Abstract

In this review a number of magnetic properties of different thin film systems are investigated as functions of the temperature and the atomic morphology. Special attention is paid to determine the influence of the collective magnetic excitations and the noncollinear magnetic structure. At finite temperatures these problems are studied within a Heisenberg model by application of a mean field approximation as well as by a many-body Green’s function theory.

First, the magnetization profiles and the magnetic ordering (Curie-) temperatures are calculated for different magnetic systems. In particular, single ferromagnetic (FM) films, coupled magnetic bilayers with two FM films, and trilayers consisting of two FM films separated by a nonmagnetic spacer layer are studied. Here the role of the magnetic fluctuations are highlighted, which are particularly important for these low-dimensional magnets. For the different systems under consideration we show that the strongly varying magnetic properties caused already by weak interlayer couplings can be explained only by taking into account the collective magnetic excitations. Hence, the effect of these excitations for two-dimensional magnets can be studied explicitly. The calculated results are partly compared with measurements.

Moreover, the thin-film magnetic structure of materials with a helical bulk magnetization is investigated. We show that due to the breaking of nearest- and next-nearest-neighbor bonds in the surface region the helical magnetic structure of, e.g., a thin Ho film becomes significantly disturbed. Even a FM phase may result in a decreasing film thickness or an increasing temperature. The possibility of a paramagnetic layer within an ordered magnetic structure is pointed out.

In addition, the spin reorientation transition of thin FM films is studied as a function of the temperature, the film thickness, and an external magnetic field. This phenomenon results from competing effective anisotropy contributions with a different dependence on the temperature, for example. Special attention is paid to investigate the influence of magnetic noncollinearities caused by an atomic roughness or a variation of the film thickness. We show that these noncollinearities result in a much broader magnetic reorientation as compared to the one of a smooth film. This feature can be considered by effective higher-order anisotropies for an otherwise collinear thin film magnetization. Approximate expressions for these quantities are presented. We show that for a strongly inhomogeneous system the magnetization profile during SRT exhibits a smooth behavior, hence is not characterized by magnetic domains separated by comparatively thin domain walls.

The magnetic reversal as induced by a transversal magnetic field is studied for the simple cases of FM and antiferromagnetic (AFM) monolayers. For the latter case we show that with increasing strength of the transversal field the magnetization component perpendicular to that field exhibits a maximum. This unexpected property is explained by the presence of quantum fluctuations which are particularly important for antiferromagnets.

Finally, the magnetic structure of coupled FM–AFM layers is studied. The interlayer coupling induces a net magnetic binding energy and thus results in an effective interface anisotropy of the FM layer. A noncollinear magnetization is induced in the AFM layers and possibly also in the FM layers close to the interface. We show that the magnetic structure of the interface layers and the resulting magnetic ordering temperature(s) depend sensitively on the lattice symmetry. The importance of the collective magnetic excitations for these coupled FM–AFM systems is also pointed out. The resulting interface anisotropy may enhance the total anisotropy of the FM subsystem. Alternatively, if the interface anisotropy competes with the intrinsic FM anisotropy, a magnetic reorientation of the FM film can take place with an increasing temperature and varying FM film thickness.

Keywords: Magnetic structure; Ordering temperature; Magnetic reorientation; Noncollinear magnetization; Ferro–antiferromagnetic interfaces

Article Outline

1. Introduction

2. Magnetization and ordering temperature of thin films

2.1. General remarks and experimental results

2.2. Single thin films

Diversion: Two layers

2.3. Antiferromagnetic thin films

2.4. Helical magnetic structure: Thin Ho films

2.5. Coupled magnetic systems

2.6. Magnetic trilayers

3. Spin reorientation transition

3.1. Experiments on the spin reorientation transition

3.2. Stripe-domain phase

3.3. Collinear magnetic reorientation

3.4. Temperature-driven magnetic reorientation

3.5. In-plane magnetic reorientation

3.6. Sixth-order uniaxial anisotropy

3.7. Field-induced spin reversal by GFT

Ferromagnetic monolayer

Anisotropic susceptibility of a FM monolayer

Antiferromagnetic monolayer

4. Noncollinear magnetic reorientation

4.1. Higher-order anisotropy coefficients of an inhomogeneous thin film

4.2. Vertically inhomogeneous thin films

4.3. Laterally inhomogeneous thin films

5. Ferromagnetic–antiferromagnetic interfaces

5.1. Magnetic reordering near ferromagnetic–antiferromagnetic interfaces

5.2. Spin reorientation of a ferromagnetic layer on an antiferromagnetic surface

6. Conclusions and outlook

Acknowledgements

Appendix A. The Heisenberg Hamiltonian

A.1. General remarks

A.2. Isotropic exchange interaction

A.3. Anisotropic interactions

A.3.1. Magnetocrystalline anisotropy

A.3.2. Exchange anisotropy

A.3.3. Magnetic dipole interaction

A.4. The full Hamiltonian

A.5. Rotation of spin operators

Appendix B. Magnetic anisotropies in thin films

B.1. General remarks

B.2. Explicit expressions

B.3. Temperature dependence of the anisotropies

Appendix C. Two-time Green’s functions

C.1. General definitions

C.2. Green’s functions for spin systems

C.3. Example: S=1/2

C.4. Subsystem structure

C.5. Mean field approximation

Appendix D. Effective fourth-order anisotropy

D.1. Two subsystems

D.2. Generalization to several subsystems

References

(43K)
Fig. 1. Sketch of the investigated layered magnetic systems. A nonmagnetic substrate may carry (a) a single magnetic film with thickness N, or (b) a magnetic bilayer consisting of two coupled films with different magnetic materials and with film thicknesses N₁ and N₂. These two systems can be covered by a nonmagnetic cap layer. Moreover, multilayer film systems have been prepared (c), consisting of alternating magnetic and nonmagnetic layers.

(39K)
Fig. 2. Measured Kerr intensity of a Co/Cu(001) thin film as a function of the relative temperature T/T_C at an applied field H=0 (solid circles) and (open circles). The Curie temperature of the single Co layer is measured to be . The insets show two hysteresis loops below and above T_C[25].

(60K)
Fig. 3. (Left) Real part of the measured ac-susceptibility during growth of a Co/Au(111) thin film at room temperature. The ac-modulation field is directed parallel (H₌, solid circles) and perpendicular (H, open circles) to the film plane [37]. (Right) Real and imaginary parts of the measured ac-susceptibility of a Fe/W(110) film with as a function of the temperature [23].

(41K)
Fig. 4. Measured relative Curie temperatures as a function of the film thickness N for different thin film systems as indicated [47].

(36K)
Fig. 5. Reduced Curie temperature of Gd(111)/W(110) thin films as a function of the film thickness N. The squares and the triangles denote T_C(N) of a smooth and of a rough Gd film, respectively. The full line is obtained from the scaling law, Eq. (1)[30].

(37K)
Fig. 6. Measured Kerr ellipticities at saturation, extrapolated to (top), and Curie temperatures (bottom) of fcc-like Fe/Cu(001) thin films as a function of the Fe coverage. I, II, and III denote different magnetic phases. The Fe magnetization is directed perpendicular to the film plane below 11 ML, and in-plane above 11 ML [17].

(37K)
Fig. 7. Measured relative Curie temperatures for Ni(111) (solid diamonds, open triangles), Ni(001) (solid triangles and circles, open diamonds and squares), and Gd(111) (open circles) thin films as a function of the film thickness d/d₀ in units of a normalized length (d₀=1  for Ni and d₀=2  for Gd). The solid lines are obtained from the scaling law, Eq. (1)[36].

(37K)
Fig. 8. Curie temperatureT_C(N) of thin homogeneous ferromagnetic films as a function of the thickness N for different symmetries of the film structure: sc(001) (q₁=1, ), fcc(001) (q₁=4, ), and fcc(111) (q₁=3, ) film faces. Here, and q₁ are the coordination numbers in the bulk magnet and between nearest-neighbor layers in the film, respectively. The results are obtained from a mean field approximation. The dotted line refers to the average coordination number estimate, Eq. (3), of the fcc (001) film. T_C(N) is given in units of the bulk Curie temperature , and N in units of atomic (mono-) layers (ML).

(34K)
Fig. 9. Curie temperature T_C(N) of fcc(001) homogeneous ferromagnetic films as a function of the thickness N. The results are calculated from a mean field approximation (MFA, dashed line), and from a Green’s function theory (GFT, solid line). The dotted line is obtained from the average coordination number estimate, Eq. (3). For the other notations we refer to Fig. 8.

(57K)
Fig. 10. Sublayer magnetization components and as functions of the temperature T for ferromagnetic films with thicknesses N and with S=1 spins as calculated within GFT [72].

(43K)
Fig. 11. Layer-dependent magnetizations m_i(T) as a function of the distance i from the surface. Different temperatures T below and above the bulk Curie temperature are assumed as indicated. The surface layer carries an exchange interaction much stronger than of the bulk. The lower graph shows the results of the upper one using a semi-logarithmic plot.

(40K)
Fig. 12. Curie temperature T_C(N) of inhomogeneous fcc(001) ferromagnetic films as a function of the thickness N. Different values of the exchange coupling of both surface layers with respect to the exchange of the bulk magnet are assumed as indicated. The results are obtained within MFA. For the other notations we refer to Fig. 8.

(19K)
Fig. 13. Sketch of sc(001) and fcc(001) AFM films. Depicted are the interlayer bonds of a given spin in a layer to the spins in the neighboring layers for both symmetries.

(46K)
Fig. 14. Magnetic structure peak (00τ) of a Ho thin film with 11 atomic layers measured by resonant magnetic scattering at various temperatures as indicated. The data are recorded at a photon energy hν=1353.2 eV [40].

(40K)
Fig. 15. Measured T_N(d) of Ho films as a function of the film thickness d (solid squares). The dashed and solid lines represent fits according to Eqs. (1) and (8), respectively. Inset: comparison to thin-film data for Gd (open circles, [30]) [40].

(75K)
Fig. 16. Rotation angles between neighboring layers as a function of the temperature T for thin Ho films. Different film thicknesses are applied: (a) N=5, (b) N=6, and (c) N=7. The ‘five-constant model’ by Nicklow et al. is used [99], yielding and . The corresponding magnetic structures are indicated by the sketches. (b) for T→T_N(6). (c) ‘o’ denotes a disordered (paramagnetic) layer.

(40K)
Fig. 17. Layer-dependent magnetizations m_i(T) as a function of the temperature T for a Ho film with N=7 layers, corresponding to the system shown in Fig. 16(c). At T_N(7) the 4th layer becomes paramagnetic, whereas the other layers stay magnetically ordered up to T_C(7).

(34K)
Fig. 18. Calculated critical temperatures T_C(N) and T_N(N) as functions of the thickness N of thin Ho films. Here the ‘six-constant model’ by Bohr et al. is used [100], yielding . The solid and dashed lines are calculated from Eqs. (1) and (8), respectively, using λ=1.72 and . ‘PM’ denotes the paramagnetic phase. The sketch depicts the ‘block-spin’ phase for located between T_N(N) and T_C(N)[40].

(36K)
Fig. 19. Profile of the rotation angle of Ho thin films with different thicknesses and for T=0. The structure is calculated within the J₁–J₂-model, using J₂/J₁=−0.29 which yields , i.e., . The range of the disturbed helical phase near the surfaces is indicated by N₀[40].

(32K)
Fig. 20. Range N₀ of the disturbed helical structure near the surfaces (open circles) as a function of the bulk period at T=0, calculated within the J₁–J₂ model. The solid line is obtained from a linear fit to the data points [40].

(66K)
Fig. 21. Layer-resolved magnetizations m_i(T) as functions of the temperature T of a ferromagnetic bilayer consisting of two different FM materials with three layers each and intralayer exchange couplings J₁₁ (layers 1–3 as indicated) and J₂₂ (layers 4–6), assuming J₂₂/J₁₁=0.5. The two constituents are coupled by the intralayer exchange coupling . S=1 spins have been used. (a) GFT calculation, (b) MFA calculation.

(60K)
Fig. 22. Layer-resolved magnetizations m₂(T) and m₅(T) of atomic layers 2 and 5 corresponding to the bilayer system as depicted in Fig. 21. A decoupled () and a coupled () bilayer is assumed, where T_C,1 and T_C,2 refer to the bare Curie temperatures of the decoupled system, and T_C to the ordering temperature of the coupled bilayer. (a) GFT calculation, (b) MFA calculation.

(25K)
Fig. 23. Sketch of a magnetic trilayer system. Two ferromagnetic films FM1 and FM2 with thicknesses and are separated by a nonmagnetic spacer layer NM with thickness . The two magnetic films are coupled by the interlayer exchange across the spacer.

(31K)
Fig. 24. Temperature shift of the magnetization curve as a function of the Cu spacer thickness for various Co/Cu/Ni/Cu(001) trilayers, which is proportional to the strength of the interlayer coupling [141]. The solid line is obtained from calculations of [153].

(62K)
Fig. 25. Coupling field H_j at the maximum strengths of the AFM coupling in Cu/Co/Cu/Co /Cu(001) trilayers as a function of the thickness of the Cu cap layer. The dashed lines denote the phase differences between the oscillations. The solid lines are obtained from calculations [146].

(44K)
Fig. 26. Sublayer magnetizations of the Ni (circles) and the Co (squares) films of a Cu/Ni/Cu(001) system (open symbols) and a coupled Co/Cu/Ni/Cu(001) trilayer (solid symbols) as probed by XMCD. The influence of the interlayer coupling results in a shift of the Ni magnetization curve estimated by [142].

(33K)
Fig. 27. Phase diagram of the magnetic phases of (a) Cu/Co/Cu(2ML)/Ni/Cu(100) and (b) Cu/Co/Fe(5ML)/Ni/Cu(100) systems at ambient temperatures. The triangles and squares are the Co and Ni critical film thicknesses, respectively, at which the paramagnetic-to-ferromagnetic order transitions occur. The Co and Ni films are both ordered (IV), or both disordered (I). Phase (II) refers to an ordered Ni and a disordered Co film, and phase (III) vice versa. The solid lines are guides to the eye [140].

(24K)
Fig. 28. Sketch of a magnetic trilayer consisting of two ferromagnetic films, FM1 and FM2, and a nonmagnetic spacer layer NM, illustrating the general problem when investigating such a system. The decreasing thickness of FM1 reduces its Curie temperature T_C,1. In contrast, the decreasing thickness of the spacer layer increases the magnetic order in FM1, since the strength of the interlayer exchange behaves approximately as [171].

(37K)
Fig. 29. Ni magnetization of a Co/Cu/Ni/Cu(001) trilayer system as a function of the temperature T. The measurements have been obtained by XMCD [142]. For the calculations within GFT we have assumed integer thicknesses next to the real ones, and exchange interactions as indicated [172].

(37K)
Fig. 30. Relative temperature shift of a Co/Cu/Ni trilayer as a function of the Ni film thickness . The results are obtained by applying the Green’s function method, assuming two small values of the interlayer exchange as indicated. The dotted line refers to MFA results, assuming a much larger value .

(54K)
Fig. 31. Calculated phase diagram of the Co/Cu/Ni trilayer system as a function of the exchange coupling of the Co film for the Co/Cu/Ni trilayer system. Here, and are the bare Curie temperatures of the decoupled trilayer, and the one of the coupled case with . The ‘quasi-critical temperatures’ and are obtained from the inflection points of the Co and Ni magnetization curves. In addition we have assumed , , and [142].

(39K)
Fig. 32. Susceptibility χ₁(T) of the layer with the lower bare Curie temperature T_C,1 of a coupled ferromagnetic trilayer for different interlayer couplings . For the common singularity of the susceptibility at T_C,1 is obtained, which is decreased by the factor 50 for a better visibility. A maximum (resonance) of χ₁(T) is obtained for , and a singularity at the ordering temperature T_C of the trilayer close to the lower bare Curie temperature T_C,2 of the other FM layer.

(46K)
Fig. 33. Measured susceptibility χ(T) as a function of the temperature T for (a) a single Ni film and (b) a Co/Cu/Ni/Cu(001) trilayer with thicknesses as indicated. The two FM layers are separated by a comparably thick Cu spacer, yielding a very weak interlayer coupling . For thinner spacer layers and correspondingly stronger no signal of χ(T) near could be detected. For the dashed line in (b) an additional magnetic field of about 147 A/m has been applied [139].

(34K)
Fig. 34. Temperature difference of the maximum of the susceptibility (full line) and the inflection of the magnetization curve m₁(T) with respect to the lower bare Curie temperature T_C,1 of the FM1 layer in a trilayer system. For the other notations we refer to Fig. 32.

(18K)
Fig. 35. Sketch of the polar spin reorientation transition between a perpendicular (left) and an in-plane (right) orientation of the thin film magnetization. This SRT can be induced by the temperature T, by the film thickness N, or by an applied magnetic field , for example. θ is the angle between the magnetization and the film normal.

(52K)
Fig. 36. Polar and in-plane (longitudinal) hysteresis loops of W/Co/Au(111) thin films for different thicknesses of the Co film at [208].

(49K)
Fig. 37. Thickness-induced magnetic reorientation of the Fe/Ag(001) (top) and the Fe/Cu(001) (bottom) thin film systems at . The diamonds represent the perpendicular, and the open circles the in-plane magnetization component. The solid lines indicate the magnetizations of single-domain films [193].

(48K)
Fig. 38. Perpendicular and in-plane components of the remanent magnetizations of Fe/Ag(001) thin films as functions of the temperature T for (left) and as functions of the film thickness N for (right) [203].

(71K)
Fig. 39. Evolution of the domain pattern with increasing temperature for a Fe/Cu(001) thin film with thickness N=5.3 grown at . Shown are the perpendicular (upper panel) and in-plane (lower panel) magnetization components [194].

(38K)
Fig. 40. Perpendicular magnetization component of a Fe/Gd(111) thin film as a function of increasing and decreasing temperature, indicating hysteresis effects of the magnetization [188].

(109K)
Fig. 41. Magnetic phase diagram of (a) W/Co/Au(111) and (b) Au/Co/Au(111) thin film systems in the temperature–thickness plane. The polarization values P=±1 correspond to the perpendicular (bright areas) and the in-plane magnetization (dark areas) [208].

(38K)
Fig. 42. Magnetic phase diagram of the Ni/Cu(001) thin film system in the temperature–thickness plane. The lines separate the paramagnetic, the perpendicular, and the in-plane magnetization phases. In the hatched area a canted magnetization is observed [242].

(19K)
Fig. 43. Sketch of in-plane magnetic reorientations between different in-plane directions of a rectangular (110)- (left), a square (001)- (middle), and a triangular (111)-film face (right).

(122K)
Fig. 44. Magnetic domain structure in wedge-shaped Co/Au(111) thin films before (left) and after (right) annealing. The black and white areas indicate domains with a perpendicular magnetization, the grey areas refer to an in-plane magnetization. The scanning area is , t_c denotes the reorientation thickness [42].

(62K)
Fig. 45. Domain wall density vs. film thickness of a Co/Au(001) thin film before (left) and after (right) annealing, referring to the films shown in Fig. 44. The hatched area denotes the thickness ranges where the SRT takes place. The solid line in the right panel results from a theoretical analysis [263], using the indicated anisotropy parameters [42].

(89K)
Fig. 46. Kerr images of a Fe/Cu(001) wedge grown at for different oxygen coverages. The black and white contrasts refer to perpendicularly magnetized domains, and the grey areas to an in-plane magnetization [200].

(43K)
Fig. 47. Effective spin magnetic moments of (a) clean and (b) CO-covered Co/Pd(111) thin films at 200 K as a function of the Co thickness. Circles and solid lines correspond to the perpendicular components, and the triangles and dashed lines to the in-plane components of the magnetization. The hatched areas indicate the regions of a perpendicular magnetization [222].

(60K)
Fig. 48. Reversible H₂-induced variation of the polar magnetization of a Ni/Cu(001) thin film with at . Depicted are (a) the polar MOKE signal, (b) the H₂ partial pressure, and (c) the H₂ coverage on the Ni surface. Note the time delay between the H₂ partial pressure and the polar MOKE signal, indicating the dynamics of the H₂-adsorption process. The dashed lines show the locations of the SRT [217].

(20K)
Fig. 49. Sketch of the magnetic stripe domain phase of a thin film. Perpendicularly magnetized domains with alternating magnetic directions are separated by Bloch-type domain walls. The directions of the in-plane magnetization component inside the domain walls also alternates.

(20K)
Fig. 50. Phase diagram in the -plane for the polar orientation of a thin film. The ‘perpendicular’ and the ‘in-plane’ phases are characterized by the polar angles θ=0 and θ=π/2, and the ‘canted’ phase by 0<θ<π/2. In the ‘coexistence’ region the perpendicular and the in-plane phase both refer to energy minima, and are separated by an energy barrier.

(53K)
Fig. 51. Behavior of the polar angle θ(K₂) as a function of the second-order magnetic anisotropy K₂ during a polar SRT for a negative (top) and a positive (bottom) fourth-order anisotropy K₄ (left panel). In the right panel the free energy F(θ) as a function of θ is depicted for K₄<0 and K₄>0. The numbers correspond to the locations indicated in the left panel. In the case K₄<0 a continuous SRT results, the canted magnetization 0<θ<π/2 refers to a minimum of the energy. For K₄>0 a discontinuous SRT is present, the perpendicular and the in-plane magnetic phases are separated by an energy barrier, resulting in a hysteretic behavior. The dotted line in the lower left panel denotes the angle of the energy maximum, i.e., the unstable solution.

(57K)
Fig. 52. The polar angle θ(T), the internal energy E(T), the entropy S(T), the free energy F(T), and the specific heat c_H(T) during a continuous polar SRT as a function of the relative temperature T/T_C in units of the Curie temperature T_C. The dashed and dotted lines refer to unstable perpendicular and in-plane magnetizations. The thin vertical lines indicate the range of the canted magnetic phase.

(33K)
Fig. 53. Anisotropy flow in the phase diagram (top), corresponding to the polar angle θ (bottom) during a continuous polar SRT. Two different values for the anisotropy parameter K₄ are assumed. The arrows indicate the direction of an increasing temperature.

(39K)
Fig. 54. Effective dipole-coupling-induced quartic in-plane magnetic anisotropy for a square monolayer as a function of the temperature T, calculated within a mean field approach for spin cluster sizes . Two different magnetic field strengths B are considered. The dot–dashed line shows for B=0 and N=4 by considering non-diagonal matrix elements of the Hamiltonian. For comparison, also the effective single-ion quartic anisotropy resulting from the spin–orbit interaction is shown (dotted line). The field energies and anisotropies are given in units of the demagnetizing energy. For details we refer to [320].

(26K)
Fig. 55. Magnetic phase diagram of the polar SRT of a ferromagnetic thin film in the K₂–K₄ plane as obtained from Eq. (15). The second-, fourth-, and sixth-order uniaxial anisotropies K_l are considered, with K₆>0. Besides the perpendicular and the in-plane magnetizations, phase I denotes the region with a canted magnetization and a smooth variation of the polar angle θ. In phase II the perpendicular magnetization coexists with the in-plane, and in phase III with the canted magnetization. The latter phase is located in the triangle-shaped region ranging between K₂=0 and the bullet. At the dashed line, referring to K₄=−K₂−K₆, the perpendicular and in-plane phases have the same energy.

(60K)
Fig. 56. Anisotropy energy E(θ) as a function of the polar angle θ. (a) E(θ) obtained from Eq. (15) for three different values of K₄ located in different regions of the phase diagram as shown in Fig. 55; a perpendicular magnetization (K₄/K₆=−1.1, solid line); a coexisting region (phase III, K₄/K₆=−1.4, dashed line); and a canted magnetization (phase I, K₄/K₆=−1.8, dashed–dotted line). We have assumed K₂/K₆=0.5 and K₆>0. (b) Electronic band structure calculations for different thicknesses m of the Pd/Co_m/Pd(111) ferromagnetic thin film system [321].

(41K)
Fig. 57. Magnetization components m_z(T,B_x) and m_x(T,B_x) of a ferromagnetic monolayer during magnetic reversal as a function of the transversal magnetic field B_x. Different temperatures T in units of the exchange interaction J are assumed as indicated. An exchange anisotropy D^z/J=0.01 along the z-direction is present, yielding the Curie temperature T_C/J=0.472. The calculations are performed within GFT.

(32K)
Fig. 58. (a) Magnetization modulus m(B_x) and (b) expectation value S⁻S⁻(B_x) of a FM monolayer as a function of the transversal magnetic field B_x for T=0. For the other notations we refer to Fig. 57.

(39K)
Fig. 59. Inverse susceptibilities and in SI units measured along the easy and hard in-plane directions of a vicinal Co/Cu(1 1 17) thin film as a function of the temperature T. In addition the solid line shows calculated within GFT using the exchange coupling and the anisotropy . For details we refer to [324].

(38K)
Fig. 60. Calculated inverse longitudinal and transversal susceptibilities and along the easy and hard directions of a FM (001) monolayer as a function of the temperature T in the paramagnetic regime (solid lines). An exchange anisotropy D^z/J=0.05 is considered, yielding the Curie temperature T_C/J=0.63 as calculated from GFT. For comparison the inverse susceptibilities are also shown as obtained from MFA (dashed lines) [324].

(15K)
Fig. 61. Sketch of the angles θ₁ and θ₂ of the two sublattices of an AFM monolayer in the presence of an exchange anisotropy along the z-direction and a transversal magnetic field parallel to the x-axis.

(60K)
Fig. 62. (a) Magnetization modulus m(T,B_x) and (b) equilibrium angle θ₀(T,B_x) of an AFM monolayer as functions of the transversal field B_x for different temperatures T below and above the Néel temperature T_N. The temperatures and interactions are given in units of the isotropic exchange J. We have assumed a spin quantum number S=1/2 and an exchange anisotropy D^z/J=0.01. The calculations are performed within GFT.

(37K)
Fig. 63. Staggered magnetization component |m_z(T,B_x)| along the easy axis as a function of the transversal field B_x for different temperatures T. For the other notations we refer to Fig. 62.

(40K)
Fig. 64. Staggered magnetization component |m_z(T,B_x)| along the easy axis as a function of the temperature T for different B_x as indicated. For the other notations we refer to Fig. 62.

(31K)
Fig. 65. Reorientation temperature T_R(B_x) as a function of the transversal magnetic field B_x for an AFM and a FM monolayer. In both cases the exchange anisotropy is chosen to be D^z/J=0.01. T_C and T_N refer to the critical temperatures of the FM and the AFM, respectively. For a better comparison B_x for the FM monolayer is multiplied by the factor 200.

(18K)
Fig. 66. Sketch of the phase diagram in the K₂–K₄ plane. Along the dashed line a continuous polar SRT between the perpendicular and the in-plane magnetization occurs via a canted phase (shaded region). The width of the transition is indicated by .

(20K)
Fig. 67. (a) Sketch of an inhomogeneous thin film system with a second-order surface anisotropy K_2s and a ‘volume’ anisotropy K_2v in the film interior layers. N is the film thickness. (b) Sketch of the equilibrium polar angles θ_i dependent on the layer i.

(34K)
Fig. 68. Effective fourth-order anisotropy resulting from the noncollinear magnetization of a smooth thin film as a function of the film thickness N. The surface anisotropy K_2s is varied by such an amount that a polar SRT occurs. Different assumptions for the volume anisotropy K_2v have been used as indicated. The anisotropies and the strength w of the dipole coupling are given in units of the exchange interaction J.

(56K)
Fig. 69. Linear (left) and double-logarithmic (right) plots of the effective fourth-order anisotropy of a smooth thin film as a function of the strength w of the dipole coupling for two film thicknesses N as indicated. The volume anisotropy is neglected here (K_2v=0). The anisotropies are given in units of the exchange interaction J.

(19K)
Fig. 70. Sketch of the laterally structured thin film. The film normal defines the z-direction, N is the film thickness, and L the periodicity along the x-direction. The periodicity along the y-direction is equal to unity. The magnetization of lattice site i varies in the yz-plane, its direction is given by the polar angle θ_i.

(41K)
Fig. 71. (Left) Sketch of the thin film structure with ‘one-atom-steps’. The sites indicated by solid circles carry the surface anisotropy K_2s, and the open circles the volume anisotropy K_2v. (Right) Corresponding effective fourth-order anisotropy as a function of the nominal film thickness N ranging from 2 ML to 4 ML. The anisotropies and the strength w of the dipole coupling are given in units of the isotropic exchange J. The lateral periodicity is chosen to be L/a₀=31, and K_2v=0.

(41K)
Fig. 72. (Left) Sketch of the thin film structure with ‘two-atom steps’. (Right) Corresponding effective fourth-order anisotropy as a function of the nominal film thickness N. Caused by the growth mode the film thickness ranges from 1 ML to 3 ML, and from 2 ML to 4 ML. For the other notations we refer to Fig. 71.

(35K)
Fig. 73. Effective fourth-order anisotropy as a function of the lateral periodicity L for different nominal film thicknesses N as indicated. The assumed growth mode refers to one-atom steps. For the other notations we refer to Fig. 71.

(53K)
Fig. 74. (a) Average polar angle during SRT as a function of the surface anisotropy K_2s. The averaging procedure is performed over all spins in the unit cell (solid line) and over the surface spins carrying the surface anisotropy K_2s (dashed line). The nominal film thickness is , and the lateral periodicity L/a₀=101. One-atom steps are present. The arrow indicates the SRT in the case of an infinite exchange J, where the total second-order anisotropy changes sign. The dotted line represents the simple cos²θ function for a collinear rotation with second- and fourth-order anisotropies, cf. Eq. (13). (b) Corresponding standard deviation σ_θ(K_2s) of the polar angle θ. For the other notations we refer to Fig. 71.

(60K)
Fig. 75. Angular profile θ(x) as a function of the lateral (x-) position and for different film layers. The nominal film thickness is . K_2s is chosen in such a way that the average polar angle is , moreover, K_2v=0. (a) θ(x) for different lateral periodicities L/a₀=31 (dotted line), L/a₀=71 (dashed line), and L/a₀=101 (solid line). The strength of the dipole coupling is put equal to w/J=2×10⁻³. (b) θ(x) for L/a₀=101 and for w/J=2×10⁻³ (solid line), w/J=1×10⁻² (dashed line), and w/J=2×10⁻² (dotted line). For the other notations we refer to Fig. 71.

(39K)
Fig. 76. Angular profile θ(x) for and L/a₀=101. Here the dipole interaction is neglected (w=0). Instead, different volume anisotropies are assumed, namely K_2v/J=−0.02 (solid line), K_2v/J=−0.1 (dashed line), and K_2v/J=−0.2 (dotted line). As before, K_2s is chosen in such a way that the average polar angle yields . For the other notations we refer to Fig. 71.

(23K)
Fig. 77. Topview sketch of (a) the sc(001) and (b) the fcc(001) bilayer system. A single FM layer (dark arrows) and a single AFM layer (grey arrows) are assumed, with two sublattices per layer. Both layers are coupled by the interlayer exchange . The angles and quantify the deviations from the undisturbed magnetic arrangement [373].

(61K)
Fig. 78. (a) Magnetizations m_i(T) and (b) equilibrium angles _0,i(T) for a sc(001) bilayer as functions of the temperature T for different values of the interlayer exchange coupling as indicated. The AFM exchange is chosen to be , hence . The temperature is given in units of the bare Curie temperature of the FM monolayer. At the sublattice reorientation temperature one obtains and [373].

(53K)
Fig. 79. (a) Magnetizations m_i(T) and (b) equilibrium angles for a fcc(001) bilayer as functions of the temperature T for different values of the interlayer exchange coupling as indicated. We have chosen , hence . For these systems two different critical temperatures and for the FM and AFM layers, respectively, are obtained. For the AFM spins relax to the undisturbed AFM arrangement [373].

(62K)
Fig. 80. Curie temperature as a function of the interlayer exchange for sc(001) (top) and fcc(001) (bottom) FM–AFM bilayers, normalized to . The results are obtained within MFA and GFT, as indicated [377].

(39K)
Fig. 81. Magnetic energy E() of FM–AFM bilayers in units of K/spin as a function of the magnetization direction of a FM particle deposited on an AFM substrate. The full line refers to an interface coupling of , and the dashed line to . and are the corresponding energy barriers representing the total magnetic anisotropy of the FM particle. For the intralayer couplings we have assumed and , referring approximately to a bulk Ni ferromagnet and a bulk NiO antiferromagnet. The corresponding anisotropies are set equal to [13].

(39K)
Fig. 82. Energy barrier of a FM particle deposited on an AFM substrate in units of K/spin as a function of the interface coupling . Different values of the exchange coupling of the AFM are assumed as indicated. For the other couplings we refer to Fig. 81. The coupling values used for that figure are indicated by the dot [13].

(41K)
Fig. 83. Total effective anisotropy per spin of the FM film as a function of the temperature T. prefers a collinear, and an orthogonal FM film magnetization with respect to the AFM magnetic direction. The exchange couplings, the intrinsic anisotropies, and the temperature are given in units of . We have assumed and , in addition and for the thicknesses of the FM and AFM films. For these values the Curie temperature is larger than the Néel temperature . The solid line (a) shows the intrinsic anisotropy for a decoupled FM film () for . The dashed line (b) refers to the bare interface anisotropy (), assuming . The presence of both anisotropic contributions results in a reduced , dot–dashed line (c). For an enhanced absolute value is obtained, dotted line (d) [372].

(45K)
Fig. 84. Total effective anisotropy per FM film spin as a function of the temperature T for different interlayer exchange couplings . The reorientation temperature T_R is defined by the change of sign of . In addition, the equilibrium angles of the FM film magnetization for and are displayed, indicating a continuous magnetic reorientation near T_R[372].

(48K)
Fig. 85. Same as Fig. 84 for different FM film thicknesses , assuming . The equilibrium angles are shown for and . For the former case a continuous SRT between the collinear and orthogonal magnetic arrangements with an intermediate canted magnetization is obtained, whereas for the latter a canted arrangement is present at low temperatures [372].

(39K)
Fig. A.1. Magnetization curves for the easy (111) and hard (100) axis of bulk Ni. The hatched area enclosed by the two curves is a direct measure for the strength of the magnetocrystalline anisotropy. The saturation magnetizations are slightly different for both directions [242].

(62K)
Fig. B.1. (Left) Measured magnetic anisotropies for Fe/W(110) thin films as a function of the inverse thickness for two different temperatures as indicated [228]. (Right) Calculated magnetocrystalline anisotropy (squares, dotted line) for Co/Cu(111) thin films as a function of the number N of Co layers. In addition the shape anisotropy resulting from the dipole interaction is given (circles, solid line), which adds with to the total anisotropy E_a(N) (diamonds, dashed line) [433].

(36K)
Fig. B.2. Measured anisotropy coefficients determined by FMR for a Ni/Cu(001) thin film with 7–8 Ni layers as a function of the temperature T. Shown are the second- and fourth-order perpendicular anisotropies K₂ and K₄, and the fourth-order in-plane anisotropy K₄[242].

(76K)
Fig. B.3. Temperature factors as functions of the relative temperature T/T_C obtained from a mean field approximation, and for classical spins. In (a) also the magnetization m(T) is depicted.

(33K)
Fig. D.1. Phase diagram of the magnetic structure with two subsystems in the K_2,1–K_2,2-plane, see text. The anisotropies are given in units of the exchange coupling J between the subsystems. In between the perpendicular and the in-plane magnetic phases a canted magnetic phase with a noncollinear structure is located. and denote the phase boundaries to the perpendicular and to the in-plane magnetizations, respectively. The dashed line refers to K_2,2=−K_2,1.

(49K)
Fig. D.2. (a) Average angle and angular deviation ε as a function of K_2,2 along the thick line indicated in Fig. D.1, assuming K_2,1/J=0.1. By variation of K_2,2 the canting angle varies from (in-plane magnetization) to (perpendicular magnetization), passing a noncollinear magnetic phase with a finite ε>0. For a better visibility ε is enlarged by a factor 10. The dotted line indicates the polar angle in the case of a collinear rotation (ε=0). (b) Corresponding energy E(K_2,2) for the noncollinear (solid line) and the collinear structure (dotted line) as shown in (a).

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