The properties of an itinerant or band ferromagnet are exemplarily shown for Fe. Calculation of the density of states as a function of energy with respect to the Fermi energy EF for Fe neglecting the exchange splitting, i. Copyright , with permission from Elsevier i. Calculation of the density of states as a function of energy with respect to the Fermi energy EF for Pd which also exhibits a large value at EF compared to Fe Reprinted from [1] p.

Copyright , with permission from Elsevier 3. Calculation of the density of states as a function of energy with respect to the Fermi energy EF for the noble metal Cu. Reprinted from [1] p. Copyright , with permission from Elsevier Fig. Left: Calculation of the spin resolved density of states as a function of energy with respect to the Fermi energy EF for Fe taking into account the exchange splitting. Right: Calculated spin resolved band structure of Fe.

Majority bands are characterized by dark points, minority bands by light points. Reprinted from [1] pp. First, the number of majority and minority electrons directly at the Fermi energy is no more identical. Second, more majority electrons below EF are present than minority electrons. The spin resolved band structure right part of Fig. This crossing can directly be probed using, e. This representation is called Slater—Pauling curve see Fig. For Fe50 Ni50 this value is very close to that of Co. Saturation magnetization of ferromagnetic alloys as a function of the electron concentration Adapted from [2] 3.

The magnetic moment per atom reaches a maximum with about 2. It assumes that the s and d bands are rigid in shape with varying atomic number. This model is of course not correct but we can understand some trends in physical properties. Fe possesses 8 valence electrons being in 3d and 4s states whereas Ni has The remaining 7.

On the other hand Ni exhibits 0. The magnetic moment of 0. These metals are found on the left side of the maximum in the Slater—Pauling curve. One example is Fe. They only have holes in the minority band and are located on the right side of the peak in the Slater—Pauling curve. One example is given by Ni. Thus, the Fermi energy moves up through the rigid bands with increasing Ni content.

However, the magnetic moment per atom may be simply calculated if the Fermi energy EF lies above the top of the spin up band, i. This equation also explains why the average moment of Co should be very close to that of Ni50 Fe Both exhibit the same valence electron concentration and thus the same value of nd. Further, this consideration explains the observation of non-integral average magnetic moments in 3d alloys. Additionally, we understand why the maximum in the Slater—Pauling curve is reached with a value of 2.

For a lower d electron concentration than 7. This situation is realized in weak ferromagnets. As already mentioned above the rigid-band model is a rather naive picture due to the assumption that the band structure and the shape of the curve describing the density of states do change with alloy composition in reality. Another problem is that this model gives the averaged value of the magnetic moment of an alloy.

It does not allow to determine the magnetic moment of each constituent individually. In the FeNi alloy closed and open squares the magnetic moment of each Fe atom as well as that of each Ni atom nearly remain constant. This latter observation directly gives evidence that the slope in the Slater—Pauling curve cannot be constant over the whole range.

Magnetic moments per atom in 3d transition metal alloys as a function of the electron concentration. The corresponding line is comparable to the Slater—Pauling curve see Fig. Data taken from [3] and [4] Problems 3. As a result magnetic long range order can occur. We will start our considerations with the interaction of two single magnetic dipoles.

Depending on the distance between the magnetic moments we distinguish between the direct and indirect exchange. If the overlap of the involved wave functions is only small e. For this class of systems the indirect exchange interaction is responsible for magnetism. But, the order temperature typically reaches values of several K for a lot of ferromagnetic materials.

Therefore, the magnetic dipole interaction is too small to cause ferromagnetism. The electrons belonging to are undistinguishable. Therefore, the wave function squared must be invariant for the exchange of both electrons. The second term is spin dependent and the important one concerning ferromagnetic properties.

If the exchange integral J is negative then ES with Jij being the exchange constant between spin i and spin j. The factor 2 is included in the double counting within the sum. Crystal and magnetic structures of MnO means of a non-magnetic ion which is located in-between. The distance between the magnetic ions is too large that a direct exchange can take place. An example of an antiferromagnetic ionic solid is MnO see Fig. There are two possibilities for the relative alignment of the spins in neighboring Mn atoms.

A parallel alignment leads to a ferromagnetic arrangement whereas an antiparallel alignment causes an antiferromagnetic arrangement. In the antiferromagnetic case the electrons with their ground state given in a can be exchanged via excited states shown in b and c leading to a delocalization. For ferromagnetic alignment with the corresponding ground state presented in d the Pauli exclusion principle forbids the arrangements shown in e and f.

Thus, no delocalization occurs. Therefore, the antiferromagnetic coupling between two Mn atoms is energetically favored as depicted in Fig. It is important that the electrons of the oxygen atom are located within the same orbital, i. The exchange is mediated via the valence electrons; it is therefore not a direct interaction. Occurrence of a super exchange interaction in a magnetic oxide. This type of exchange coupling is long range and anisotropic which often results in complicated spin arrangements. Additionally, it possesses an oscillating behavior.

Positive values light gray area lead to a ferromagnetic coupling whereas negative ones dark gray area result in an antiferromagnetic arrangement 46 4 Magnetic Interactions moments. One example is represented by rare earth metals with their localized 4f electrons. A detailed discussion is given in Chap. A schematic overview is given in Fig. Ferromagnets exhibit magnetic moments which are aligned parallel to each other.

In antiferromagnets adjacent magnetic moments are oriented in Fig. Magnetic moments in a spin glass are frozen out with a random orientation. A helical or spiral arrangement is given if the magnetic moments are aligned parallel in a plane but the direction varies from plane to plane in such a way that the vector of the magnetic moment moves on a circle or a cone, respectively. The relative magnetization is therefore given by cf. This procedure can be used to obtain the critical temperature TC graphically. The derivation of 5. Using the expression for the critical temperature see 5.

Now, we want to look for the behavior of the spontaneous magnetization near the critical temperature TC which was found to be see 5. This result does not correctly describe the situation for all ferromagnets. For low temperatures the physics behind is more complicated compared to this simple picture and will be discussed in Chap. Due to see 5. The phase transition has been vanished. TC scales with the strength of the exchange interaction. The latter situation is schematically shown in Fig.

Equation 5. Above the transition temperature the magnetic susceptibility can be expressed as: 1 5. Thus, the argument of the Brillouin function can be approximated as expressed in 5. In this situation the magnetization of the sublattices amounts to: c 5. Thus, the total magnetization remains constant. Two forces are acting against each other. Above TN all spins are free to rotate thus loosing a preferred orientation. Possible types of antiferromagnets in the simple cubic form are shown in Fig. Type A results in a layered structure which each layer being ferromagnetically ordered.

It is called topological antiferromagnet. Type B leads to a chain-like arrangement of the spins. An antiferromagnetic arrangement of nearest neighbors often occurs in materials which couple via the superexchange interaction, e. MnO details are discussed in Chap. Temperature dependence of the magnetic left and reciprocal magnetic susceptibility right for antiferromagnetic materials. The Bethe—Slater curve describes the relation of the exchange constant with the ratio of the interatomic distance rab to the radius of the d shell rd The exchange interaction between neighboring magnetic moments being described in the Heisenberg Hamiltonian can lead to a parallel or antiparallel alignment, i.

A spontaneous magnetization occurs below a critical temperature. At high temperatures the magnetic susceptibility exhibits a Curie—Weiss behavior with a negative paramagnetic Curie temperature. The simplest characterization, but already satisfactory for the fundamental understanding, is given by the assumption of two magnetic sublattices with 5. Thus, the total magnetization does not vanish as in the antiferromagnetic case. Whereas in the ferromagnetic and antiferromagnetic case the inverse susceptibility behaves as a linear function of the temperature this situation changes to a hyperbolic behavior for a ferrimagnetic system.

Using 5. Temperature dependence of the reciprocal magnetic susceptibility for ferromagnetic, antiferromagnetic, and ferrimagnetic material 5. Below the transition temperature TC each sublattice exhibits a spontaneous magnetization given by 5. This becomes obvious for the following situation. Substituting in 5. It is characterized by a parallel alignment of the spins within each layer, i. In this situation we have a ferromagnetic or an antiferromagnetic alignment, respectively, between adjacent layers. On the other hand the equation is solved by: J1 5.

Let us discuss the behavior if helical arrangement is present.

## Fundamentals of magnetism

The energies for ferro-, antiferro-, and helimagnetic arrangement amount to see 5. The spins are frozen out below a critical temperature with a statistical distribution of their directions. This spin glass state only occurs in a limited concentration 68 5 Collective Magnetism susceptibility 0. Data taken from [5] range of the magnetic ions. It must be high enough on the one hand to ensure an interaction via the RKKY coupling but low enough on the other hand in order to prevent the formation of clusters or the presence of a direct coupling of the magnetic moments.

Spin glasses exhibit a sharp maximum in the temperature dependence of the magnetic susceptibility see Fig. The temperature T0 which the susceptibility exhibits a maximum at becomes higher with increasing amount of Fe ions and depends on the concentration: 5. Problems 5. Problems 69 5. J2 6 Broken Symmetry The occurrence of a spontaneously ordered state at low temperatures is a fundamental phenomenon in solid state physics. Examples are ferromagnetism, antiferromagnetism, and super conductivity. The parameters which are responsible for this transition can be forces or pressure; mostly it is induced by a varying temperature.

Below a critical value melting temperature the transition to the solid state occurs which is related to the breaking of the symmetry. Contrarily, the solid shown in Fig. Phase transition between the liquid a and solid state b. Above the critical temperature Curie temperature TC the system possesses a complete rotational symmetry; all directions of classical spins or magnetic moments are equivalent see Fig. Below TC a preferential alignment is present. A rotational symmetry only occurs around the direction of magnetization; this directly proves that the symmetry is broken.

An important aspect is the fact that the symmetry of these systems cannot be changed gradually. This can be understood using thermodynamical considerations. At low temperatures the ordered ground state leads to a minimum free energy. At high temperatures F is minimized by a large value of the entropy S which is the disordered state. Landau Theory The free energy of a ferromagnetic system is described by a function of the order parameter using a power series in M. The ground state can be determined by minimizing the free energy F. The system is stable above TC if the magnetization vanishes.

The resulting temperature dependence of the magnetization in the region near the critical temperature is shown in Fig. The advantage of this approach using mean 6. Free energy F M for temperatures below, at, and above the critical temperature TC concerning 6. Magnetization M as a function of temperature T near the critical temperature TC concerning 6. Heisenberg Model This alternative approach is implemented in a microscopic model which only takes into account interactions between nearest neighbors.

The summation is carried out only for nearest neighbor atoms i and j. In the ground state all spins are uniformly oriented in one direction, i. As a consequence defects can spontaneously be created. Thus, a long-range ordering cannot occur which results in a critical temperature identical to zero. These considerations are not only valid for the one-dimensional Ising model but for most of the models in one dimension. Schematic representation of a defect in a two-dimensional Ising model Two-Dimensional Ising Model The spins are arranged on a two-dimensional lattice.

A defect results in an increase of energy as well as of entropy both scaling with the length of the boundary of the defect as schematically shown in Fig. The exact solution for the two-dimensional Ising model was given by Onsager in Three-Dimensional Ising Model Without going in further detail, an ordered state also occurs for temperatures above 0 K, i. The regime of the phase transition is called the critical regime. With increasing temperature the degree of ordering is decreased due to excitations concerning the order parameter.

For crystals lattice vibrations can be correlated with phonons. For ferromagnets spin waves are related to magnons. The boundary represents a defect. In crystals the defect may be a dislocation or grain boundary. In ferromagnets it is a domain wall. Thus, exact solutions are not possible and approximations must be carried out. Critical Exponents for Ferromagnetic Systems The most commonly used critical exponents describing ferromagnetic systems are: 6.

For temperatures T near TC but T 6. Using 6. Consequently, only two independent critical exponents occur whereas the other ones are determined by the scaling laws. Table 6. The same result is obtained using quantum mechanical considerations. Dispersion relation of a magnon in a one-dimensional chain a Fig. Relative spontaneous magnetization of a ferromagnet as a function of the reduced temperature. Especially, we have discussed the energy as a function of the magnitude of the magnetization M but we have neglected the energy dependence on the direction of M.

The Heisenberg Hamiltonian is completely isotropic and its energy levels do not depend on the direction in space which the crystal is magnetized in. However, real magnetic materials are not isotropic. One illustrating example is given by spin waves which are quantized by magnons see Chap. The number of magnons diverges in one and two dimensions, i.

The electron orbitals are linked to the crystallographic structure. Therefore, there are directions in space which a magnetic material is easier to magnetize in than in other ones easy axes or easy magnetization axes. The spin-orbit interaction can be evaluated from basic principles. The magneto crystalline energy is usually small compared to the exchange energy. But the direction of the magnetization is only determined by the anisotropy because the exchange interaction just tries to align the magnetic moments parallel, no matter in which direction.

Due to the reduced symmetry only the indices 1 and 2 are indistinguishable. Under these conditions the term in second order amounts to: 7. The magneto crystalline energy density for cubic materials was given by cf. One example for this situation is bcc-Fe. Magnetization curves of a bcc-Fe, b fcc-Ni, and c hcp-Co. The easy magnetization axes are []- for Fe, []- for Ni, and []-directions for Co. Magneto Crystalline Anisotropy for Tetragonal and Hexagonal Materials The magneto crystalline anisotropy energy per volume for this class of materials was cf.

Now, the easy magnetization axis lies within the []-plane and the []-direction becomes the hard magnetization axis see right part of Fig. If one anisotropy constant Table 7. One example for positive K1 and K2 is Co at low temperatures see Table 7. Thus, the []-direction is the easy magnetization axis.

The rotation from the []-direction towards the basal plane occurs between K and K due to the temperature dependence of K1 and K2 see Fig. The easy magnetization axis is oriented along the []-direction. With increasing amount of Co the anisotropy constant K1 monotonously decreases. This allows to determine the easy, medium, and hard magnetization axes as a function of the stoichiometry using Table 7. It is obvious that with increasing amount of Co the easy magnetization axis changes from []- to []-directions whereas the hard axis behaves conversely.

The medium magnetization axis remains along the []-direction. As discussed above the anisotropy constants additionally depend on the temperature which may alter the magnetization axes as shown in Table 7. Table 7. With increasing amount of Co the easy magnetization axis changes from []- to []-directions whereas the hard axis behaves conversely. The situation becomes more complex if B is oriented parallel to each sublattice magnetization. At B2 saturation is reached.

Thus, the ferromagnetic phase is directly reached. But, an overall isotropic behavior concerning the energy being needed to magnetize it along an arbitrary direction is only given for a spherical shape. This phenomenon is known as shape anisotropy. The calculation is rather complicated for a general shape. It becomes more easy for symmetric objects which is shown in the following. N is a diagonal tensor if the semiaxes a, b, and c of the ellipsoid represent the axes of the coordination system. This situation is only valid for a sphere.

## Books and Media

Equation 7. The prerequisites are a disordered distribution of the atoms in the crystal lattice see Fig. During cool-down or rapid thermal quenching the high-temperature state is frozen out under retention of the oriented magnetization direction. A magnetically induced anisotropic directional short-range order is created as schematically shown in Fig.

In the following a binary alloy with two type of atoms A and B will be discussed. Using 7. Therefore, this type of induced anisotropy is called roll-magnetic anisotropy. Its occurrence is schematically explained for the example of an FeNi - alloy. In this fcc-like alloy a cube texture is present with the -plane in the roll plane and the []-direction being parallel to the roll direction.

Such a sheet exhibits a uniaxial magnetic anisotropy with its easy axis in the plane but perpendicular to the rolling direction see Fig. Magnetization parallel to the rolling direction takes exclusively place through domain rotation giving rise to a linear magnetization curve until saturation is reached. The explanation is given in the following for the example of an A3 B-type superlattice which exhibits a crystal structure shown in Fig. During rolling a plastic deformation takes place.

One part of the crystal slips relative to another part along a gliding plane which is parallel to a -plane for this fcc-like structure see Fig. One part of the crystal is displaced by one atomic distance which results in the creation of BB-type pairs that are not present in the undisturbed crystal. The distribution of these bonds is anisotropic producing a unidirectional anisotropy.

M H roll direction H Fig. Domain structure left and corresponding magnetization curve due to the rolling procedure right A B Fig. Diagram indicating the appearance of BB pairs due to a single step slip along the -plane gray shaded area in an A3 B-type superlattice A further induced anisotropy is given by the exchange anisotropy. This type is discussed on p.

In the following an elastic degree of freedom is additionally allowed. Vice versa, magnetic properties can alter elastic properties. The responsibility of this interplay is the magneto elastic interaction. The magneto crystalline anisotropy energy per volume of cubic systems was given by cf.

As a consequence the cubic unit cell spontaneously deforms to a tetragonal system below the Curie temperature due to a decrease of the energy Ecrys. Thus, the total anisotropy energy density amounts to: 7. The correlation between magnetostriction, elastic, and magneto elastic constants are: B1 2 7. In this situation 7. Due to the Table 7. Therefore, Ni contracts along all three directions if it is becomes magnetic.

This results in a rather complex behavior. Thus, this particular alloy represents a cubic material with isotropic magnetostriction. Additionally, the magnetostriction nearly vanishes. In the following low-dimensional systems are discussed concerning the anisotropy which is related to these interfaces.

The factor of two is due to the creation of two surfaces. The second term exhibits an inverse dependence on the thickness d of the system. Rewriting 7. Due to the shape anisotropy K V is negative. This can directly be seen by the negative slope which results in an in-plane magnetization. The zero-crossing occurs at a positive value K S. The relative amount of the surface contribution increases with decreasing thickness followed by a spin reorientation transition towards the surface normal below dc.

This behavior is exemplarily illustrated in Fig. The slope allows to determine K V. The zero crossing amounts to 2K S. Adapted from [9]; used with permission. Copyright , American Institute of Physics to the surface plane right part. A further phenomenon which is caused by the surface anisotropy is the rotation of the easy magnetization axis within the surface.

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Problems 7. Determine Ki as a function of Ki. Determine the ratio of the induced magnetic anisotropy constants for the three situations. Thus, a demagnetized sample consists of domains each ferromagnetically ordered with a vanishing total magnetization. The boundary between neighbored domains are domain walls. In this chapter general considerations on the behavior of magnetic domains and domain walls are made for macroscopic systems exhibiting a lower length scale of the order of microns.

Properties concerning magnetic domains of lowdimensional systems will be discussed in Chaps. Each of them exhibits the saturation magnetization. It is important that the magnetization direction of single domains each along the easy axis are not necessarily parallel. Domains are separated by domain boundaries or domain walls. Thus, a movement of domain walls only occurs which requires low energy. The boundary movements can be reversible as well as irreversible.

Magnetization of an Ideal Crystal In an ideal crystal which does not contain any defects the magnetization of a demagnetized ferromagnet starts with reversible domain wall movements. Subsequently, reversible rotation processes can occur. The magnetization curve which is obtained for pure rotation processes is shown in Fig.

Magnetization of a Real Crystal A real magnetic system additionally possesses crystal defects which results in a domain wall potential. Curve 1 : Pure domain wall movements in an ideal crystal. Curve 2 : Pure rotation processes in an ideal crystal.

Curve 3 : Combination of domain wall movements and rotation processes in an ideal crystal. Curve 4 : Behavior of a real crystal 8.

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The enlargement enables to observe the Barkhausen jumps defects. These so-called Barkhausen jumps are directly observable in the hysteresis loop see Fig. The graphic representation is known as the hysteresis curve or hysteresis loop see Fig. Hysteresis loop schematically shown 8 Magnetic Domain Structures 8. The perpendicular direction of a domain wall corresponds to the bisecting line between the directions of the magnetization of neighbored domains. Domains on the side plane of a Co crystal From [10] used with permission Fig. But, the magneto crystalline anisotropy favors a short length.

Thus, both energy contributions try to move the domain wall width into the opposite direction. The magneto crystalline anisotropy energy cf.

## fundamentals of magnetism

Domain walls exhibit a continuous rotation of the magnetization vector between two domains. The question arises what happens at the surface due to the reduction of the symmetry. Let us start with a sample which exhibits one single domain as the initial state see Fig. In order to lower the energy Bloch walls are created see Fig. Only a small number of domain walls occurs within the crystal. From [10] used with permission the domain closure energy but in the bulk a wide pattern is favored to save domain wall energy. The branching process connects the wide and the narrow domains in a way that depends on crystal symmetry and in particular on the number of available easy directions.

Sketch of the iterated generation of domains towards the surface 8 Magnetic Domain Structures Fig. The easy axis is perpendicular to the surface. With increasing thickness from the left to the right the degree of branching increases. From [10] used with permission A cross section of the example Co was already shown in Fig.

The top view is given in Fig. Almost as important is magnetic anisotropy in these materials. Most details of the domain patterns are therefore determined by the surface orientation relative to the easy magnetization directions. Several cases must be distinguished.

From the simplest case, a surface with two easy axes, to strongly misoriented surfaces with no easy axes the domain patterns become progressively more complicated. In positive anisotropy materials, i. The -surface of a negative anisotropy material, i. The bcc-Fe surface exhibits two -like directions both representing easy magnetization axes Fig. Domains on a surface of silicon-stabilized Fe.

From [10] used with permission simple. It consists of domains being magnetized parallel and antiparallel to the easy []-direction see Fig. The bcc-Fe surface exhibits one -like direction which represents an easy magnetization axis and a -like direction being a hard magnetization axis Fig. Domains on a largely undisturbed -oriented Fe crystal which contains small amounts of Si. The isolated lancets as well as the short kinks in the main walls are connected with internal transverse domains.

From [10] used with permission be present without the misorientation these domains are known as supplementary domains. In the examples given below the magnetization is always assumed to follow strictly the easy magnetization directions. From [10] used with permission Fig. One example for the occurrence of lancet domains was already given in Fig.

Surface domains on strongly misoriented surfaces are not at all representative for the underlying bulk magnetization. The interior domains can only be inferred from subtle features and the dynamics of the surface pattern. The striking fat walls are no real walls but traces of internal domains. The pattern mainly consists of basic domains along the one of the easy axes which are covered by shallow closure domains magnetized along the other axis. The cross section in c demonstrates how the charges on the surface are compensated 8.

The pattern mainly consists of basic domains along one of the easy axes which are covered by shallow closure domains being magnetized along the other axis. The cross section c demonstrates how the magnetic charges on the surface are compensated by charges at the lower boundaries of the shallow domains. Wall energy is saved in the second step by slightly opening the basic domains d. From [10] used with permission by charges at the lower boundaries of the shallow domains.

Therefore, we have analyzed the static or nearly static situation. This chapter deals with the consequences of rapid changes of H, i. If H2 is, for example, in the regime where irreversible changes of the magnetization occur Mn is relatively large. If H2 is in the regime of reversible rotation processes of the magnetization Mn is relatively small. A semi-logarithmic plot of Mn t as a function of t for Fe containing a small amount of carbon demonstrates agreement with experiment see Fig. Magnetism in Reduced Dimensions — Atoms. Magnetism in Reduced Dimensions — Clusters.

Magnetism in Reduced Dimensions — Nanoparticles. Magnetism in Reduced Dimensions — Nanoscaled Wires. Magnetism in Reduced Dimensions — Multilayers. Back Matter Pages Magnetism Nanoscience Solid state physics Surface physics Thin films and nanostructures thin film. Angewandte Physik Germany. The lectures are self-supported on each topic, so that no particular academic background is strictly necessary.

The supporting slides are made available to the students and the broad community through our repository. Many aspects of the School are designed to favor interactive learning : questions are encouraged during the course of the lectures, as well as during specially scheduled question sessions typically 8h ; students attend at least two practicals analytics, experimental or computer-based 2h or more each; a library of books pertaining to all aspects of Magnetism is made available. Besides fundamental lectures, several lecturers are selected from companies.

They provide an overview of their working topic and conditions.

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They will be available for discussion with all academic members who consider turning to the industry. Topics Each European School on Magnetism provides a broad spectrum of fundamentals on Magnetism, possibly complemented by a series of lectures focused on a dedicated topic that changes from school to school. Attendees The school is addressed mainly to young scientists , both PhD students and post-docs.