The next magnetic property studied in this section is the magnetocrystalline anisotropy energy (MAE). The MAE can be determined from the magnetic force theorem <cit.>, or by calculating the energy difference of the magnetisation directions. In this thesis the MAE is calculated using the latter approach. The equation used to calculate the MAE in this work is as follows:

\begin{equation}
\Delta E_{MAE} = E_{[010]} - E_{[001]},
\label{eq.39}
\end{equation}

where $E_{[001]}$ and $E_{[010]}$ correspond to the total energies of the magnetisation directions [001] and [010] of the supercell, respectively.

In Magnetocrystalline Anisotropy Energy (MAE) calculations the orientation of magnetisation along the crystallographic directions plays an important role. In Fig. <ref> the resulting energy differences based on DFT+$U$ and LSDA calculations are shown for changes from the easiest magnetisation axis [001] to the hardest axis [010]. The directional changes from [001] to [010] are shown for the different compounds. LSDA is used because Erdmann et al. <cit.> showed that GGA+$U$ sometimes gives inaccurate results when calculating anisotropy.

Calculated energy differences of magnetocrystalline energy for the shift of the crystallographic direction from [001] to [010] for all compounds considered in MAE calculation. LSDA results are shown on the left and DFT+$U$ results on the right. For the change from [001] to [010], angles 0, 22.5, 45, 67.5 and 90° are taken into consideration. For NdFe$_{11}$Ti only the values of the angles 0 and 90° are shown.

From the total energies and the resulting energy differences the anisotropy constant $K_{1}$ and the anisotropy field $H_{a}$ can be calculated. The anisotropy energy can be written as:

\begin{equation}
\Delta E_{MAE} = K_{1} \cdot sin(\phi)^{2},
\label{eq.40}
\end{equation}

where $\phi$ is the angle between the direction of magnetisation and the axis that is easiest to magnetise. In this formulation only the leading term containing $K_{1}$ is shown, being sufficiently accurate in this case. All considered 1:12 phases were found to exhibit uniaxial anisotropy ($K_{1} > 0$). The higher order anisotropy constants are negligible because $K_{2}$ is almost two orders of magnitude smaller than $K_{1}$. Thus makes $K_{1}$ sufficient to describe the magnetocrystalline anisotropy (MCA) (for further details see <cit.>). A resulting positive (or negative) value for $K_{1}$ corresponds to uniaxial (or planar) anisotropy.

The anisotropy field $H_{a}$ corresponds to the upper limit of the coercivity and can be calculated using Eq. <ref>. $H_{a}$ reflects an important property of an permanent magnet and should be as high as possible. At this point it is important to mention that the values calculated here for $K_{1}$ and $M_{S}$ are valid for T = 0 K. This excludes the temperature dependence of $H_{a}$. The calculated anisotropy constant $K_{1}$ and the anisotropy field $H_{a}$, calculated by GGA+$U$ and LSDA, are shown in Tab. <ref>.

Calculated theoretical magnetocrystalline anisotropy constant $K_{1}$ according to Eq. <ref>, anisotropy field $H_{a}$ according to Eq. <ref> and magnetic hardness factor $\kappa$ from Eq. <ref> for stable compounds with available experimental data. The values are given for both GGA and LSDA treatment for the exchange–correlation potential. DFT+$U$ ($U$ = 6 eV) results are reported in parenthesis for Nd containing compounds. The experimental value of $K_{1}$ and $\kappa$ of well known hard magnets are also given for comparison.

Alloy                                                             c]@c@$K_{1}$ GGA              c]@c@$K_{1}$ LSDA              c]@c@$K_{1}$ experi-           c]@c@$H_{a}$ GGA         c]@c@$H_{a}$ LSDA        c]@c@$H_{a}$ experi-           c]@c@$\kappa$ GGA              c]@c@$\kappa$ LSDA              c]@c@$\kappa$ experi-

(MJ/m$^{3}$)                                                 (MJ/m$^{3}$)                                                  mental                                                            (T)                                                     (T)                                                     mental                                                                                mental

(MJ/m$^{3}$)                                                                                (T)
Nd$_{2}$Fe$_{22}$Ti$_{2}$                                         0.67                                                             1.68                                                              0.61* <cit.>                                                 0.80                                                        2.00                                                        0.72* <cit.>                                                 0.49                                                              0.77                                                               0.52* <cit.>

(NdFe$_{11}$Ti)                                                                                1.41´ <cit.>                                                                                1.86´ <cit.>                                                                                0.88´ <cit.>

1.78$^{b}$ <cit.>                                                                                1.9$^{a}$ <cit.>

2.0$^{a}$ <cit.>
Zr$_{2}$Fe$_{22}$Ti$_{2}$                                         2.44                                                             2.47                                                                                3.22                                                        3.27                                                                                1.03                                                              1.04

ZrNdFe$_{22}$Ti$_{2}$                                             2.16                                                             2.41                                                                                2.69                                                        3.01                                                                                0.92                                                              0.97

ZrNdFe$_{20}$Co$_{2}$Ti$_{2}$                                     1.97                                                             2.08                                                                                2.44                                                        2.58                                                                                0.87                                                              0.90

(Zr$_{0.3}$Nd$_{0.7}$)$_{2}$(Fe$_{0.75}$Co$_{0.25}$)$_{23}$Ti                                                                                1.41$^{c}$ <cit.>                                                                                1.7$^{c,d}$ <cit.>                                                                                0.71$^{c}$ <cit.>

Nd$_{2}$Fe$_{14}$B                                                                                4.9$^{e}$ <cit.>                                                                                1.54$^{e}$ <cit.>

Sm$_{2}$Co$_{17}$                                                                                4.2$^{e}$ <cit.>                                                                                1.89$^{e}$ <cit.>

SmCo$_{5}$                                                                                17.0$^{e}$ <cit.>                                                                                4.40$^{e}$ <cit.>

Alnico 5                                                                                0.32$^{e}$ <cit.>                                                                                0.45$^{e}$ <cit.>

* * Theoretical GGA+$U$ references are represented by *.
* ´ Theoretical LSDA references are represented by ´.
* $^{a}$ Derived from magnetisation curve at 300 K.
* $^{b}$ Single Crystal, Physical Property Measurement System (PPMS) measured at 10 K.
* $^{c}$ Calculated from source data.
* $^{d}$ Calculated from experimental $\mu_{0}M_{S}$ using the law of approaching saturation.
* $^{e}$ Properties measured at 300 K.

In the work of Erdmann et al. <cit.>, an increase in the anisotropy constant with increasing Ti concentration was observed. Based on this, the MAE of the likely stable compounds RFe$_{11}$Ti (R: Zr, (ZrNd), Nd) and (ZrNd)Fe$_{10}$CoTi are calculated.

In case of the NdFe$_{11}$Ti compound, the DFT (GGA+$U$) calculation yields an anisotropy constant of 0.67 MJ/m$^{3}$. This is a significant underestimation of the experimental value of Herper et al. <cit.> with 1.78 MJ/m$^{3}$. Therefore, the local spin density approximation (LSDA) was used to study the MCA. With LSDA, a significant increase of $K_{1}$ is obtained with 1.68 MJ/m$^{3}$. This improvement is in accordance with the work of Erdmann et al. <cit.>, providing values of 0.61 (GGA+$U$) and 1.41 MJ/m$^{3}$ (LSDA).

According to the calculations, the substitution of Nd with Zr improves the MAE compared to NdFe$_{11}$Ti. The quaternary 1:12 compound (Zr,Nd)Fe$_{11}$Ti exhibits $K_{1}$ values of 2.16 (GGA+$U$) and 2.41 MJ/m$^{3}$ (LSDA), which are both larger than the experimental value of 1.78 MJ/m$^{3}$ of NdFe$_{11}$Ti. The complete Zr-based 1:12 phase with ZrFe$_{11}$Ti leads to another small increase in the $K_{1}$ value to 2.44 (GGA+$U$) and 2.47 MJ/m$^{3}$ (LSDA). Due to the lack of literature, an estimation of the Zr-containing phase is rather difficult to assess. In comparison the NdFe$_{11}$Ti phase shows a larger difference between the GGA+$U$ and LSDA values for $K_{1}$ than the ZrFe$_{11}$Ti phase. It should be noted that the difference between the LSDA value of (Zr,Nd)Fe$_{11}$Ti of 2.41 MJ/m$^{3}$ and the values of ZrFe$_{11}$Ti of 2.44 (GGA+$U$) and 2.47 (LSDA) MJ/m$^{3}$ is very small. It appears that the GGA+$U$ method becomes more accurate as the Zr content increases. In case of the (Zr,Nd)Fe$_{11}$Ti compound, the GGA+$U$ value can be estimated to be more precise.

In case of the effect of Co on $K_{1}$, it can be seen that the values decrease. This decrease can be seen by comparing the values of (Zr,Nd)Fe$_{11}$Ti with 2.16 (GGA+$U$) and 2.41 (LSDA) MJ/m$^{3}$ with those of (Zr,Nd)Fe$_{10}$CoTi with 1.97 (GGA+$U$) and 2.08 (LSDA) MJ/m$^{3}$. A rather small lowering effect of Co on $H_{a}$ and on $K_{1}$ was found by Suzuki et al. <cit.> as well.

The values of the anisotropy constant $K_{1}$ and the anisotropy field $H_{a}$ are shown in black and red, respectively. The values calculated with the GGA+$U$ functional are shown as circles, while those using the LSDA functional are shown as squares.

Furthermore, if the $K_{1}$ values of RFe$_{11}$Ti (R: Y, Zr, Ce and Nd) are compared, the following trend of the GGA+$U$ values can be observed with Nd (0.67 MJ/m$^{3}$) $<$ Y (0.92 MJ/m$^{3}$ <cit.>) $<$ Ce (2.08 MJ/m$^{3}$ <cit.>) $<$ Zr (2.16 MJ/m$^{3}$). From the LSDA values the same trend can be observed with Nd (1.68 MJ/m$^{3}$) $<$ Y (1.73 MJ/m$^{3}$ <cit.>) $<$ Ce (2.12 MJ/m$^{3}$ <cit.>) $<$ Zr (2.41 MJ/m$^{3}$). From these trends, Zr causes the greatest increase in the value of $K_{1}$. Moreover, the value obtained for the Zr-containing compounds can be considered rather reasonable, since it is not too far from that of the mentioned Ce compound. All considered quaternary compounds from this work and for comparison those from Erdmann et al. <cit.>, show the following trend of GGA+$U$ (and LSDA) treatments with YNd with 0.91 (1.65) <cit.> $<$ CeNd with 1.35 (1.61) <cit.> $<$ ZrNd with 2.16 (2.41) MJ/m$^{3}$.

As can be seen in Eq. <ref>, the anisotropy field $H_{a}$ is calculated from the anisotropy constant $K_{1}$ and the saturation magnetisation $M_{S}$. As in case of $K_{1}$, there is no literature for comparison of Zr-containing compounds. For NdFe$_{11}$Ti, the values can be compared with the experimental values obtained by Bouzidi et al. <cit.> with 1.9 T and Akayama et al. <cit.> with 2.0 T. It shows that the calculated value obtained from GGA+$U$ treatment of 0.80 T is well below literature. The LSDA treatment leads to a value of 2.00 T and is therefore in good agreement. For ZrFe$_{11}$Ti, (Zr,Nd)Fe$_{11}$Ti and (Zr,Nd)Fe$_{10}$CoTi, the GGA+$U$ (and LSDA) treatments yielded values of 3.22 (3.27), 2.69 (3.01) and 2.44 (2.58) T, respectively. These values follow the same mentioned trends as for $K_{1}$. Similar trends can be seen when comparing the Y and Ce compounds from Erdmann et al. <cit.>. For the quaternary compounds, the value increases with GGA+$U$ (LSDA) treatment as follows: YNd with 1.19 (2.33) <cit.> $<$ CeNd with 1.73 (2.30) <cit.> $<$ ZrNd with 2.69 (3.01) T.

This section concludes that the MCA including $K_1$ and $H_a$ of the ZrNdFe$_{22}$Ti$_{2}$ compound is promising. According to calculations, the values are even higher than for quaternary Y and Ce compounds. This needs future experimental verification of this compound.

§.§ Hardness Factor

With the anisotropy constant $K_{1}$ and the saturation magnetisation $M_{S}$, the magnetic hardness factor $\kappa$ can be calculated. This is done using the following equation:

\begin{equation}
\kappa = \sqrt{\frac{K_{1}}{\mu_{0}M_{S}^{2}}}. \tag{\ref{eq:3}}
\end{equation}

The resulting $\kappa$ value can be used to estimate the potential of a material as a permanent magnet. A magnetic compound can be classified as a soft ($\kappa < 0.1$), semi-hard ($0.1 < \kappa < 1$) or hard ($\kappa > 1$) magnet. With $\kappa > 1$ the material is suitable as a permanent magnet regardless of its shape, whereas in the semi-hard range ($0.1 < \kappa < 1$) a shape/dimension limitation as with Alnico magnets <cit.> has to be considered.

The $\kappa$ values calculated according to Eq. <ref> are shown in Tab. <ref>. Comparing the values from the earlier work of Erdmann et al. <cit.> of NdFe$_{11}$Ti (0.52 (GGA+$U$), 0.88 (LSDA)) with the newly calculated values (0.49 (GGA+$U$), 0.77 (LSDA)), a good theoretical agreement can be seen. Looking at the values of RFe$_{11}$Ti (R: Y, Zr, Ce and Nd), it is shown that Zr with 1.04 increases the $\kappa$ value less in case of LSDA than Y or Ce with 1.18 <cit.> and 1.31 <cit.>, respectively. Comparing the Nd (0.77) and Zr (1.04) compounds, the $\kappa$ value of Zr is higher. In case of GGA+$U$ treatment the value of Zr with 1.03 is higher than that of Nd with 0.49 and Y with 0.71 <cit.>, but lower than that of Ce with 1.12 <cit.>. Thus, Zr improves the hardness factor in comparison to the Nd compound, but increases the value less than Y or Ce.

The calculated $\kappa$-values of the quaternary compound (Zr,Nd)Fe$_{11}$Ti are 0.92 (GGA +$U$) and 0.97 (LSDA). As there is no direct literature on the Zr compound, it has to be classified as a semi-hard magnet for the time being, as it gives values $\kappa < 1.00$. In comparison to the quaternary compounds (Y,Nd)Fe$_{11}$Ti (0.67 (GGA+U) and 1.02 (LSDA)) and (Ce,Nd)Fe$_{11}$Ti (0.83 (GGA+U) and 1.02 (LSDA)) <cit.>, the Zr compound has a comparable or slightly lower hardness factor. It can be seen that GGA+$U$ gives better values in case of Zr than for Y or Ce. This is indicated by the smaller difference between the GGA+$U$ and LSDA values in case of Zr.

Calculated hardness factors $\kappa$ for GGA+$U$ (squares) and LSDA (circles) functionals. The magnetic hardness factors of well known magnets are shown by horizontal dashed lines.

As in the case of $K_{1}$ and $H_{a}$ values, the Co substitution lowers the $\kappa$ value. Since the $\kappa$ values of (Zr,Nd)Fe$_{10}$CoTi are 0.87 (GGA+$U$) and 0.90 (LSDA), both are lower than those of the (Zr,Nd)Fe$_{11}$Ti compound with 0.92 (GGA+$U$) and 0.97 (LSDA). As there is no comparable literature for $K_{1}$, $H_{a}$ and $\kappa$ for (Zr,Nd)Fe$_{11}$Ti or (Zr,Nd)Fe$_{10}$CoTi, these values can only be roughly compared with the similar  (Zr$_{0.3}$Nd$_{0.7}$)(Fe$_{0.75}$Co$_{0.25}$)$_{11.5}$Ti$_{0.5}$ compound. The literature sample reaches 1.41 MJ/m$^{3}$ for $K_{1}$, 1.7 T for $H_{a}$ and a $\kappa$ value of 0.71 <cit.>. Therefore, the values of the quaternary and quinary compounds of this work are significantly higher than those of the mentioned literature compound. The deviations can be explained by a higher Zr content as well as a higher Ti and lower Co concentration of the chosen compounds. Thus, a good accordance with the limited available literature can be assumed.

In summary, the hardness factor of the ZrNdFe$_{22}$Ti$_{2}$ compound is lower than the values of the quaternary compounds containing Y and Ce. This is because the calculations yield values for $\kappa$ that are just below 1. However, the computed values of 0.92 (GGA+$U$) and 0.97 (LSDA) are very close to $\kappa = 1$, requiring experimental verification.

§ CONCLUSION AND OUTLOOK

In summary, starting from the promising findings on the phase stability and magnetic properties of Nd-lean permanent magnets such as (X,Nd)Fe$_{12-y}$Ti$_{y}$ (X: Y and Ce; y: $0 \leq y \leq 1$) <cit.>, systematic investigations on the intrinsic magnetic properties of (Zr,Nd)Fe$_{12-y}$Ti$_{y}$ compounds were performed. Zr has been chosen as an alternative to Nd, being an abundant, less expensive and less critical material to be procured. Since there was no available literature on the chosen Zr-containing phases, only the NdFe$_{12-y}$Ti$_{y}$ compounds were compared with literature directly. However, the Zr-containing phases were compared with the similar (Zr$_{0.3}$Nd$_{0.7}$)(Fe$_{0.75}$Co$_{0.25}$)$_{11.5}$Ti$_{0.5}$ compound. As Co is substituted together with Zr in the literature <cit.>, this is also discussed by considering the compound (Zr,Nd)Fe$_{10}$CoTi. The resulting good agreement of the literature with the calculated values gives great confidence in the presented magnetic properties of investigated compounds.

The used methodology correctly predicts the expected overall magnetisation trend of the 1:12 phases, with the values of the NdFe$_{12-y}$Ti$_{y}$ compounds being larger than those of the ZrFe$_{12-y}$Ti$_{y}$ (y: $0 \leq y \leq 1$) compounds. Experimentally determined values for the saturation magnetisation $\mu_{0}M_{S}$ of the NdFe$_{11}$Ti compound are in the range of 1.38 to 1.70 T <cit.> depending on the measurement temperature and method. The calculated values using DFT (DFT+$U$) are 1.68 (1.69) T. In case of ZrFe$_{11}$Ti and (Zr,Nd)Fe$_{11}$Ti compounds there is no direct literature to compare. The $\mu_{0}M_{S}$ values calculated by DFT (DFT+$U$) are 1.51 T for ZrFe$_{11}$Ti and 1.60 (1.62) T for (Zr,Nd)Fe$_{11}$Ti. A comparison of the value of (Zr,Nd)Fe$_{11}$Ti with a similar compound such as (Zr$_{0.3}$Nd$_{0.7}$)(Fe$_{0.75}$Co$_{0.25}$)$_{11.5}$Ti$_{0.5}$ with the value of $\mu_{0}M_{S}$ of 1.63 T <cit.> shows a good agreement.

Next to the magnetisation calculations promising $|BH|_{max}$ values were obtained for (Zr,Nd)Fe$_{11}$Ti, whose DFT (DFT+$U$) value is 510 kJ/m$^{3}$ (525 kJ/m$^{3}$). These values are higher than that of Sm$_{2}$Co$_{17}$B with 294 kJ/m$^{3}$ <cit.>. Moreover, the quaternary compound (Zr,Nd)Fe$_{11}$Ti exceeds the values of 509 kJ/m$^{3}$ and 490 kJ/m$^{3}$ for (Y,Nd)Fe$_{11}$Ti and (Ce,Nd)Fe$_{11}$Ti <cit.> as calculated from Erdmann et al.. It must be noted that even higher $|BH|_{max}$ values can be obtained by using a lower Ti content in the quaternary compound (Zr,Nd)Fe$_{11.5}$Ti$_{0.5}$. The calculated values of the (Zr,Nd)Fe$_{11.5}$Ti$_{0.5}$ compound are 600 and 603 (DFT+$U$) kJ/m$^{3}$. With these values, the quaternary Zr-containing compound clearly exceeds the value of Nd$_{2}$Fe$_{14}$B with 512 kJ/m$^{3}$ <cit.>. The substitution of Co into the (Zr,Nd)Fe$_{11}$Ti compound at 7.7 at.%, giving the (Zr,Nd)Fe$_{10}$CoTi compound, results in a slight increase in saturation magnetization and a minimal decrease in the $|BH|_{max}$ value.

In addition, mean field approximation (MFA) was used to calculate the Curie temperature, where a systematic overestimation was observed in the ferromagnetic (FM) state. Nevertheless, the values obtained are in quantitative agreement with the literature. As Miyake et al. <cit.> reported, better results are obtained for the REFe$_{11}$Ti compounds using the local moment disordered (LMD) state which can be supported by NdFe$_{12-y}$Ti$_{y}$ compounds (y: $0 \leq y \leq 1$). In case of the ZrFe$_{12-y}$Ti$_{y}$ (y: $0 \leq y \leq 1$) compounds, the FM states appear to be more applicable. Note that FM-based calculations tend to overestimate (margin of error for the ternary compounds is about 45%). The calculated Curie temperatures of the Nd-lean quaternary compound (Zr,Nd)Fe$_{11}$Ti are 783 (FM) and 652 (LMD) K. In consequence, this compound has a higher $T_{C}$ than Nd$_{2}$Fe$_{14}$B (588 K). The $T_{C}$ can further be increased by Co substitution. According to the calculations an increase of about 85 K with a substitution of about 7.7 at% Co from (Zr,Nd)Fe$_{11}$Ti to (Zr,Nd)Fe$_{10}$CoTi is achievable.

For NdFe$_{11}$Ti, the exchange correlation function based on the generalised gradient approximation (GGA) fails to calculate the magnetocrystalline anisotropy constant $K_{1}$, as already reported in the literature <cit.>. This is also the reason why the Local Spin Density Approximation (LSDA) has been used to obtain more accurate data. As the Zr concentration increases, GGA+$U$ supplies better results. The addition of Zr (Co) leads to an increase (small decrease) of the magnetocrystalline anisotropy. The found effects of Co substitution are supported by the work of Suzuki et al. <cit.>. From $K_{1}$ and $\mu_{0}M_{S}$, the hardness factor $\kappa$ can be calculated, which indicates the potential of a material to develop into a permanent magnet for which $\kappa > 1$ is desired. The 50% Nd-lean quaternary compound (Zr,Nd)Fe$_{11}$Ti gives a theoretical $\kappa$ value of 0.97, classifying it as a semi-hard magnet. In order to obtain a higher $\kappa$ value, nitrogen substitution is a suitable option, as shown in previous works <cit.>.

In summary, the thesis shows that the replacement of the critical RE element Nd by more abundant Zr in ThMn$_{12}$ compounds provides possible and promising RE-TM magnets, which have comparable or only slightly lower magnetic properties than the currently utilized Nd-Fe-B magnets. Thus, the new quaternary (ZrNd)Fe$_{24-y}$Ti$_{y}$ compounds can be considered as a potential alternative to the Nd-Fe-B magnets. In parallel, the findings of this work suggest the investigation of the suggested quarternary ZrNdFe$_{24-y}$Ti$_{y}$ hard/semi-hard magnets, in order to confirm the theoretical calculations via experimental measurements.

As an outlook to this study, further investigation of other yet uninvestigated elements, besides Zr, using a combination of DFT and KKR calculations can be considered for substitution with Nd atoms. Beyond that, an addition or another substitution to Zr, such as nitrogenation, may be possible to compensate for the weaknesses of the (ZrNd)Fe$_{24-y}$Ti$_{y}$ compounds. An alternative approach could involve the expansion of research to encompass other established magnetic structures, such as the 2:14 or 2:17 phases, in order to explore the potential decrease in RE elements within these compounds. The quaternary compounds studied have similar T$_{C}$'s as the Nd-Fe-B magnets, which makes them very attractive from a technological point of view. Subsequently, they could be used in almost the same range of applications as the Nd-Fe-B magnets, when slightly lower magnetic properties are acceptable.

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titleJacob’s ladder of density functional approximations for the exchange-correlation energy

journaltitleCanadian Journal of Physics
titleAccurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis

journaltitlePhysical Review Letters
titleGeneralized gradient approximation made simple

journaltitlePhysical Review B
titleAccurate and simple analytic representation of the electron-gas correlation energy

journaltitlePhysical Review B
titleAtoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation

journaltitlePhysical Review B
titleProjector augmented-wave method

journaltitlePhysical Review B - Condensed Matter and Materials Physics
titleFrom ultrasoft pseudopotentials to the projectoraugmented-wave method

journaltitleRapid Communications
titleSoft self-consistent pseudopotentials in a generalized eigenvalue formalism

Cham, Switzerland

Springer International Publishing AG

titleSemiconductor Physics

journaltitleJournal of Physics C: Solid State Physics
titleA local exchange-correlation potential for the spin polarized case. I

titleHandbook of Magnetism and Advanced Magnetic Materials

journaltitlePhysical Review B
titleDensity-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulators

journaltitlePhyical Review B
titleElectron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study

journaltitlePhyical Review B
titleHubbard model in infinite dimensions

journaltitleJournal of Magnetism and Magnetic Materials
titleLocal Spin Density Functional Approach to the Theory of Exchange Interactions in Ferromagnetic Metals and Alloys

journaltitleJournal of Physics F: Metal Physics
titleStudy of the 4f and valence band density of states in rare-earth metals. I. Theory of the 4f states

journaltitleSSRN Electronic Journal
titleMagnetic Properties of NdFe$_{11}$Ti and YFe$_{11}$Ti, from Experiment and Theory

journaltitleComputational Materials Science
titleEfficiency of ab-initio total energy calculations formetals and semiconductors using a plane-wave basis set

journaltitlePhysical Review B - Condensed Matter and Materials Physics
titleEfficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set

titleRare earth magnetism: structures and excitations

journaltitlePhysica XIII
titleOn the Calculation of the Energy of a Bloch Wave in a Metal

journaltitlePhysical Review
titleSolution of the schrödinger equation in periodic lattices with an application to metallic lithium

journaltitleProgress of Theoretical Physics
titleA Reformulation of the Coherent Potential Approximation and Its Applications

journaltitlePhysica B+C
titleResidual resistivity of Ni-Fe, Ni-Cr and other ferromagnetic alloys

New York

Pergamon Press

titleCalculated Electonic Properties of Metals

journaltitleJournal of Physics D: Applied Physics
titleBand structure theory of magnetism in 3d-4f compounds

journaltitleNew Journal of Physics
title$Ab$ $initio$ screening methodology applied to the search for new permanent magnetic materials

journaltitleScientific Reports
titleTheoretical screening of intermetallic ThMn$_{12}$ type phases for new hard-magnetic compounds with low rare earth content

journaltitlePhysical Review Materials
titleAb initio phase stabilities of Ce-based hard magnetic materials and comparison with experimental phase diagrams

journaltitleJournal of Alloys and Compounds
titleThermal decomposition of ThMn$_{12}$-type phase and its optimum stabilizing elements in SmFe$_{12}$-based alloys

journaltitleApplied Sciences
titleThermodynamic Modeling of Formation Enthalpies of Amorphous and Crystalline Phases in Zr, Nd, and Ce-Substituted Fe-Si Systems

journaltitlePhysical Review B
titleStatic equation of state of bcc iron

journaltitlePhysical Review B
titleStructure and magnetism in bcc-based iron-cobalt alloys

journaltitlePhysical Review B
titleEquation of state of cobalt up to 79 GPa

journaltitlePhysical Review B
titleThermal equations of state for titanium obtained by high pressure-temperature diffraction studies

journaltitleThe Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics
titleThe electronic structure of the metals of the first transition period

journaltitleMechanics of Materials
titleAcoustic velocities, elasticity, and pressure-induced elastic softening in compressed neodymium

journaltitleThe Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics
titleMagnetic saturation in alloys of neodymium at low temperatures

journaltitlePhysical Review B
titleThermal equations of state of the $\alpha$, $\beta$, and $\omega$ phases of zirconium

journaltitlePhysical Review B
titleHigh-magnetic-field magnetization and magnetoresistance of a transition-metal-base spin glass: Zr-Mn

journaltitleJournal of Magnetism and Magnetic Materials
titlePhysical properties of nitrogenated RFe$_{11}$Ti intermetallic compounds (R=Ce, Pr and Nd) with ThMn$_{12}$-type structure

University Grenoble

Ph.D. thesis

titleRole of interstitial elements on permanent magnet alloys based on rare earth elements and iron: Synthesis, structural study, spectroscopic analysis in relation to magnetic properties

journaltitlePhysical Review Materials
titleBayesian optimization of chemical composition: A comprehensive framework and its application to RFe$_{12}$-type magnet compounds

journaltitleJournal of Alloys and Compounds
titleAb-initio study of the electronic structure and magnetic properties of Ce$_{2}$Fe$_{17}$

journaltitleThe Journal of The Minerals, Metals & Materials Society
titleRare-Earth Lean Hard Magnet Compound NdFe$_{12}$N

journaltitleActa Materialia
titleFirst-principles investigation of Nd(Fe,M)$_{12}$(M = K-Br) and Nd(Fe,Cr,Co,Ni,Ge,As)$_{12}$: Possible enhancers of Curie temperature for NdFe$_{12}$ magnetic compounds

journaltitlePhysical Review B - Condensed Matter and Materials Physics
titleNitrogen as the best interstitial dopant among X=B, C, N, O and F for strong permanent magnet NdFe$_{11}$TiX: First-principles study

journaltitleJournal of Applied Physics
titleMagnetic and crystallographic properties of novel Fe‐rich rare-earth nitrides of the type RTiFe$_{11}$N$_{1-\delta}$ (invited)

journaltitleJournal of Applied Physics
titleA magnetic and Mössbauer spectral study of the spin reorientation in NdFe$_{11}$Ti and NdFe$_{11}$TiH

Oxford University Press

titleSimple Models of Magnetism

journaltitleJournal of Physics: Condensed Matter
titleRecent developments in hard magnetic bulk materials

journaltitleJournal of Physics F: Metal Physics
titleA first-principles theory of ferromagnetic phase transitions in metals

journaltitlePhysical Review B
titleLocal moment disorder in ferromagnetic alloys

journaltitleScience and Technology of Advanced Materials
titleUnderstanding and optimization of hard magnetic compounds from first principles

Cambridge U.K.

Cambridge University Press

titleElectrons at the Fermi Surface

journaltitlePhysics of the Solid State
titleMagnetic anisotropy of YFe$_{11}$Ti and its hydride

journaltitleJournal of Materials Science: Materials in Electronics
titleLow-field magnetocaloric effect of NdFe$_{11}$Ti and SmFe$_{10}$V$_{2}$ compounds

journaltitlePhysical Review B
titleIntrinsic magnetic properties in R(Fe$_{1-x}$Co$_{x}$)$_{11}$Ti$Z$ (R = Y and Ce; $Z$ = H, C, and N)

tocsectionList of Abbreviations

List of Abbreviations

RE - Rare-earth

TM - Transition metal

$m_{tot}$ - Total magnetic moment

µ$_{0}$ - Magnetic constant

µ$_{B}$ - Bohr magneton

$H$ - Magnetic field

$H_{a}$ - Anisotropy field

$H_{c}$ - Coercivity field

$H_{d}$ - Internal magnetic field

$M$ - Magnetisation

$M_{r}$ - Remanence magnetisation

$M_{S}$ - Magnetisation saturation

$M(T)$ - Finite temperature magnetisation

$T_{C}$ - Curie temperature

$|BH|_{max}$ - Maximum energy product

MCA - Magnetocrystalline anisotropy

MAE - Magnetocrystalline anisotropy energy

$\Delta _{MAE}$ - Anisotropy energy

$K_{1}$ - Anisotropy constant 1$^{st}$ order

$\kappa$ - Hardness factor

HF - Hartree-Fock

post-HF - post Hartree-Fock

DFT - Density functional theory

DFT +U - Density functional theory with applied Hubbard U correction

DMFT - Dynamical mean field theory

LDA - Local Density Approximation

LSDA - Local Spin Density Approximation

GGA - Generalised Gradient Approximation

PW91 - Functional by Perdew and Wang

PBE - Functional by Perdew, Burke and Ernzerhof

PAW - Projector Augmented Wave

USPP - Ultrasoft Pseudopotentials

VASP - Vienna ab initio Simulation Package

KKR - Korringa-Kohn-Rostoker

SCF - self-consistent field

FM - Ferromagnetic

LMD - Local Moment Disorder

DLM - Disorder local moment

MFA - Mean-field approximation

ASA - Atomic sphere approximation

CPA - Coherent potential approximation

MJW - Moruzzi, Janak and Williams

BCC - Body-Centered Cubic

HCP - Hexagonally close packed

TB-LMTO-ASA - Tight-binding linear-muffin-tin-orbital atomic-sphere-approximation

DOS - Density of states

XRD - X-Ray diffraction

PPMS - Physical property measurement system

VSM - Vibrating sample magnetometer

Hiermit versichere ich an Eides statt, dass ich diese Arbeit selbstständig verfasst und keine
anderen als die angegebenen Quellen und Hilfsmittel benutzt habe. Außerdem versichere
ich, dass ich die allgemeinen Prinzipien wissenschaftlicher Arbeit und Veröffentlichung,
wie sie in den Leitlinien guter wissenschaftlicher Praxis der Carl von Ossietzky Universität
Oldenburg festgehalten sind, befolgt habe.

Nico Yannik Merkt