Macroscopic properties II

Transcription

1 Paolo Allia DISAT Politecnico di Torino acroscopic properties II

2 acroscopic properties II Crucial aspects of macroscopic ferromagnetism Crystalline magnetic anisotropy Shape anisotropy Ferromagnetic domains and domain structures Domain walls The technical magnetization process agnetic losses agnetization dynamics: ferromagnetic resonance

3 agnetic anisotropy: crystalline The Heisenberg exchange hamiltonian is isotropic: it depends on the mutual directions of spins S i and S j only: H i j J ij S i S j i j J ij S xi S xj S yi S yj S Real magnetic materials, however, exhibit magnetic anisotropy, which is essentially caused by the spin-orbit coupling. Spin-orbit coupling implies that the direction of spin depends on the orientation of electron orbits zi S zj Orbital states are determined by the electronic configurations and are influenced by crystal symmetry too therefore, magnetic anisotropy depends on the interplay between shape of electron orbitals and crystal symmetry 3

4 Sketchy examples of actual orbital shapes: Interplay between 3d orbitals and crystal structure (for simple cubic symmetry): 4

5 The crystal field already defined in connection with quenching of orbital momentum - is deemed responsible for the emergence of magnetic anisotropy (single-ion anisotropy). The energy is indeed strongly affected by the actual arrangement of neighboring atoms/ions. Such a localized-spin picture applies to 3d-insulators & 4f-metals mostly. 5

6 For transition metal ions, magnetic anisotropy is usually treated by examining the crystal field splitting and adding spin-orbit coupling as a perturbation. In fact, crystal-field splitting is larger than spin-orbit coupling for 3d electrons. Instead, in rare-earth ( RE ) metals and RE-intermetallic compounds, crystal-field splitting is weak compared to spin-orbit energy. Therefore, magnetic anisotropy of RE metals/intermetallics can be very large. Phenomenology of the magnetic crystalline anisotropy In view of the difficulties in producing a general theory of magnetic anisotropy, from first principles starting from the Heisenberg hamiltonian, it is often preferred to draw a macroscopic, phenomenological picture of the measured effects. 6

7 agnetocrystalline anisotropy: observations Favored/unfavorable directions of the magnetization vector are usually termed easy/hard axes. 7

8 Types of magnetocrystalline anisotropy Uniaxal anisotropy Co crystals show negligible anisotropy in the basal plane. Therefore, Co exhibits uniaxial anisotropy with a preference for magnetization to point along the c axis. Uniaxial anisotropy energy (per unit volume) is expressed as a power series containing even powers of sin q (energy does not change value under space inversion) E U K n n sin n q K K sin q K sin 4 q... 8

9 Usually, K > K. The c-axis is the easy axis if K >. The anisotropy constants K, K are strongly temperature-dependent; they even intersect well above room temperature. K changes of sign at even higher T. E k U l l g l 9

10 Anisotropy field In uniaxial materials, the hard-axis magnetization process is described adding the Zeeman energy (- s H sin θ) to the uniaxial anisotropy energy. Neglecting the K term, one gets: E E U S H sinq K sin q SH sinq The equilibrium condition is found setting the derivative of the energy with respect to q equal to zero: E K sinq cosq q K sinq H S S H cosq The field needed to rotate the S vector by 9 away from the easy axis (sin q = ) is K H S. The same quantity accounts for the strength of the anisotropy in terms of an internal fictitious field: the anisotropy field.

11 Cubic anisotropy In cubic crystals, symmetry considerations require that the anisotropy take the form: E A K K... K where the direction cosines i are defined in terms of the Euler angles as: sinq cos; 3 sinq sin; 3 cosq In cubic systems the [ ], [ ], [] crystallographic axes are equivalent. Usually K > K therefore, [] (edge of the cube) is the easy axis when K > as in Fe; [] (principal diagonal of the cube) is the easy axis when K < as in Ni

12 Room-temperature values of K, K and temperature dependence of K,K,K 3 for Fe and Ni monocrystals. Ki ( T) K () i S S ( T) () l( l) l= for uniaxial anisotropy l=4 for cubic anisotropy S

13 agnetic anisotropy: shape B H Discontinuities of magnetization act as sources of demagnetizing field H d. Generally speaking, H d =-N/. The demagnetization factor N depends on body s shape. For an ellipsoid, three demagnetization factors can be defined: N x + N y + N z = H H H dx dy dz N N N x y z x y z 3

14 4 The magnetic energy (per unit volume) is given by: z z y y x x d d d N N N H General rule: if a>b>c, then N x < N y < N z For an ellipsoid of revolution, N x = N y = ( - N z )/ and q cos 3 4. z s z z y y x x d N const N N N Shape anisotropy is uniaxial; easy axis is always the long axis of ellipsoid.

15 Important limiting cases Spherical sample: N x = N y = N z =/3 no shape anisotropy Thin films: one dimension (z) is much smaller than the other two dimensions; according to the general rule N z >> N x,n y, and N z, N x N y. d s cos q The energy is minimized for q = p: the magnetization usually lies in the film s plane at equilibrium. The only way to get a perpendicular equilibrium magnetization in a film is to use a material with an extremely high crystal anisotropy and easy axis along z. icrowires: N z, N x N y /. 4 d s cos q The magnetization is spontaneously directed along the wire axis (even if this is bent). 5

16 Ferromagnetic domains & domain structures agnetic poles (of field H!) may appear at sample surfaces H d d H d The magnetostatic energy for a uniformly magnetized body can be written as: E E d d body H H space d d dv dv where the second integral is to be performed over the space, inside and outside the body. 6

17 For a uniformly magnetized macroscopic body the overall magnetostatic energy can be significant. It can be effectively reduced with the nucleation of magnetic domains of antiparallel magnetization such as those shown in the figure, cases (b) and (c): Positive and negative magnetic poles at the sample surface are finely distributed, and the external field decreases. A quick calculation: consider a slab of Co, with s =.8 T aligned along z by crystal anisotropy. Taking N z one gets, in the single-domain configuration (a): d = s / =.3 6 J/m 3. In the presence of domains of width D, one gets instead: d.3 6 D which can be much smaller. 7

18 Of course, D cannot become equal to zero as suggested by the last formula, because a magnetic domain wall (DW) must exist between two adjacent domains of antiparallel magnetization. Nucleation of a DW requires paying an energy cost, because DW s store both exchange and anisotropy energy. Domain wall thickness and stored energy Within a DW the magnetization points along a direction not corresponding to an easy axis and a quasi-continuous rotation of the vector occurs there. The wall thickness is the space length required for a full rotation of a given angle to occur. The full rotation angle between domains of antiparallel magnetization is p (8 domain wall). 8

19 Domain wall thickness is determined by minimizing the sum of exchange and magnetic anisotropy energies. Exchange energy Exchange energy is a minimum inside each domain. Within a DW instead: exch J SiS j JS cos const. JS The exchange energy per unit area of the DW is: E exch const N a. JS where N is the number of n.n. distances a contained in the DW thickness: d= Na. 9

20 Approximating p/n (homogeneous rotation) one gets: E exch const. JS p Na i.e., the DW should be infinitely large to minimize exchange energy. However, a DW stores anisotropy energy also. Anisotropy energy A rough estimate of the stored anisotropy energy per DW unit surface is given by the product K V (where V is the material s volume within the DW) divided by the DW surface S. The volume V in turn is just NaS. Therefore, E an K Na i.e., the DW should be infinitely narrow to minimize anisotropy energy. Of course, a trade-off between competing energies will occur.

21 The total stored energy (per unit wall surface) is E p const. JS KNa Na The DW thickness is therefore found by putting the derivative E/N equal to zero: E N p JS JS Ka d Na p N a K a and the stored DW energy (per unit wall surface) is: JS a K E wall p Example: Co JS /a =.6 - J/m K = J/m 3 d.7-8 m = 7 nm E wall.5 - J/m

22 Bloch walls vs. Néel walls Across the thickness of a 8 dw there are virtually no magnetic poles ( inhomogeneities of the magnetization in a direction perpendicular to the wall). Of course, some free magnetic poles are created at the top and bottom of the wall on opposite surfaces of a bulk material, where the dw terminates; positive and negative poles they are however separated by a macroscopic distance and the associated magnetostatic energy is negligible agnetic poles appear at the surface in correspondance of a 8 dw (The stray field generated by the poles provides a way to visualize the dw by observing the magnetic micropowders accumulated there)

23 Bloch walls vs. Néel walls In thin films, the magnetostatic energy of the wall significantly increases as a result of these free poles created of film surfaces. In order to reduce this magnetostatic energy, the spins inside the wall may perform their 8 rotation in such a way as to minimize the density of magnetic poles, which leads to the rotation of spins in the plane of the surface. Such a wall is called a Néel wall. Néel wall (below) becomes more stable than Bloch wall (above) below some critical film thickness (data refer to Permalloy [a soft NiFe alloy]). 3

24 Domain walls patterns agnetic domains reduce the magnetostatic energy of a magnetized body; however, each pair of adjacent domains involves the presence of a dw which stores energy. The actual number of magnetic domains found in a material at equilibrium (no applied field) is dictated by a trade-off between DW and magnetostatic energies. In the simple case treated above, and using energy densities (in J/m 3 ): d wall wall E E wall wall s s D (totalsurface) / volume L x L D x D L y L L E D y z wall L z E D wall where E wall is a known quantity: domain wall width D JS a K E wall p and we look for the 4

25 The magnetic domain width at equilibrium is found by minimizing the total energy density: Ewall s D Example: Co D s =.8 T E E wall wall.5 - J/m s d = 7 nm D D D = -4 m = m Ewall D In actual materials, there is a wide variety of equilibrium magnetic domain patterns; domain shape and width is dictated by the complex interplay of different energies s 5

26 Domain disappearance in fine particles agnetic domains have a typical width; therefore they do not develop in a ferromagnetic body of sufficiently small size. Which is the critical size below which a particle no longer exhibits multiple domains ( i.e., it becomes a single-domain particle)? This is important for applications of fine particles in permanent magnets and recording media. For the single-domain state to be stable, the energy needed to create one domain wall spanning a spherical particle of radius r, E wall pr, must exceed the magnetostatic energy E d = /3 s V (the demagnetizing factor being /3 in this case): r p crit JS a K 9 p 4 pr crit JS a 4p 9 K s s r 3 crit For Co, r crit 5 nm; for SmCo 5 with K u = 7 J/m 3, r crit m. The effect called superparamagnetism of single-domain, magnetic nanoparticles is beyond the scope of these lectures. 6

27 The technical magnetization process in bulk ferromagnetic materials A ferromagnetic body in the absence of applied magnetic field is spontaneously divided into many domains. The initial macroscopic magnetization of the body is zero (or almost zero) because of the mutually compensating contributions from antiparallel domains. An applied field modifies the starting configuration and a net magnetization of the material is measured. 7

28 The technical magnetization process in bulk ferromagnetic materials Al low fields, favored domains (those whose local magnetization is closest to the applied field direction) grow at the expenses of the other unfavorable ones according to a sort of principle of survival of the fittest. At higher fields, the magnetization rotates coherently towards the field direction 8

29 Domain wall displacement DW displacement occurs because of the force (pressure) brought about by the external field and involves a continuous rotation of local magnetization with time. The displacement can be hindered by defects (point defects, dislocations, inclusions, stress centers, ). The wall moves in a complex, multivalley potential energy landscape which is the source of intrinsic irreversibility of this motion. 9

30 In general, both magnetization mechanisms, i.e., DW displacement and coherent rotation of the vector s have intrinsic irreversibility characters. As a consequence, if the magnetization process is done (as usual) under the effect of an alternating magnetic field, it displays hysteresis. The hysteresis loop s area has the meaning of the energy (per unit volume) which must be provided from outside in order to perform the loop; i.e., this is the energy dissipated by the material in the magnetic loop. Consider a toroidal core with area A and length l. Energy loss over a period T: E T i( t) V ( t) dt where i(t) is the eddy current flowing in the toroid and V is the electromotive force 3

31 The quantities in the integrand function can be written as: i(t) H(t)ds V(t) A db dt H(t)l Ampere s law Faraday s law so that E T i( t) V ( t) dt la T H( t) db dt dt Al H( t) db If the loop is performed at higher frequency, the electromotive force increases and the dissipated energy too (the loop becomes wider). The mesoscopic source of eddy currents is the motion of domain walls; their motion is non-uniform because of the multi-valley character of the energy potential landscape, resulting in quick jumps forward followed by stasis. This increases the losses (db/dt becoming exceedingly high during a jump) 3

32 icrowave magnetization dynamics and ferromagnetic resonance Under a magnetic field H, the free magnetization vector precedes around the field axis: d s H dt B g When H is applied along the z axis, the solution is: x mcos t y msin t z const =H being the Larmor frequency, m is the projection of x, y on the z axis. 3

33 icrowave magnetization dynamics and ferromagnetic resonance In a ferromagnetic material there exists an internal field (exchange field) which is of the order of several hundred koe. Therefore. 5 5 s - (f 9 Hz). Of course the magnetization vector will eventually relax towards the field direction on the time scale of magnetometer measurement because of dissipation processes. If it is assumed - as usual in these cases - that the rate of relaxation is proportional to the amount by which the moment is out of equilibrium, addition of the loss term results in: d dt sz H τ is the longitudinal relaxation time. Similarly the transverse components relax toward zero, but with a different (transverse) relaxation time: z z s d dt s x, y H These are the phenomenological Bloch equations. x, y x, y 33

34 icrowave magnetization dynamics and ferromagnetic resonance It is possible to show that the Bloch equations are compatible with a general equation of motion containing a phenomenological damping term. Two suggested forms, due to Landau -Lifschitz and Gilbert respectively, are: d dt d dt H H H d dt is the Gilbert damping, which is of the order.-.; with s -, is the of the order of the nanosecond. Note that the resonance condition of the ferromagnets is altered by magnetostatic fields associated with the sample shape. The components of the internal magnetic field become H i j = H-N j j, with N j = demagnetizing factor and j = x, y, z. 34