that the magnetic and
exchange fields are pointing in the z direction. This leads to a decoupling of
the re-written Schrödinger equation into the form of the two equations

$(-\frac{\hbar^{2}}{2m}\nabla^{2}+\nu_{Coul}(r)+\nu(r)+B_{z}(r)+\nu_{xc}^{(+)}(r))\psi_{i}^{(+)}(r)=\epsilon_{i}^{(+)}\psi_{i}^{(+)}(r),$
(3.27)
$(-\frac{\hbar^{2}}{2m}\nabla^{2}+\nu_{Coul}(r)+\nu(r)-B_{z}(r)+\nu_{xc}^{(-)}(r))\psi_{i}^{(-)}(r)=\epsilon_{i}^{(-)}\psi_{i}^{(-)}(r).$
(3.28)

In these equations, $\nu_{Coul}$ marks the classical Coulomb potential and
$\nu_{XC}^{+,-}$ the exchange-correlation potential in accordance to the spin
up or spin down part of the diagonal density matrix. Both equations need to be
solved independently [59].

With these two equations, all kind of magnetic structures (ferromagnetic,
antiferromagnetic or ferrimagnetic) can be calculated in the collinear case.
The spin density and the spin moment can be obtained with the following two
equations, respectively

$m(r)=-\mu_{B}\sum_{\alpha,\beta}\psi_{\alpha}^{(+)}(r)\sigma_{\alpha\beta}\psi_{\beta}(r),$
(3.29)

and

$M_{spin}=\int m(r)\,dr=\int(n^{(+)}(r)-n^{(-)}(r))\,dr.$ (3.30)

The quality of the calculated magnetic moment depends on the exchange-
correlation functional used. In most cases, GGA yields better results than
LSDA [7].

###  3.12 Treatment of f-electrons

Due to large self-interaction errors, strongly correlated f-electrons cannot
be represented sufficiently in DFT. The functionals of LDA and GGA are not
able to fully cancel the electronic self-interaction in the Hartree term,
which leads to the phenomenon that the electron sees itself. This creates
false repulsion and supports electrons to artificially delocalize. This is
especially a problematic case for partially filled f states for the RE
elements Pr-Eu and Tb-Yb. The exceptional cases, which are Ce and Gd, are
handled reasonably well. As in the case of $\alpha$-Ce, the f-electrons are
delocalized and Gd has an half filled f-shell. However, in case of
$\gamma$-Ce, this exception is not valid anymore as the f-electrons are
strongly localized. In order to handle these elements, different methods
beyond DFT can be used. Two of these methods are DFT+U [60, 61] and the
dynamical mean-field theory (DMFT) [62] of which only the first one is used in
this thesis.

With DFT+U, an energy contribution called the Hubbard U repulsion energy is
added. This energy term originates in the Hubbard model, where the Hamiltonian
describes moving particles on a lattice. It adds two contributions to the the
Hamiltonian in form of a hopping term, that describes the probability of a
particle to hop from one site to another, and a penalty energy that represents
the on-site repulsion. The on-site repulsion is commonly named as U. With the
addition of U, the artificial delocalization due to the self interaction
errors are tried to be removed.

The according energy functional can be expressed as

$D^{DFT+U}[n]=E^{DFT}[n]-E^{dc}[n_{i}]+E_{U}[n_{i}],$ (3.31)

with $E^{dc}$ as a correction term to avoid double counting of correlation
effects and $E^{U}$ as a repulsion term in the form of
$E^{U}[n_{i}]=\frac{1}{2}U\sum_{i\neq j}n_{i}n_{j}$ that is added to the
occupancies of $n_{i}$.

The DFT+U approach is mostly implemented in form of the method by
Liechtenstein [63] or by Dudarev [61]. As only the latter one is used in this
thesis, only the Dudarev approach will be described and can be formulated as

$E^{DFT+\textit{U}}_{Dudarev}=E^{DFT}+\frac{U_{eff}}{2}\sum_{I,\sigma}\sum_{i}n_{i}^{I,\sigma}(1-n_{i}^{I,\sigma}.$
(3.32)

The effective U parameter $U_{eff}$ is determined as the on-site Coulomb
interaction parameter and is crucial for the method. To define $U_{eff}$,
density of states calculations (DOS) can be performed. In case of NdFe11Ti the
experimental findings for the f peak results in a value of 4.65 eV below the
Fermi energy for $U_{eff}$.

##  Chapter 4 Computational Details

###  4.1 Vienna ab initio Simulation Package Calculations

In this thesis, all first-principles calculations were performed in the
framework of spin-polarized density functional theory (DFT) using the Vienna
ab initio Simulation Package (VASP) [64, 65]. The projector-augmented wave
method (PAW) was used as implemented in VASP. Exchange-correlation effects
were treated within the generalized gradient approximation (GGA) of Perdew-
Burke-Ernzerhof (PBE) [49]. In addition to this, the performance of the local
spin density approximation (LSDA) [48] was examined for magnetocrystalline
anisotropy (MAE) calculations in case of the clean ternary and quarternary
compounds. For LSDA-MAE calculations, the optimized geometry obtained through
the relaxation with the GGA-PBE functional was used as input. For clearance,
calculations in this work, except for $T_{C}$ calculations that are referred
to as DFT (DFT+U) are treated using the GGA (GGA+U) functional, and the
treatment with LSDA is always mentioned explicitly.

The 4f-electrons of both Ce and Nd elements were treated carefully, which is
not straightforward in DFT. In the case of Ce, the single f-electron was
treated explicitly as valence state. The treatment of correlated f-electrons
in DFT requires the Hubbard U correction. However, Ce-Fe compounds exhibit
different behavior due to the hybridization of Ce-4f with Fe-3d-electrons.
Therefore, similar to works done by Sözen et al. [52, 53], no Hubbard U
correction scheme was applied for Ce-4f-electrons. In the case of Nd, the
f-electrons were examined as in-core state for the DFT scheme. This treatment
yielded good mechanical and elastic properties, however dramatically failed
for the magnetic properties due to the missing f-electrons. Hence, additional
DFT+U treatment via the Dudarev method [61] was considered for the Nd case
with Hubbard U=6 eV. It has to be noted that in this work the orbital
contribution of the Nd-4f electrons to the magnetic moment are not treated and
only the spin magnetic moments are considered for the calculation of the
different properties. Therefore, the total magnetic moment refers only to the
total spin magnetic moment.

The Brillouin zone was sampled with a k-point mesh described by a
$\Gamma$-centered grid, considering 10$\times$10$\times$8 meshes for
supercells containing 26 atoms (28 for N contained cases). The cutoff energy
for the plane wave basis used was 500 eV, and the width of the smearing
parameter was 0.1 eV. The convergence criteria within the self-consistent
field (SCF) scheme was set to be 10-5 eV for all considered alloys. The values
of all these input parameters provided an energy convergence with an error
equal to or smaller than 1 meV/atom.

For magnetocrystalline anisotropy calculations, first a collinear self-
consistent calculation with the mentioned input details was performed. The
obtained geometry and magnetic configuration were used as input in non-
collinear calculations that take the spin-orbit coupling into account. In each
non-collinear run the magnetic moment was aligned along one of the
crystallographic directions [001],[100],[010] and [110] to obtain the
different energies for each direction.

###  4.2 Korringa-Kohn-Rostoker Calculations

In order to calculate the Curie temperature $T_{\text{C}}$, the exchange
interaction energies $J_{ij}$ for converged ferromagnetic (FM) and local-
moment-disorder (LMD) states of the 1:12 phases by the Liechtenstein method
were computed [63]. The magnetic couplings were calculated using density
functional theory following the Korringa-Kohn-Rostoker (KKR) [66, 67] Green’s
function method based AkaiKKR [68] code -also known as MACHIKANEYAMA- with the
implemented atomic sphere approximation (ASA) incorporating coherent potential
approximation (CPA) [69, 70]. Continuous concentration changes for both RE and
TM sublattices were considered on the basis of KKR-CPA. All KKR calculations
were based on the local density approximation (LDA) [39, 42], taking the
exchange-correlation function as parametrized by Moruzzi, Janak and Williams
(MJW) [71]. In this work, the scattering was considered up to d-scattering
($l_{max}=2$) in the systems and all f-electrons were put in the valence state
on basis of the open-core approximation [72, 73].

##  Chapter 5 Results and Analysis

###  5.1 Determination of the Hubbard U Parameter by Density of States
Calculations

Due to the strong localization of the 4f-electrons in Nd atoms, the DFT+U
approach is implemented in this work for all Nd containing compounds. To
determine the proper value for the Hubbard U correction, the density of states
(DOS) investigations were performed for the NdFe11Ti compound from U=1 to 8
eV, which is given in Fig. 5.1.

Figure 5.1: Calculated density of states (DOS) for the NdFe11Ti compound. The
Dudarev approach is used with the U parameter ranging from 1 to 8 eV. Vertical
dotted line represents for the Fermi level. The 4f peaks (red lines) change
according to the U parameter and are located for small U near the Fermi
energy.

As shown, the applied Hubbard U parameter only changes the Nd-4f states and
has no considerable impact on the Fe-3d states. It is also noticeable that
small U parameters, such as U=1-4 are not capable to reproduce localized 4f
states, except for U= 3 eV. The localization of 4f-electrons only becomes
visible once U$\geq$5 eV. Once U=5 eV, the localized 4f-electrons peak is
located $\sim$ 4 eV lower than the Fermi energy. In addition to this,
increasing the U parameter from 5 to 7 eV also does not change the location of
the 4f peak significantly.

The spectroscopic properties of NdFe11Ti are reported by Lang et al. [74], and
it is noted that the 4f peak locates around 4.65 eV below the Fermi energy. In
addition, the calculated magnetic moments of Nd atoms can be compared with the
experimentally reported ones. In the case of U=6 eV, Nd atoms carry -3.31
$\mu_{\text{B}}$ magnetic moment. Experimentally, it is reported to be 3.47
$\mu_{\text{B}}$ [75]. In theoretical calculations, an increasing U parameter
does not modify the magnetic moments significantly, and already a promising
agreement is obtained for U=6 eV for both spectroscopic and magnetic
properties. Therefore, we considered 6 eV as the Hubbard correction in case of
this study. In addition, Herper et al. [75] have shown that the most accurate
description of magnetic properties is achieved with a Hubbard U value of U
between 5 and 6 eV for NdFe11Ti. In this case, considered U=5 eV, is also
applied for binary NdFe12 and quaternary compounds such as (Nd,Y)Fe11Ti as
well. Since all the considered compounds belong to the ThMn12 structure,
special DOS calculations for each compound have not been performed.

###  5.2 Calculation of Lattice Constants and Cell Volume

Figure 5.2: Schematic representation of the 2 formula unit (f.u.) (26 atoms)
body-center-tetragonal ThMn12 (I4/mmm, space group number 139) 1:12 phases.
(a) Unstable RFe12, (b) 1 Ti atom substituted (3.8 at.%) R2Fe23Ti
(RFe11.5Ti0.5) and (c) 2 Ti atom substituted (7.7 at.%) and nitrogenated
supercell RFe11TiN (R2Fe22Ti2N2). Ti atoms substituted in energetically
favorable 8i sites and 2 N atom considered to fully occupy interstitial 2b
site.

Besides the prototype and most well-known compound of Nd2Fe14B [76, 77],
another class of typical compounds that fulfills the technological
requirements for hard-magnetic applications is of the form RFe12-yTMy. The
schematic representation of the RFe12 compound is given in Fig. 5.2(a) which
is thermodynamically not stable. However, alloyed transition metal elements,
for instance, a small amount of Ti with y$\geq$0.7, can stabilize these
compounds [33, 78]. Structural parameters of CeFe11Ti are obtained from
neutron powder diffraction by Isnard et al. [79]. Ce atoms occupy the 2a site
and Fe atoms are distributed over the three sublattices 8i, 8j, and 8f, which
is similar for NdFe11Ti and YFe11Ti as well.

For a theoretical study of the Ti solubility in the RFe12 phase (R=Y, Ce and
Nd), it is expected that there will be no antisites at the R(2a) sites and
that the addition of Ti will correspond to the composition RFe12-yTiy.
Starting by substituting a single Ti atom in a 26-atom supercell (2 f.u.),
which corresponds to 3.8 at.% Ti, its substitution in different sites, namely,
8i, 8j, and 8f is evaluated. Sözen et al. [52] found that the lowest energy is
found for Ti at the 8i site for all the considered RFe12 phases (see Fig.
5.2(b)) and it is followed by 8j and 8f. The solution enthalpy for the 8j and
8f sites are by 19 and 29 meV/atom higher for YFe12-yTiy, respectively. This
is also very similar for Ce and Nd-based 1:12 compounds, where Ti substitution
for 8j site is 20 and 18 meV/atom and for 8f site is 29 and 27 meV/atom
higher, respectively. In the case of the second Ti atom substitution,
corresponding to 7.7 at.%, all possible sites and configurations by keeping
the first Ti atom at the 8i site are checked. Again, for the second Ti atom
energetical preference appeares to be the 8i site, see Fig. 5.2(c). The
calculated theoretical Ti preferences agree with the experimental works [79,
9, 80, 81, 82] which revealed that Ti atoms are exclusively located at 8i
sites. This strong 8i site preference of Ti atoms is mainly due to the larger
Wigner-Seitz radius of the 8i site compared to 8j and 8f sites [83]. The 8j
and 8f sites have similar energies, due to their similar local coordination
[84].

Table 5.1: Calculated lattice constant, cell volume and bulk modulus of the
considered alloys and nitrogenated cases using DFT. In the case of Nd
containing stable compounds DFT+U has been considered only for Nd 4f-electrons
with U=6 eV and is given in parenthesis. N is assumed to occupy all available
N(2b) sites in the 2 f.u. supercell with 2 atoms (see Fig. 5.2 c)). For
instance, Y-based 1:12 phases have the following compositions after
nitrogenation: YFe12N, Y2Fe23Ti2N2 and YFe11TiN.

Alloy |  | Lattice constants (Å)
---
a b c
| Cell volume
---
clean
(Å3)
| Experiment
---
cell volume
clean
(Å3)
| Cell volume
---
with N
(Å3)
| Experiment
---
cell volume
with N
(Å3)
| Bulk modulus
---
clean
(GPa)
| Bulk modulus
---
with N
(GPa)
|  Y2Fe24
---
(YFe12)
8.452 | 8.452 | 4.683 | 167.26 |  167.47* [85] | 173.55 |  | 133.14 | 146.19
|  Y2Fe23Ti
---
(YFe11.5Ti0.5)
8.471 | 8.457 | 4.702 | 168.43 |  | 174.78 |  | 130.61 | 145.60
|  Y2Fe22Ti2
---
(YFe11Ti)
8.501 | 8.452 | 4.731 | 169.94 |  173.5a [86] | 175.98 |  178.75a [86] | 128.88 | 144.73
|  Ce2Fe24
---
(CeFe12)
8.504 | 8.504 | 4.653 | 168.25 |  168.06* [85] | 174.65 |  | 128.48 | 142.73
|  Ce2Fe23Ti
---
(CeFe11.5Ti0.5)
8.512 | 8.522 | 4.669 | 169.33 |  | 175.73 |  | 126.68 | 143.71
|  Ce2Fe22Ti2
---
(CeFe11Ti)
8.527 | 8.530 | 4.690 | 170.57 |  174.35a [87] | 176.78 |  179.1a [87] | 126.16 | 144.30
|  Nd2Fe24
---
(NdFe12)
8.549 | 8.549 | 4.670 | 170.65 |  180.37* [88] | 177.60 |  183.36* [88] | 127.96 | 145.55
|  Nd2Fe23Ti
---
(NdFe11.5Ti0.5)
| 8.552
---
(8.561)
| 8.570
---
(8.543)
| 4.686
---
(4.692)
| 171.73
---
(171.81)
|  | 178.81
---
(178.26)
|  | 127.17
---
| 144.94
---
|  Nd2Fe22Ti2
---
(NdFe11Ti)
| 8.561
---
(8.517)
| 8.584
---
(8.619)
| 4.708
---
(4.706)
| 173.01
---
(172.92)
|  176.2a [89]
---
180.35a [86]
| 179.95
---
(179.42)
183.35a [86] |  | 127.20
---
| 143.48
---
|  NdYFe24
---
8.501 | 8.501 | 4.676 | 168.97 |  | 175.64 |  | 130.88 | 145.67
|  NdYFe23Ti
---
| 8.511
---
(8.534)
| 8.512
---
(8.527)
| 4.695
---
(4.719)
| 170.06
---
(170.24)
|  | 176.83
---
(176.47)
|  | 128.77
---
| 145.02
---
|  NdYFe22Ti2
---
| 8.528
---
(8.511)
| 8.524
---
(8.499)
| 4.719
---
(4.704)
| 171.54
---
(171.75)
|  | 178.01
---
(177.65)
|  | 127.94
---
| 143.88
---
|  NdCeFe24
---
8.525 | 8.525 | 4.662 | 169.44 |  | 176.25 |  | 129.11 | 142.92
|  NdCeFe23Ti
---
| 8.538
---
(8.542)
| 8.540
---
(8.534)
| 4.678
---
(4.679)
| 170.55
---
(170.66)
|  | 177.37
---
(176.90)
|  | 127.40
---
| 143.23
---
|  NdCeFe22Ti2
---
| 8.544
---
(8.535)
| 8.558
---
(8.559)
| 4.700
---
(4.695)
| 171.83
---
(171.59)
|  | 178.41
---
(177.85)
|  | 127.31
---
| 143.02
---

  * •

Theoretical references are represented by *.

  * •

a X-ray diffraction (XRD) analysis at 300 K.

Knowing the structure of Ti contained ternary 1:12 phases, one can calculate
physical properties such as lattice constants and bulk moduli as given in Tab.
5.1. It has been captured an increasing lattice constant and cell volume trend
as Nd2Fe24-yTiy > Ce2Fe24-yTiy > Y2Fe24-yTiy for 0$\leq y\leq$2 in agreement
with experimental data [90, 79, 91, 81]. As expected, Ce doped quaternary
compounds, NdCeFe24-yTiy, have a larger volume than Y doped cases,
NdYFe24-yTiy. The influence of each substitution can be seen in Fig. 5.3,
where the resulting trends are shown for GGA calculated values. Note that Y
and Ce substitution has been considered only for the RE(2a) site. The effect
of nitrogen has been investigated with 2 N atoms in a 2 f.u. 28 atom
supercell, see Fig. 5.2(c). N atoms are doped into the midpoint between RE
atoms along the c axis. The interstitial N expands the volume by $\sim$3.7 %
for Y2Fe24-yTiy and Ce2Fe24-yTiy and $\sim$4.1 % for the rest of the
compounds, which agrees with experimental data [81, 92, 87, 90]. As mentioned
above, DFT+U treatment is performed for Nd-contained compounds to include the
influence of the 4-f-electrons (calculated values are reported in parenthesis
in Tab. 5.1), and a similar volume trend compared to the in-core treatment in
DFT is observed as well. In addition, it has been captured that increasing Ti
concentration in all considered compounds causes a reduction in bulk modulus.
The softening effect of Ti addition is also the case for most of the
nitrogenated compounds, except Ce2Fe24-yTiyN2.

Figure 5.3: Schematic representation of the cell volume trends for two
different substitutions, which are Ti substitution on Fe sublattice and Ce and
Y substitutions in Nd sublattice, via GGA calculations. Purely Nd-containing
compounds are depicted in black while compounds only including Y and Ce are
shown as blue and red, respectively. The quarternary compounds containing Nd,Y
and Nd,Ce are illustrated as cyan and orange, respectively. The resulting
trend for each substitution is further clarified by arrows.

###  5.3 Total Magnetic Moment and Magnetization

As a first intrinsic magnetic property, the magnetization of the considered
compounds has been calculated. It has to be distinguished between the total
magnetic moment $m_{tot}$ and saturation magnetization $M_{S}$. The former one
is calculated via the summation of each atom’s magnetic moment and is
expressed in units of the Bohr magneton $\mu_{\text{B}}$ per formula unit,
while the latter one is in units of Tesla (T) which is estimated from the
density of the compound $\rho$, the Avogadro constant NA and the molecular
weight $M_{\text{alloy}}$ via

$\mu_{0}M_{S}=m_{tot}\mu_{B}\frac{\rho\text{N${}_{A}$}}{M_{\text{alloy}}}.$
(5.1)

Theoretical total magnetic moments and saturation magnetization with a
comparison of available experimental data are given in Tab. 5.2. Knowing that
the considered PBE functional tends to overestimate the magnetic moments [93],
the calculated values are within the expected range. It is important to note
that having accurately relaxed structures (check Tab. 5.1) does not guarantee
correct reproduction of magnetic properties. For instance, in the case of
YFe11Ti an experimental magnetic moment is reported as 18.4
$\mu_{\text{B}}/$f.u. [37], whereas the calculated value in this work is 21.96
$\mu_{\text{B}}/$f.u meaning $\sim$19 % overestimation. Similar deviation is
found for CeFe11Ti with $\sim$21% overestimation. Experimentally, the Ce-based
1:12 phase has a lower $m_{tot}$ than the Y-based 1:12 phase. This is due to
the magnetic moments carried by RE elements. In both cases, Fe and Ti atoms
have rather similar magnetic moments, but Ce has -0.76 $\mu_{\text{B}}$ where
Y carries -0.31 $\mu_{\text{B}}$ magnetic moment.

In the case of Nd2Fe22-yTiy compounds, DFT calculations with f-electrons in
core treatment yield rather good $m_{tot}$ values. As an example, for NdFe11Ti
it is calculated to be 21.84 $\mu_{\text{B}}$, and an experimental value is
measured as 21.27 $\mu_{\text{B}}$ [86]. However, in this treatment
4f-electrons are assumed as frozen in core. Treating the Nd 4f-electrons as
fully localized with DFT+U calculations yields 18.92 $\mu_{\text{B}}$ total
magnetic moment and 1.28 T saturation magnetization, which is experimentally
measured as 1.38 T. With the DFT+U approximation a reduced magnetization is
seen because the contribution from the Nd ion is about -3.31 $\mu_{\text{B}}$
(mainly from f-electrons) and it is -0.28 $\mu_{\text{B}}$ for simple DFT due
to the missing f orbital. Nevertheless, for both treatments, acceptable values
compared to experimental data are achieved.

Y and Ce substitution to the Nd-based 1:12 phase decreases the total magnetic
moments for DFT treatment because both Y (-0.31 $\mu_{\text{B}}$) and Ce
(-0.76 $\mu_{\text{B}}$) have lower magnetic moments than the Nd (-0.28
$\mu_{\text{B}}$) ion. However, the situation is reversed in the case of DFT+U
treatment, since the inclusion of localized f-electrons yields more negative
magnetic moments for the Nd ion by -3.31 $\mu_{\text{B}}$. The calculated
magnetic moments of Nd atoms are not influenced considerably, once the
f-electrons are localized with Hubbard U$\geq$ 5 eV. Therefore, having larger
U values is not changing the calculated magnetic moments of the RE site. For
instance, in case of NdFe11Ti a value of U=5 eV results in a total magnetic
moment of 18.93 $\mu_{\text{B}}/$f.u., while a value of U=8 eV leads to 18.92
$\mu_{\text{B}}/$f.u. In addition, for each considered compound increasing Ti
concentration reduces the total magnetization due to the antiferromagnetic
alignment of Ti atoms against Fe atoms with an average of -1.1
$\mu_{\text{B}}$.

Table 5.2: Calculated magnetic moments at rare-earth site $m^{2a}_{tot}$,
total magnetic moments $m_{tot}$ ($\mu_{\text{B}}/f.u$) and saturation
magnetization $\mu_{0}M_{S}$ (T) with comparison against available
experimental data. In the case of Nd contained compounds DFT+U with U= 6 eV is
applied to Nd 4f-electrons only and results are given in parenthesis. 2 N
atoms have been considered in the N(2b) site in the 2 f.u. supercell for the
nitrogenated cases.

Alloy |  | $m_{tot}^{2a}$
---
clean
($\mu_{\text{B}}/f.u.$)
| $m_{tot}$
---
clean
($\mu_{\text{B}}/f.u.$)
| $m_{tot}$
---
experiment
clean
($\mu_{\text{B}}/f.u.$)
| $\mu_{0}M_{S}$
---
clean
(T)
| $\mu_{0}M_{S}$
---
experiment
clean
(T)
| $m_{tot}$
---
with N
($\mu_{\text{B}}/f.u.$)
| $m_{tot}$
---
experiment
with N
($\mu_{\text{B}}/f.u.$)
| $\mu_{0}M_{S}$
---
with N
(T)
| $\mu_{0}M_{S}$
---
experiment
with N
(T)
Y2Fe24 | -0.30 | 26.23 |  24.30a [37] | 1.83 |  1.66a [37] | 28.21 |  | 1.89 |
Y2Fe23Ti | -0.31 | 23.95 |  | 1.66 |  | 25.83 |  | 1.72 |
Y2Fe22Ti2 | -0.31 | 21.96 |  |  18.3b [94]
---
18.4a [37]
18.57c [86]
19.0f [95]
19.97e [75]
20.3c [96]
1.50 |  |  1.24b [94]
---
1.26a [37]
1.26c [86]
1.29f [95]
1.36e [75]
1.38c [96]
23.38 |  21.75c [86] | 1.55 |  1.42c [86]
Ce2Fe24 | -0.79 | 25.75 |  26.55∗ [97] | 1.78 |  1.89∗ [97] | 27.73 |  29.66∗ [77] | 1.85 |  1.98∗ [77]
Ce2Fe23Ti | -0.75 | 23.35 |  24.68∗ [97] | 1.61 |  1.70∗ [97] | 25.24 |  | 1.67 |
Ce2Fe22Ti2 | -0.76 | 21.15 |  |  17.4d [79]
---
18.62c [87]
22.25f [81]
1.45 |  |  1.19d [79]
---
1.27c [87]
1.55f [81]
22.69 |  |  21.69c [87]
---
25.19∗ [77]
1.50 |  |  1.43c [87]
---
1.66∗ [77]
Nd2Fe24 | -0.27 | 26.43 |  |  27.2∗ [98]
---
29.15∗ [77]
31.20∗ [88]
1.80 |  |  1.73∗ [98]
---
1.99∗ [77]
2.01∗ [88]
28.73 |  |  31.40∗ [77]
---
32.40∗ [88]
1.88 |  |  2.06∗ [77]
---
2.06∗ [88]
Nd2Fe23Ti |  | -0.28
---
(-3.30)
| 24.04
---
(21.35)
|  | 1.63
---
(1.45)
|  | 26.35
---
(23.14)
|  | 1.72
---
(1.51)
Nd2Fe22Ti2 |  | -0.28
---
(-3.31)
| 21.84
---
(18.92)
|  21.27c [86]
---
21.90b [89]
23.43e [75]
24.10∗ [99]
24.50∗ [77]
25.24f [81]
26.30∗ [88]
| 1.47
---
(1.28)
|  1.38c [86]
---
1.48b [89]
1.58e [75]
1.63∗ [99]
1.65∗ [77]
1.70f [81]
1.70∗ [88]
| 23.87
---
(20.86)
|  23.22c [86]
---
26.84∗ [99]
26.86∗ [77]
27.2∗ [88]
| 1.55
---
(1.35)
|  1.56c [86]
---
1.74∗ [99]
1.74∗ [77]
1.73∗ [88]
NdYFe24 | -0.27 (Nd), -0.30 (Y) | 26.33 |  | 1.82 |  | 28.49 |  | 1.89 |
NdYFe23Ti |  | -0.28 (Nd), -0.30 (Y)
---
(-3.31 (Nd), -0.30 (Y))
| 23.99
---
(22.78)
|  | 1.64
---
(1.56)
|  | 26.11
---
(24.64)
|  | 1.72
---
(1.63)
NdYFe22Ti2 |  | -0.28 (Nd), -0.31 (Y)
---
(-3.53 (Nd), -0.31 (Y))
| 21.91
---
(20.47)
|  | 1.49
---
(1.39)
|  | 23.66
---
(22.10)
|  | 1.55
---
(1.45)
NdCeFe24 | -0.27 (Nd), -0.76 (Ce) | 26.09 |  | 1.80 |  | 28.31 |  | 1.87 |
NdCeFe23Ti |  | -0.28 (Nd), -0.74 (Ce)
---
(-3.31 (Nd), -0.75 (Ce))
| 23.74
---
(22.39)
|  | 1.62
---
(1.53)
|  | 25.86
---
(24.24)
|  | 1.70
---
(1.60)
NdCeFe22Ti2 |  | -0.28 (Nd), -0.77 (Ce)
---
(-3.31 (Nd), -0.77 (Ce))
| 21.52
---
(19.93)
|  | 1.46
---
(1.35)
|  | 23.28
---
(21.57)
|  | 1.52
---
(1.41)

  * •

Theoretical references are represented by *

  * •

aMössbauer measurement at 77 K, bMössbauer measurement at 4.2 K cExtraction
sample magnetometer (ESM) analysis at 1.5 K, dExtraction sample magnetometer
(ESM) analysis at 5 K, eSingle crystal, physical property measurement system
(PPMS) measured at 10 K, fVibrating sample magnetometer (VSM) analysis at 4.2
K.

The effect of interstitial nitrogenation on the magnetic properties, such as
magnetization, Curie temperature and magnetocrystalline anisotropy energy of
REFe12-xTix (RE: Y, Ce, Nd and Sm) compounds has been studied experimentally
[86, 87], and a dramatic impact is commonly reported.

In the case of magnetization, an improvement of $\sim$8% for the magnetic
moments of all the considered compounds is calculated and given in Tab. 5.2.
For the Y-based 1:12 phase, YFe11TiN, the magnetic moment is increased by 1.42
$\mu_{\text{B}}$ compared to the N-free compound. This is followed by 1.54 and
2.03 $\mu_{\text{B}}$ increments for CeFe11TiN and NdFe11TiN, respectively.
Experimentally, the impact of nitrogenation on YFe11Ti magnetisation is
reported to be a 3.2 $\mu_{\text{B}}$ increase by Yang et al. [86]. A similar
improvement of 3.1 $\mu_{\text{B}}/f.u.$ is measured by Pan et al. [87] for
CeFe11Ti and 2.0 $\mu_{\text{B}}/f.u.$ increase for NdFe11Ti is reported by
Yang et al. [86]. Even though, the experimental improvement of $\sim$17% for
YFe11Ti and CeFe11Ti is higher than our theoretical improvement, the increase
due to nitrogenation is properly shown and still in good agreement. For the
experimental improvement of NdFe11Ti of $\sim$9%, a very good agreement with
the calculated value in this work can be seen.

The effect of nitrogenation on Y and Ce substituted quaternaries are rather
similar. According to the DFT (DFT+U) calculations, the magnetic moment
increase for both (Nd,Y)Fe11TiN and (Nd,Ce)Fe11TiN is $\sim$1.75 (1.63)
$\mu_{\text{B}}$. It is also noticeable that increasing Ti concentration
reduces the impact of N. For instance, the N effect on Ti-free YFe12N is 1.98
$\mu_{\text{B}}$, once a Ti atom is added (Y2Fe23TiN2), the impact reduces to
1.88 $\mu_{\text{B}}$, and it is further reduced to 1.42 $\mu_{\text{B}}$ once
Ti concentration increases (YFe11TiN).

As given in Tab. 5.1, nitrogenation expands the volume of all the considered
compounds, and as mentioned above increases the magnetic moments. In order to
understand the underlying reason, if it is based on the magnetovolume effect
or a purely chemical effect, we have performed a set of simulations as given
in Fig. 5.4. Already calculated N-free optimized R2Fe24-yTiy compounds are
given in black circles, and optimized and nitrogenated R2Fe24-yTiyN2 are given
in black squares. Then, the latter optimized structure is kept and the N atoms
are removed (R2Fe24-yTiyE2). In this set of calculations, expanded volumes are
considered but N is removed and further cell relaxation in self-consistent
calculations is restricted, given as black triangles. The change in the
magnetic moment from the circles to the triangles corresponds to the
magnetovolume effect which is associated with the expanded volume of
R2Fe24-yTiyE2 compared to R2Fe24-yTiy by $\sim$4%. This is also colored light
grey. On the other hand, the change of magnetic moments from triangles to
squares corresponds to the chemical effect which is colored by dark grey. As
shown for the considered compounds, the magnetovolume effect has a major role
in the improvement of magnetic moments by nitrogenation.

Figure 5.4: Total magnetic moments of the clean (RFe24-yTiy), N doped
(RFe24-yTiyN2) and N removed but volume kept fixed, and atoms are allowed to
relax cases (RFe24-yTiyE2). Since DFT+U calculations do not change the results
significantly, only DFT results are given. The light(dark) grey background
coloring shows the magnetovolume (chemical) effect. The blue circles show
theoretical data [97, 88] for the clean alloys from others, the red circles
show experimental data [37, 96, 81, 92] for the clean alloys, the blue squares
show theoretical nitrogenated cases [97, 77, 88], and the red squares show the
experimental nitrogenated cases [92].

###  5.4 Maximum Energy Product

One of the performance measures of permanent magnets is the maximum energy
product |BH|max. It is an important figure-of-merit for the strength of
permanent magnetic materials, and a magnet must be shaped for the most
efficient use in such a way that its operating point is close to the |BH|max
point. It is the absolute upper limit of magnetostatic energy stored in free
space by a permanent magnet of unit volume [100]. By calculating the
saturation magnetization $M_{S}$, one can calculate the |BH|max for an ideal
square hysteresis loop as given by Ref. [10]

$|BH|_{max}=\frac{\mu_{0}M_{S}^{2}}{4},$ (5.2)

where $\mu_{0}$ is the vacuum permeability ($\mu_{0}=4\pi\times 10^{-7}$NA-2).

Calculated |BH|max values of the considered compounds for clean and
nitrogenated cases are given in Fig. 5.5. As a search criteria for a promising
hard magnetic compound a |BH|max higher than 400 kJ/m3 is important. As shown,
the highest |BH|max values are generally obtained for Ti-free compounds, RFe12
(RE: Y, Ce and Nd). However, they are unstable and Ti is needed for
thermodynamic stabilization, as mentioned above. Although it is needed to
stabilize the 1:12 phases, Ti substitution sacrifices a considerable amount of
|BH|max. An average of 3.8 at.% Ti concentration (1 Ti atom in 2 f.u.
supercell) reduces the maximum energy product by $\sim$120 kJ/m3 and the loss
is $\sim$100 kJ/m3 once the Ti concentration is increased to 7.7 at.% (2 Ti
atom in 2 f.u. supercell).

Due to the missing experimental |BH|max measurements, a comparison for the
theoretical results can not be made. Nevertheless, Körner et al. [77]
calculated the |BH|max for various 1:12 phases by using tight-binding linear-
muffin-tin-orbital atomic-sphere approximation (TB-LMTO-ASA). According to
their findings, the |BH|max of CeFe11Ti is 396 kJ/m3 and NdFe11Ti is 438
kJ/m3, while in case of this work they are 416 and 431 kJ/m3, respectively.
This is a good theoretical agreement. In the case of DFT+U treatment, 324
kJ/m3 is found for NdFe11Ti with U=6 eV. Since, DFT+U treatment yields
localized 4f-electrons with total -3.3 $\mu_{\text{B}}$ magnetic moment, it
reduces $m_{tot}$ and |BH|max compared to DFT, which treats f-electrons as in
core.

Quaternary Y and Ce alloying to Nd-based 1:12 magnets have promising |BH|max
values. In the case of Y substitution, the maximum energy product increases
from 324 to 384 kJ/m3 for (Nd,Y)Fe11Ti, while it increases to 365 kJ/m3 for
(Nd,Ce)Fe11Ti. As mentioned before, nitrogenation increases the total
magnetization, and this similarly affects |BH|max. For each considered
compound nitrogenation increases |BH|max by $\sim$7-12%. As an example,
nitrogenation improves |BH|max for NdFe11Ti $\sim$45 (41) kJ/m3 for DFT (DFT+U
with U= 6 eV) in the calculations of this thesis and the effect is calculated
to be 49 kJ/m3 by Körner et al. [77].

Fig. 5.5 also includes the |BH|max values of the most well known hard magnets.
The highest energy product is achieved by Nd2Fe14B with 512 kJ/m3. In the case
of DFT+U calculations the most promising quaternary compound is calculated to
be NdYFe23Ti with 484 kJ/m3, which is 28 kJ/m3 lower than |BH|max of Nd2Fe14B.
In the case of NdCeFe23Ti it is 465 kJ/m3. As mentioned earlier, quarternaries
containing a higher concentration of Ti have a lower |BH|max, such as 384 and
365 kJ/m3 for NdYFe22Ti2 and NdCeFe22Ti2, respectively. Note that
nitrogenation of these compounds further increases the calculated values by
$\sim$40 kJ/m3. Therefore, achieving such high energy products with about 50%
less Nd would be technologically and economically very valuable.

Figure 5.5: Theoretical maximum energy product |BH|max values for considered
compounds and nitrogenated cases. In addition to conventional DFT, DFT+U with
U=6 eV has been considered for stable Nd-contained alloys. Experimental values
[10, 101] of most common hard magnets are given as horizontal dotted lines.

###  5.5 Curie Temperature

In addition to a strong magnetization, a high Curie temperature $T_{\text{C}}$
is another desired intrinsic magnetic property. The well-known Nd2Fe14B magnet
is famous for its strong saturation magnetization at room temperature,
however, it is also known to have a relatively low $T_{\text{C}}$ (588 K [10])
compared to more traditional RE-based permanent magnets such as Sm2Co17 (838 K
[10]). Finding an Nd-lean RE magnet with a better balance between saturation
magnetization and Curie temperature thus deserves extensive efforts.

In order to calculate Curie temperatures of the considered compounds, an
effective spin Heisenberg model is employed and solved by the mean-field
approximation (MFA). In the Heisenberg model, the Hamiltonian for the spin-
spin interaction can be written as $\mathcal{H}=-\sum_{i\neq
j}J_{ij}\vec{S_{i}}\vec{S_{j}}$, where $\vec{S_{i}}$($\vec{S_{j}}$) is the
spin at site $i(j)$ and $J_{ij}$ is the exchange coupling constant between
sites $i$ and $j$, which is calculated by AkaiKKR [68]. The absolute value of
the magnetic ordering temperature, $T_{\text{C}}$, derived from the MFA is
$k_{\text{B}}T_{\text{C}}=\frac{3}{2}J_{0}$, where $k_{\text{B}}$ is the
Boltzmann factor and $J_{0}$ is the total effective exchange that is the sum
of $J_{ij}$ connecting a given lattice site to all remaining ones.

In the MFA, the systematic overestimation of $T_{\text{C}}$ is a well-known
issue, therefore, in addition to the FM state, the local moment disorder (LMD)
approach was considered as a second magnetic state as well. In the framework
of the KKR-CPA method, the concept of LMD state is conveniently used to
describe the paramagnetic state of a ferromagnet above $T_{\text{C}}$ [102]
(note that LMD is also called disorder local moment (DLM) [103]).
Nevertheless, magnetic moments survive and do not vanish. As an advantage of
the KKR based on Green’s function theory, the exchange interaction energies
for such situations and the Curie temperature for both FM and LMD states can
be calculated.

Figure 5.6: Calculated Curie temperatures $T_{\text{C}}$ for the clean and
nitrogenated 1:12 phases. Exchange interaction energies have been calculated
with both ordered ferromagnetic (FM) and disordered local moment disorder
(LMD) states, and the input lattice information is taken from our
theoretically relaxed calculations (check Tab. 5.1). Experimental data is
taken from [86, 104].

The calculated Curie temperatures are given in Fig. 5.6 in comparison with
available experimental data for clean and nitrogenated compounds and well-
known permanent magnets. As shown, the calculated FM ground state
approximation in this work overestimates about 40% for most of the cases as
expected. However, calculating the $T_{\text{C}}$ by considering the LMD as
the ground state, the agreement improves significantly. The underlying reasons
for the overestimation of $T_{\text{C}}$ for the FM ground state are two-fold:
$i)$ the problem in describing the delocalized electronic state in the
magnetism of intermetallics on the basis of localized degrees of freedom.
$ii)$ exchange couplings between the local moments are calculated for the
ground state, and this assumption does not need to be held for high
temperatures near the $T_{\text{C}}$.

Nevertheless, Miyake et al. [105] calculated the $T_{\text{C}}$ for various
RE-based compounds for the 2:14, 2:17 and 1:12 phases (namely RE2Fe14B,
RE2Co14B, RE2Fe17, RE2Co17 and RE2Fe22Ti) and reported that the performance of
the LMD treatment is not always better than the FM case. For instance, in the
case of RE2Fe14B, the $T_{\text{C}}$ values are systematically overestimated
for FM treatment and agreement appeared to be better for the LMD case.
However, in RE2Co14B, the $T_{\text{C}}$ results from the FM ground state are
much better than the underestimated LMD values. In the case of the RE2Fe17
phase, both FM and LMD treatments fail to reproduce $T_{\text{C}}$ by
substantial overestimation, while in the case of RE2Co17, the $T_{\text{C}}$
is well produced for the FM treatment. In agreement with this work, they also
reported an overestimation of $T_{\text{C}}$ for the FM treatment in
RE2Fe22Ti2 compounds, while the LMD treatment gives much better agreement with
respect to experiment. As reported by Miyake et al. [105], the FM ground state
yields generally overestimation (or good agreement as in the case of
RE2Co14B), but there is a complicated dependence of the LMD performance on the
crystal structure and the transition metal element. This may originate from a
difference in spin fluctuations close to the magnetic transition and needs to
be further analyzed.

As in the case of $m_{tot}$ and |BH|max, increasing Ti concentration reduces
$T_{\text{C}}$ for all considered compounds shown in Fig. 5.6. The average
decrease of $T_{\text{C}}$ due to the Ti addition (1 Ti atom=3.8 at.%) is
calculated to be $\sim$45 K. Experimentally it is reported that among the
ternaries, CeFe11Ti has the lowest $T_{\text{C}}$ with 487 K [104]. This is
followed by YFe11Ti with 540 K [86] and NdFe11Ti with 570 K [86]. The
theoretical LMD results in this work perfectly reproduce the experimental
trend with the calculated $T_{\text{C}}$ values of 550, 591, and 688 K, for
CeFe11Ti, YFe11Ti, and NdFe11Ti, respectively.

Y and Ce substitution on Nd-based 1:12 phase cause a decreasing
$T_{\text{C}}$. According to our LMD calculations, the Curie temperature of
(Nd,Y)Fe11Ti is 595 K, and it is 593 K for (Nd,Ce)Fe11Ti, which means $\sim$95
K reduction compared to NdFe11Ti. However, it must be noted that the
experimental $T_{\text{C}}$ of well-known magnets are also shown in Fig. 5.6,
and both Y and Ce doped Nd-based quaternary compounds still have higher
$T_{\text{C}}$ than Nd2Fe14B. Therefore, one can conclude that Nd-lean 1:12
phases have slightly lower but still comparable Curie temperatures as compared
to Nd2Fe14B.

The effect of N doping on the considered compounds has been investigated as
well. Although the FM treatment overestimates the calculated $T_{\text{C}}$,
the difference between clean and nitrogenated compounds can be compared
($\Delta
T_{\text{C}}^{\text{N}}=T_{\text{C}}^{\text{RFe${}_{12-y}$Ti${}_{y}$N}}-T_{\text{C}}^{\text{RFe${}_{12-y}$Ti${}_{y}$}}$).
As shown in Fig. 5.6, it is experimentally reported that N increases the
$T_{\text{C}}$ for all ternaries RFe11Ti (RE: Y, Ce and Nd). This improvement
is measured to be 199 [86], 278 [104] and 170 [86] K for YFe11Ti, CeFe11Ti and
NdFe11Ti, respectively. Our FM treatment also approves such an increase after
nitrogenation with 186, 239 and 152 K for Y, Ce and Nd-based 1:12 phases,
respectively. Besides a very promising quantitative agreement between our
theoretical work and experimental data, the order of N impact is calculated
correctly as well. According to the experiment, N has the highest impact on
the Ce-based 1:12 phase, then it is followed by Y and Nd-based 1:12 phases
which is correctly reproduced in our calculations.

###  5.6 Finite Temperature Magnetization

The temperature-dependent stability of the magnetic phases has been addressed
in this thesis as well. Based on the developed computational schemes by Kuzmin
et al. [106] and Körmann et al. [107], one can calculate the finite
temperature magnetization curves of various elements and compounds by using
total magnetic moments and Curie temperature as input parameters. For the
Kuzmin plot (labeled as MF in the diagrams), the following equation is used to
determine the finite temperature magnetization.

$m(\frac{M(T)}{M_{0}})=[1-s(\frac{T}{T_{C}})^{\frac{3}{2}}-(1-s)(\frac{T}{T_{C}})^{p}]^{\frac{1}{3}}.$
(5.3)

The finite temperature magnetization $M(T)$ is depicted as the reduced
magnetization $m(\frac{M(T)}{M_{0}})$ and is calculated with the temperature
$T$, the Curie temperature $T_{C}$, the magnetization at 0 K $M_{0}$, and the
two variables $s$ and $p$. The variable $p$ is most of the times set to be
$\frac{5}{2}$ like for body centered cubic (bcc) Fe [106]. The variable $s$
can then be a value between 0 and $\frac{5}{2}$ and is varied until a good fit
to the experimental data is achieved. In the improved mean field approach by
Körmann et al. [107] on the example of bcc Fe, the variable $p$ is set to a
constant value of 4. This results in a better agreement for their considered
example of bcc Fe as well as for the considered compounds of NdFe11Ti and
YFe11Ti in Fig. 5.7. The resulting mean-field plots, as shown in Fig. 5.7,
show a good agreement with the experimental data by Herper et al. [75],
Bozukov et al. [108], and Tereshina et al. [80]. In case of the NdFe11Ti and
YFe11Ti compounds, the variable $p$ is set as described before to
$\frac{5}{2}$, and the variable $s$ leads to good results with a value of
about 1.0. In both cases, the difference between the data calculated with the
Kuzmin and the Körmann approach is only marginal and both approaches reveal a
good agreement with the experimental data. For the CeFe11Ti alloy, the
variable $p$ is chosen much bigger with a value of 8 in case of the data of
Bozukov et al. [108], and a value of 6 in case of the data by Akayama et al.
[81]. This change was applied only to the Kuzmin approach because changing the
value of 4 in the equation of Körmann leads to the same equation as used by
Kuzmin. In this case, a very small value of $s=0.05$ is chosen to get the best
possible results. With these variables and changes, all of the calculated
finite temperature plots are in good accordance with the experimental data.

Figure 5.7: Finite temperature magnetization calculated on basis of the
Körmann and Kuzmin approaches. For comparison the experimental data by Herper
et al. [75], Bozukov et al. [108], Akayama et al. [81] and Tereshina et al.
[80] are shown. For the NdFe11Ti and YFe11Ti compounds the Kuzmin and the
Körmann approaches are used. For the CeFe11Ti compound the Kuzmin approach
with $p$=8 (red) and $p$=6 (blue) are depicted.

###  5.7 Magnetocrystalline Anisotropy

Magnetocrystalline anisotropy (MCA) is a critical magnetic property and must
be addressed for a material once it has been tended to be used as a permanent
magnet. Therefore, as the last intrinsic property, the magnetocrystalline
anisotropy, which has a key importance for the performance of hard magnets is
investigated. The MCA can be calculated by employing the magnetic force
theorem [109] and by total energy differences. In this study, the latter one
has been considered and magnetocrystalline anisotropy energy (MAE) is defined
as

$\Delta E_{\text{MAE}}=E_{[110]}-E_{[001]},$ (5.4)

where $E_{[110]}$ and $E_{[001]}$ are the total energies of the considered
supercells for the magnetization being oriented along the [110] and [001]
directions, respectively.

For MAE calculations, the orientation of the magnetization along the
crystallographic directions plays an important role. In Fig. 5.8, the
resulting energy differences on basis of DFT and DFT+U calculations for
changes from the easy magnetization axis [001] to the directions [110] and
[010] are shown. DFT+U treatment leads to an increase of the energies of $\sim
0.07$ MJ/m3 for the clean and $\sim$ 0.1 to 0.2 MJ/m3 for the nitrogenated
case. The influence of interstitial nitrogenation is depicted with the change
from circles to squares with an energy increase of $\sim$ 0.4 MJ/m3. For the
different direction changes from [001] to [110] and [001] and [010], shown in
black and red, respectively, a significant energy difference of $\sim 0.4$
MJ/m3 is visible. Due to the higher energy difference, the change to [110] is
mainly focused on in this work.

Figure 5.8: Calculated energy differences of the magnetocrystalline energy for
the shift of the crystallographic direction from [001] to [110](black) and
[100] to [010] (red) for the clean (circles) and nitrogenated (squares)
NdFe11Ti compound. On the left side are DFT and on the right side DFT+U
results shown. For the change from [001] to [110] the angles 0, 22.5, 45, 67.5
and 90° are considered. For the change from [001] to [010] only the 0 and 90 °
angles are considered.

Based on the calculated magnetocrystalline anisotropy energy one can find the
anisotropy constant $K_{1}$ and the anisotropy field $H_{a}$. The anisotropy
energy can be written as an expansion whose leading term is [10]

$\Delta E_{\text{MAE}}=K_{1}\sin(\phi)^{2},$ (5.5)

where $\phi$ is the angle between the magnetization direction and the easy
axis. Further terms in the expansion reflect the crystal symmetry, but for
many purposes, it is enough to consider only the leading term. A positive
(negative) $K_{1}$ value corresponds to uniaxial (planar) anisotropy.

The anisotropy field $H_{a}$ is the upper limit for the coercivity and can be
estimated from [10]

$H_{a}=\frac{2K_{1}}{\mu_{0}M_{S}}.$ (5.6)

$H_{a}$ is an important property for permanent magnets, and a high value is
needed for hard magnetic applications. Nevertheless, it must be noted that the
calculated $K_{1}$ and $M_{S}$ values correspond to $T=$0 K, therefore, the
$T$ dependence of $H_{a}$ is excluded.

The calculated magnetocrystalline anisotropy constant $K_{1}$ and anisotropy
field $H_{a}$ are given in Tab. 5.3 for the clean compounds. As listed, all
the considered 1:12 phases have uniaxial anisotropy ($K_{1}$>0). This
linearity yields that higher-order anisotropy constants are negligible. We
have examined our considered compounds and find that $K_{2}$ is almost 2
orders in magnitude smaller than the lowest order anisotropy constant,
$K_{1}$, which makes it sufficient to describe the MCA (for more details
regarding the anisotropy constants see Ref. [91]).

Table 5.3: Calculated theoretical magnetocrystalline anisotropy constant
$K_{1}$ according to Eq. 5.5, anisotropy field $H_{a}$ according to Eq. 5.6
and magnetic hardness factor $\kappa$ from Eq. 5.7 for stable compounds with
available experimental data. Values are given for both GGA and LSDA treatments
for the exchange-correlation potential and DFT+U (U= 6 eV) results are
reported in parenthesis Nd contained compounds. The experimental value of
$K_{1}$ of well known hard magnets is also given for a comparison.

Alloy |  | $K_{1}$
---
GGA
(MJ/m3)
| $K_{1}$
---
LSDA
(MJ/m3)
| $K_{1}$
---
experiment
clean
(MJ/m3)
| $H_{a}$
---
GGA
(T)
| $H_{a}$
---
LSDA
(T)
| $H_{a}$
---
experiment
clean
(T)
Y2Fe23Ti | 0.81 | 1.65 |  | 0.98 | 2.28 |
Y2Fe22Ti2 | 0.92 | 1.73 |  |  0.85a [91]
---
1.89b [91]
1.91c [80]
2.0c [95]
2.1d [75]
1.22 | 2.76 |  |  2.2e [110]
---
4.1f [110]
4.8a [94]
5.7c [94]
Ce2Fe23Ti | 1.67 | 1.91 |  | 2.08 | 2.70 |
Ce2Fe22Ti2 | 2.08 | 2.12 |  |  1.4e [79]
---
1.78g [87]
1.9g [111]
4.3e [81]
2.88 | 3.40 |  |  1.5h [87]
---
1.7e [112]
2.3e [81]
3.5i [87]
Nd2Fe23Ti |  | 0.37
---
(0.39)
1.32 |  |  | 0.45
---
(0.54)
2.12 |
Nd2Fe22Ti2 |  | 0.54
---
(0.61)
1.41 |  |  0.94e [113]
---
1.78d [75]
| 0.74
---
(0.96)
2.57 |  |  1.9e [113]
---
2.0e [81]
NdYFe23Ti |  | 0.63
---
(0.66)
1.53 |  |  | 0.77
---
(0.85)
2.28 |
NdYFe22Ti2 |  | 0.74
---
(0.91)
1.65 |  |  | 1.00
---
(1.31)
2.75 |
NdCeFe23Ti |  | 0.87
---
(0.94)
1.43 |  |  | 1.07
---
(1.23)
2.13 |
NdCeFe22Ti2 |  | 1.27
---
(1.35)
1.61 |  |  | 1.75
---
(2.00)
2.71 |
Nd2Fe14B |  |  |  4.9j [11] |  |  |
Sm2Co17 |  |  |  4.2j [11] |  |  |
SmCo5 |  |  |  17.0j [11] |  |  |
Alnico 5 |  |  |  0.32j [11] |  |  |

  * •

Theoretical references are represented by *

  * •

a Mössbauer measurement at 300 K, b Mössbauer measurement at 77 K, c Mössbauer
measurement at 4.2 K, d Single Crystal, Physical Property Measurement System
(PPMS) measured at 10 K, e Derived from magnetization curve at 300 K, f
Derived from magnetization curve at 4.2 K, g Determined by Kerr microscopy at
300 K, h X-ray diffraction measurement at 300 K, i X-ray diffraction
measurement at 1.5 K, j Properties measured at 300 K.

As given in Tab. 5.3, an increasing Ti concentration also increases the
anisotropy constant. Since periodic boundary conditions are used, the presence
of Ti can be an artifact in an unevenly distributed small supercell. The
missing real random distribution of Fe/Ti at 8i site also results in slightly
non-equivalent basal directions [100] and [010].

In the case of YFe11Ti, the DFT (GGA) approximation yields the anisotropy
constant as 0.92 MJ/m3. This is a clear underestimation of low temperature
experimental values reported in the range of 1.89-2.1 MJ/m3 [114, 80, 95, 75].
In addition, a DFT based underestimated MCA is also reported by Ke et al. [85]
and Herper et al. [75]. Therefore, the MCA is further examined by using LSDA.
Significant improvement is achieved by using LSDA with 1.73 MJ/m3, which
agrees better to experimental data.

The GGA results for CeFe11Ti already yield a very promising $K_{1}$ against
experimental data. Known reported experimental values are ranging from 1.4 to
4.3 MJ/m3 [79, 81] based on the measurement technique and temperature.
Nevertheless, the calculated $K_{1}$ with 2.08 MJ/m3 fits well to these
experiments. The LSDA treatment for Ce-based compounds does not have a
significant impact, and the $K_{1}$ is calculated to be 2.12 MJ/m3.

As mentioned earlier, 4f-electrons treatment in general DFT is not
straightforward and based on the choice of the 4f localization MCA can yield
different results. According to DFT (GGA) calculations in this study, where
f-electrons are treated in core and no hybridization takes place between 4f
and valence states, the anisotropy constant $K_{1}$ is calculated to be 0.54
MJ/m3 for the NdFe11Ti compound. Once we reduce the level of localization by
applying the Hubbard U correction to the on-site Coulomb interactions $K_{1}$
is predicted to be 0.61 MJ/m3 for U=6 eV. In comparison to the experimental
value of 1.78 MJ/m3 [75], both cases underestimates the MCA. As in the case of
Y-based compounds, the LSDA treatment for NdFe11Ti yields better MCA results
with respect to an experimental value of 1.41 MJ/m3.

According to the calculations in this work, both Y and Ce substitutions have
improved the MCA compared to the NdFe11Ti case. In case of Y substituted 1:12
phase (Nd,Y)Fe11Ti, GGA (GGA+U) has a $K_{1}$ value of 0.74 (0.91) MJ/m3 and
it is further improved for the Ce substituted case in (Nd,Ce)Fe11Ti with 1.27
(1.35) MJ/m3. Although, the GGA based calculations underestimate the MCA of
NdFe11Ti with respect to measurements, it is known that experimentally
reported YFe11Ti and CeFe11Ti $K_{1}$ values (1.89 [114] and 2 MJ/m3 [111])
are higher than the $K_{1}$ corresponding to NdFe11Ti (1.78 MJ/m3 [75]).
Therefore, such a theoretical increase is accurately captured for both GGA
(GGA+U) and LSDA calculations. In addition, for all considered compounds MCA
is found to be uniaxial ($K_{1}$>0) independently from the choice of exchange-
correlation functionals, although GGA calculations for Y and Nd-based
compounds strongly underestimate MCA. This is a good agreement with
experimental data.

As given in Eq. 5.6, the anisotropy field $H_{a}$ can be calculated in a
combination of the anisotropy constant $K_{1}$ and the saturation
magnetization $M_{S}$. In addition, as in the case of $K_{1}$ there is only
available experimental data for the ternary compounds. In the case of the
YFe11Ti compound the GGA calculations yield 1.22 T, which underestimates the
low-temperature experimental values, reported to be in the range of 4.1 to 5.7
T [110, 94], and room temperature experimental values ranging from 2.2 to 4.8
T [94, 110]. The LSDA treatment leads to a significant improvement with a
calculated value of 2.76 T, although there is still a deviation compared to
low-temperature experimental data.

The anisotropy field $H_{a}$ of the CeFe11Ti compound is already well
described by the GGA calculations with a value of 2.88 T. The experimental
findings are reported in the range of 1.5 to 3.5 T [87, 112, 81] based on
measurement technique and temperature. LSDA treatment yields a slightly higher
value of 3.40 T, which also fits into the range of the reported experimental
values and is near the value of 3.5 T measured by Pan et al. [87] with X-ray
diffraction at 1.5 K. Therefore, the LSDA treatment does not have a
significant impact as in the case of the Y and Nd-based ternaries.

In the NdFe11Ti calculations, the DFT (GGA) approach, with 4f-electrons
treated as in core and no hybridization taking place, a value of 0.74 T is
calculated. Applying the Hubbard U correction to the on-site Coulomb
interactions the result increases to 0.96 T for U=6 eV. Comparing these
results to the experimental findings of 1.9 T by Bouzidi et al. [113] and 2.0
T by Akayama et al. [81], a clear underestimation of our calculations is
observed. Nevertheless, the LSDA treatment leads to an significant increase
with 2.57 T, which better agrees with the reported experimental values.

In the case of quaternaries, the partial Nd substitution leads to an
improvement for $H_{a}$ compared to NdFe11Ti ternary. For the Y substituted
1:12 compound NdYFe22Ti2 GGA (GGA+U), calculations result in a $H_{a}$ value
of 1.0 (1.31) T. In the case of Ce substitution NdCeFe22Ti2, the increase is
even larger with a value of 1.75 (2.00) T. As shown before, the LSDA treatment
yields larger $H_{a}$ values for Y and Ce substituted quaternaries with 2.75
and 2.71 T, respectively. Since there are no experimental values to compare
the theoretical findings of this work with for the considered quaternaries, it
can only be deduced that the calculated $H_{a}$ values for the ternaries have
a good agreement with experiments which is encouraging for the quaternary
findings.

Figure 5.9: Influence of interstitial nitrogenation on the anisotropy constant
$K_{1}$ (black) and anisotropy field $H_{a}$ (red). Treatment with the GGA
(GGA+U) functional is depicted as squares (circles). Clean compounds are shown
as empty and nitrogenated ones as filled symbols.

The calculated influence of interstitial nitrogenation on $K_{1}$ and $H_{a}$
for our GGA and GGA+U calculations is shown in Fig. 5.9. A general trend of an
increase for $K_{1}$ and $H_{a}$ due to the addition of N is visible for all
compounds. For instance, $K_{1}$ calculated via DFT for the Y-based ternary
compound YFe11Ti is improved by 0.60 MJ/m3 to a value of 1.52 MJ/m3. In case
of the CeFe11Ti compound, the calculated value of 2.08 MJ/m3 for the clean
case is raised by N doping to a result of 2.52 MJ/m3. The anisotropy field is
consequently also increased due to nitrogenation to 3.36 T. For NdFe11Ti the
nitrogenated $K_{1}$ value is computed with DFT (DFT+U) to be 0.92 (1.0)
MJ/m3. The according result for $H_{a}$ is calculated as 0.96 (1.47) T.

The impact of N on the quarternary compounds NdYFe22Ti2 NdCeFe22Ti2 is similar
to the ternary ones. For $K_{1}$, an increase from 0.74 (0.91) to 1.21 (1.27)
MJ/m3 and 1.27 (1.35) to 1.72 (1.78) MJ/m3 for the respective compounds is
observed. Since the calculation of the anisotropy field is based on the
anisotropy constant, the increasement of $K_{1}$ leads to an improvement of
the anisotropy field up to values of 1.56 (1.75) and 2.26 (2.52) T,
respectively.

For all considered compounds, an average increase of $\sim 0.5$ MJ/m3 due to
the additon of N atoms into the 2b site can be recorded. The biggest
individual increase is seen for the above mentioned YFe11Ti compound with an
improvement of 0.60 MJ/m3.

###  5.8 Hardness Factor

From the first-principles calculations, the magnetic hardness factor can be
evaluated from the expression of saturation magnetisation $M_{S}$ and
magnetocrystalline anisotropy constant $K_{1}$ by

$\kappa=\sqrt{\frac{K_{1}}{\mu_{0}M_{S}^{2}}}.$ (5.7)

This parameter indicates the potential for a material to be developed into a
permanent magnet independently of the shape when $\kappa$>1\. In addition, a
material can be considered as a semi-hard (0.1 <$\kappa$< 1) and soft magnet
($\kappa$<0.1) accordingly [11].

The calculated magnetic hardness factors $\kappa$ are depicted in Fig. 5.10
for both GGA (GGA+U) and LSDA functionals. As mentioned above, for Y and Nd-
based compounds, a better $K_{1}$ agreement with respect to experimental data
by LSDA calculations was achieved. Therefore, LSDA based $\kappa$ values are
more reliable, and it is noticeable that for all considered compounds
$\kappa$>1\. Herper et al. [75] calculated the magnetic hardness factor for
YFe11Ti for both GGA and LSDA and found $\kappa$=0.63 and 1.23, respectively.
This agrees with the calculated values in this work, which are 0.71 (GGA) and
1.18 (LSDA), respectively.

For CeFe11.5Ti0.5 and CeFe11Ti Snarsky-Adamsky et al. [97] reported
theoretical values, calculated using the full-potential local-orbital code
from 2018 (FPLO18) with the PBE functional, of 0.52 and 0.79 respectively.
While these values are slightly smaller than the GGA results of 0.90 and 1.12
in this study, the trend of an increasing hardness factor with an increase of
Ti concentration is in good agreement.