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History of rare earth permanent magnets
The first discovered permanent magnets were the lodestones that provided a stable magnetic field. Until the turn of the 19th century, magnets were weak, unstable and made of carbon steel. Some improvements were made with the discovery of cobalt magnet steels in Japan in 1917. Then, the performances of permanent magnets have been continuously improved since the discovery of the alnicos (Al/Ni/Co alloys) in the 30’s . This evolution is represented by the increase in the maximum energy product (BH)max. This latter is a figure of merit for permanent magnets and represents the maximum energy density that, for a magnet of given volume, can be transformed into work in a machine that uses the magnet. Fig. 2 shows the rapid improvement in the performances of magnets encountered in the middle of the 60’s when the first generation of transition metals and rare-earth alloys, such as Sm-Co systems, was developed. However, in the late 70’s, the price of Co increased drastically due to an unstable supply situation in the Democratic Republic of Congo. At that time, Sm-Co permanent magnets showed the highest (BH)max and the research community was then forced to replace these magnets. A few years later, in 1984, Nd-Fe-B based permanent magnets were developed for the first time by Sagawa et al.  using powder metallurgy techniques at Sumitomo Special Metals, and in parallel by Croat et al. ,  using melt-spinning technique at General Motors. As shown by the below graph, in almost a century, (BH)max has been enhanced, starting from ≈1 MGOe for steels at the early part of the century, to ≈56 MGOe for Nd-Fe-B magnets during the past twenty years , .
Moreover, for the same energy density, newly developed Nd-Fe-B magnets enable an important reduction in volume for their applications, compared to former systems. Today, more than 80% of rare earth permanent magnets implemented by end users are Nd-Fe-B magnets. 
While the maximum energy product represents the strength of a magnet, the resistance to demagnetization is crucial for the design of electrical machines, regarding the operating temperature.
From context to coercivity
This second characteristic, named coercivity, will be introduced in more details in the following since large efforts in magnet industry and research groups have been engaged in the three last decades to improve this property. To further optimize magnetic properties of Nd-Fe-B magnets, emphasis of research is nowadays to better understand the link between magnetization reversal (coercivity) mechanisms and microstructure in these materials. 
Intrinsic and extrinsic properties of permanent magnets
Rare earth permanent magnets are mainly constituted by ferromagnetic materials that exhibit remarkable basic properties. These magnetic properties are generally seen as “intrinsic” since they are completely determined by the atomic composition and the structure of the ferromagnetic phases. It is important to keep in mind that standard magnet characteristics, widely considered for design purposes, are rather “extrinsic” since the magnet performances are strongly affected by the process parameters . In this section, the basic properties of hard ferromagnetic materials are briefly recalled in order to point out the influence of the microstructure on them.
Intrinsic magnetic properties
Definition of the macroscopic magnetization
The macroscopic magnetization of a magnetic material is denoted M and corresponds to the volume density of internal magnetic moments. This amount is therefore given in A.m2/m3, thus in A/m, while the polarization of the material is given by J = µ0M, with J in Tesla and µ0 = 4π x 10-7 T.m/A the permeability of free space. When an external magnetic field H (in A/m or Oe) is applied on the material, the magnetic induction B is expressed as B = µ0(H+M), with B in Tesla . The macroscopic characterization of magnetic materials generally consists in the measurement of the evolution of B (or J) as a function of H. This reveals how magnetization develops in the material and helps determining the most energy-favorable configurations of the magnetization distribution. In this frame, permanent magnets display specific features that are introduced below.
A ferromagnetic material displays a spontaneous macroscopic magnetization that comes from the ordering of individual microscopic magnetic moments. These latter are carried by atoms and result from the summation of spin and orbital moments of electrons. Transition metals (Fe, Co…) display the largest magnetic moment per atom (at room temperature). The ordering is related to the exchange interaction occurring between the magnetic moments.
A material is hard ferromagnetic when the microscopic magnetic moments are preferentially oriented along a specific crystallographic direction. This property depends on the magnetocrystalline anisotropy of the atomic lattice. These three features (high magnetic moment, ordering and anisotropy) are required for hard magnets and are fulfilled with the magnetic phase that constitutes them.
In these materials, neighboring magnetic moments are strongly coupled through exchange interactions. Exchange occurs between electronic orbitals and induces an internal energy minimization when the moments are aligned in parallel directions (ferromagnetism) or in antiparallel directions (ferrimagnetism). Basically, the exchange energy is given by = ∑ , for which the summation is extended to all couples of microscopic magnetic moments. However, exchange interactions are a short range effect that develops at a distance roughly equal to the lattice parameter. The amount A characterizes the microscopic exchange stiffness and is related to the shape of electronic orbitals and the crystal structure.
This trend towards parallel or antiparallel distribution of the magnetic moments gives rise to magnetic ordering at finite temperature and zero field. This results in the occurrence of the macroscopic spontaneous magnetization MS. Furthermore, ferromagnetic ordering is limited by the thermal agitation and disappears above the Curie temperature TC (in K). As expected, the Curie temperature is proportional to the exchange stiffness: ≈ [Eq. 1] with a0 the lattice parameter of the considered structure. The exchange stiffness is expressed in J/m and determines also the exchange length Lex (in nm). This quantity is the length below which atomic exchange interactions dominate dipolar interactions. It is given by: = √ 2 [Eq. 2]
Magnetic anisotropy corresponds to the existence of energetically favorable directions for magnetization, related to the crystalline axes (magnetocrystalline anisotropy). In a uniaxial crystallographic system, the anisotropy energy Ea is defined as: ( ) = 1 2( ) + 2 4( ) + 3 6( ) + ⋯ Ki is the i-th order anisotropy constant in MJ/m3 and θ is the angle between the magnetization direction and the easy axis. Only the first term is generally considered for Nd-Fe-B systems. In the absence of external magnetic field, magnetization will preferentially lie along the z-easy axis, with either the positive or negative orientation. The energy needed to align magnetization along any direction perpendicular to the easy axis is the anisotropy energy.
Magnetic domains form in a magnetic material as a result of the magnetostatic energy reduction. They are regions in which magnetization is uniform, while its direction may vary from one domain to another. Magnetic domains are also called Weiss domains. Between two magnetic domains of opposite magnetization, the magnetization vector has to change its direction. The transition area is called a domain wall. A particular type is the Bloch domain wall for which magnetization rotates in the plane of the domain wall. 
The transition length of magnetization reversal is called domain wall width and noted δW (in nm). In the particular case of a Bloch domain wall, it can be approximated by: ≈ √ [Eq. 4]
Moreover, the associated domain wall energy γW (in J/m2) can be expressed as: ≈ √ 1 [Eq. 5]
The hysteresis loop is the most common characterization of a magnetic material, underlying many processes that imply magnetic domains. Extrinsic magnetic properties that can be determined from the hysteresis loop depend strongly on the above introduced parameters.
Extrinsic magnetic properties
Hysteresis loop of a permanent magnet: remanence and coercivity
When an external magnetic field H is applied to a permanent magnet which is originally in a demagnetized state, the magnetization M follows the initial magnetization curve that increases rapidly and then approaches an asymptotic value called the saturation magnetization MS. When the magnetic field is decreased from the saturated state, the magnetization gradually decreases and at zero field strength, it reaches a non-zero value called the remanent magnetization or remanence MR. Further increase of the magnetic field in the negative sense results in a continued decrease of magnetization, which finally falls to zero. The absolute value of the field at this point is called the coercive field or coercivity HC. It represents the resistance to demagnetization of the permanent magnet. Another definition says that the magnetic susceptibility (i.e. the quantity dM/dH) is maximal at the coercivity point. The curve in the second quadrant from MR to the zero magnetization state is referred as the demagnetization curve. Further increase of H in the negative sense results in a decrease of M until reaching the -MS value. When H is then reversed again to the positive sense, M increases again and the loop is closed (see Fig. 3). 
This dependence of the magnetization as a function of the applied magnetic field constitutes the hysteresis loop of a permanent magnet. The evolution of the magnetic induction B or of the polarization J with H are also hysteresis loops. Hard magnets are difficult to demagnetize: they exhibit a larger coercive field and thus a broader hysteresis loop than soft magnets.
Microstructure related magnetic properties
Extrinsic magnetic properties depend on intrinsic properties and on microstructure. The remanence MR is directly proportional to MS, depends on the porosity of the material and on the degree of alignment of magnetic easy axes of the hard magnetic phase. The coercive field HC depends on the magnetocrystalline anisotropy, on the presence of defects in the microstructure and on other microstructural features such as the grain size, determined by the fabrication process. , 
HC determines if the magnet is hard and MR directly impacts the maximum energy product (BH)max (in kJ/m3 or MGOe). This latter is a figure of merit for permanent magnets. On the hysteresis loop (see Fig. 3), (BH)max is the area of the largest rectangle that can be inserted under the demagnetization curve. Its maximum possible value for an ideal system is µ0MS2/4. 
Magnetization reversal mechanisms
Coherent rotation: the Stoner-Wohlfarth model
The Stoner-Wohlfarth model  describes magnetization reversal in a ferromagnetic crystal by coherent rotation involving all magnetic moments (i.e. without formation of domains). In such system, the magnetic moments are considered to remain parallel. As a result, the exchange interaction is neglected in the model and the total energy of the system is the summation of the Zeeman energy (coming from the interaction with the external field Hext and that tends to align the moments along Hext), and the magnetocrystalline energy that prevents the moments from deviation from the easy axes. Magnetization is considered to be homogeneous and the applied field is along the easy axis direction, as shown in Fig. 4.
In the case of uniaxial systems with strong magnetocrystalline anisotropy, there is only one easy axis for magnetization. During reversal, magnetization changes its direction, but not its magnitude. Cases (1) and (3) correspond to two energy minima, whereas case (2) is the hard axis magnetization configuration. The external field required to reverse the magnetic moment and defined as the anisotropy field HA is given by: = 2 1 [Eq. 6] µ0
The anisotropy field in Nd-Fe-B magnets, calculated from K1 and MS intrinsic properties, is around 8 T at room temperature. In practice for Nd-Fe-B magnets, the measured coercivity is only about 20-30 % of the theoretical anisotropy field given by the Stoner-Wohlfarth model. This discrepancy is known as Brown’s paradox  and is attributed to the presence of defects in the microstructure that exhibit locally reduced magnetocrystalline anisotropy . This has been understood by introducing the concept of the activation volume that represents the smallest volume in which magnetization reversal begins before macroscopic propagation.
Nucleation vs pinning controlled magnetization reversal
Magnetization reversal consists of two steps: it begins at defects, corresponding to the nucleation of reversed domains, and then propagation of these reversed domains within the entire microstructure occurs. Depending on their respective field values, either nucleation or propagation could be the process that triggers magnetization reversal and limits coercivity.
After nucleation, reversal may propagate in the entire system for a given magnetic field value: it is in this case controlled by nucleation. Alternatively, the reversed domain may be pinned at magnetic heterogeneities: reversal is, in this case, controlled by pinning (i.e. propagation-driven reversal). Fig. 5 depicts magnetization configuration in the case of nucleation-pinning reversal:
Case (1) corresponds to saturation. The direction of the applied field is then reversed and nucleation starts at (2) with the formation of a small domain with reversed magnetization and its respective domain wall. At (3), this latter starts to move and then encounters defect points that act as pinning centers for the domain wall. At (4), a bigger field value is applied for the depinning of the domain wall, to finally achieve saturation in the opposite direction.
Two models based on the micromagnetic approach have been proposed to describe coercivity in Nd-Fe-B permanent magnets and to determine the mechanism controlling magnetization reversal. They will be presented in Section I.5.2.
Table of contents :
I. From context to coercivity
I.1. History of rare earth permanent magnets
I.2. Intrinsic and extrinsic properties of permanent magnets
I.2.1. Intrinsic magnetic properties
I.2.2. Extrinsic magnetic properties
I.3. Magnetization reversal mechanisms
I.3.1. Coherent rotation: the Stoner-Wohlfarth model
I.3.2. Nucleation vs pinning controlled magnetization reversal
I.4. Nd-Fe-B sintered magnets: fabrication and microstructure
I.4.1. Microstructure of a Nd-Fe-B sintered magnet
I.4.2. Industrial production process: sintering
I.4.3. Application fields and limitations
I.4.4. State of the art: current strategies to improve coercivity
I.5. Micromagnetic simulations and models
I.5.1. Micromagnetic simulations (Landau-Lifshitz-Gilbert formalism)
I.5.2. Micromagnetic and global models
I.5.3. Main results of micromagnetic simulations
I.6. The grain boundary diffusion process (GBDP) in Nd-Fe-B sintered magnets
I.6.1. Benefits of core-shell microstructure
I.6.2. State of the art
I.6.3. Micromagnetic simulations on core-shell structures
I.7. Problematic of the thesis
II. Fabrication of Nd-Fe-B sintered magnets, characterization and numerical methods
II.1. From ribbon to green compact
II.1.2. Hydrogen decrepitation to coarse powder
II.1.3. Jet milling
II.1.4. From powder to green compact
II.2. Sintering furnace
II.2.1. Sintering heat treatment
II.2.2. Post-sinter annealing (PSA) heat treatment
II.3. GBDP on Nd-Fe-B sintered magnets
II.3.1. Sample preparation for GBDP
II.3.2. Diffusion heat treatment
II.4. Characterization methods
II.4.1. Magnetic characterization
II.4.3. Microstructural characterization
II.5. Numerical methods
II.5.1. FEMME software package
II.5.2. Flux 3D software
III. Coercivity of polycrystalline hard magnets
III.2. Study of collective magnetostatic effects: experimental approach
III.2.1. Model for the demagnetization field correction
III.2.2. Experimental protocol and results
III.2.3. Analysis and model improvement
III.3. Study of collective magnetostatic effects: numerical approach
III.3.1. Closed-circuit configuration simulation
III.3.2. Open-circuit configuration simulation
III.3.3. Discussion about collective effects
III.3.4. Experimental validation
III.4. Magnetostatic coupling in heterogeneous magnets
III.4.1. J-H curve of a two-grain-population magnet
III.4.2. J-H curve of duplex magnets
IV. Experimental and computational study of magnetization reversal in Dy-Co diffused Nd-Fe-B sintered magnets
IV.1. Magnetic properties in the as-sintered state and after post-sinter annealing (PSA)
IV.1.1. Experimental results
IV.1.2. FEMME simulations: sintering vs PSA
IV.2. Magnetic properties after GBDP and post-diffusion annealing (PDA)
IV.2.1. GBDP using intermetallic compound vs eutectic alloy
IV.2.2. Influence of diffusion time on magnetic properties
IV.2.3. Influence of diffusion temperature on magnetic properties
IV.2.4. Influence of PDA on magnetic properties
IV.2.5. M(T) measurements
IV.2.6. Characterization of microstructure and coercivity profiles
IV.2.7. FEMME simulations: core-shell model
V. Discussion: coercivity of graded magnets
V.1. Modelling of diffusion profiles
V.1.1. Diffusion model hypothesis
V.1.3. Impact on coercivity profile
V.1.4. Diffusion of Co
V.2. Polycrystalline model applied to Dy-diffused thick magnets
V.2.1. Description of the geometrical model
V.2.2. Results: grain reversal patterns in a graded sample
V.2.3. Results for other diffusion conditions
Conclusions and prospects