Introduction to magnetism of confined systems P. Vavassori CIC nanogune Consolider, San Sebastian, Spain; nano@nanogune.eu
Basics: diamagnetism and paramagnetism Every material which is put in a magnetic field H, acquires a magnetic moment. In most materials M = H (M magnetic dipole per unit volume, magnetic susceptibility. paramagnetism M diamagnetism M Each atom has a non-zero magnetic moment m The moments are randomly oriented (T); H arranges these moments in its own direction. E appl = - m 0 M. H H temperature k b T m= - m B (L + gs) orbital and spin angular momenta In soilds m - gm B S (crystal field) H Each atom acquires a moment caused by the applied field H and opposed to it (Larmor frequency). m= 0 e.g., noble gas. nano@nanogune.eu
Ferromagnetism There are materials in which M is NOT proportional to H. M may be, for example, non-zero at H = 0. M in these materials is not even a one-valued function of H, and its value depends on the history of the applied field (hysteresis). saturation magnetization M S Limiting hysteresis curve: all the points enclosed in the loop are possible equilibrium states of the system. remanence coercive field M H With an appropriate history of the applied field one can therefore end at any point inside the limiting hysteresis loop. H Fe, Co, Ni, alloys also with TM and RE nano@nanogune.eu
Origin of hysteresis In ferromagnetic materials the magnetic moments of the individual atoms interact strongly with each other creating an order against the thermal fluctuations. The magnetization of a sample may be split in many domains. Each of these domains is magnetized to the saturation value M s but the direction of the magnetization vector may vary from one domain to the other at H = 0. The interaction between the magnetic moments is not dipolar (too weak); it is electrostatic (Coulomb) determined by correlaction effects (Quantum mechanics): symmetry of the electrons wavefunction and Pauli principle Hund s rule E ex = -(1/V) ij Jij Si. Sj (Heisember hamiltonian H = - ij Jij S i. S j ) (short range interaction ij is over the nearest neighbors and V is the unit cell volume) J ij is the exchange integral J ij > 0 ferromagnetic order J ij < 0 anti-ferromagnetic order nano@nanogune.eu
Phase transition ferromagnet paramagnet Exchange interaction provides the magnetic order against the thermal fluctuations M s (T) M S T C T Above a critical temperature called Curie temperature (T C ) all ferromagnets become regular paramagnets M S = 0 at H = 0 M S (T C -T) Since T < T C = ½ mean field theory (identical average exchange field felt by all spins) This temperature for anti-ferromagnets is called Néel temperature (T N ) nano@nanogune.eu
Why magnetic domains? Energy densities In vacuum u = B 2 /2m 0 Inside a material u =1/2 m 0 M s 2 Total energy U = udt All space a) b) c) The field created outside the magnet in cases a) and b) costs B 2 /2m 0 Joules/m 3, thus case c) is the one energetically favoured. This is due to the finite size of the magnet, so it is an effect of lateral confinement. nano@nanogune.eu
Magnetostatic energy It can be shown (Maxwell equations) that the dipolar magnetostatic energy E m can be espressed as: E 1 m m0 3 0 2 M r Hd r d r sample Where H d is called, variously, demagnetizing field, magnetostatic field or dipolar field. Energy due to the interaction of each dipole with the field H d created by all the other dipoles. The complication arises from the fact that H d [M(r)]. H d can be calculated like a field in electrostatics as due to the magnetic charges (bulk - M and surface n M ). The only difference is that they never appear isolated but are always balanced by opposite charges. + + + + M n M > 0 H d - - - - n M < 0 nano@nanogune.eu
Magnetostatic energy: examples Infinite ferromagnet uniformly magnetized M = 0 (M is uniform everywhere) and n M = 0 (no borders) H d = 0 E m = 0 n M < 0 n M > 0 - - - d + + + M = 0 (M is uniform everywhere) but n M 0 (borders) H d = -M 0 E m > 0 M = 0 (M is uniform everywhere) and n M = 0 (M parallel to the borders) H d = 0 E m 0 nano@nanogune.eu
Where M inside a domain is pointing to? Anisotropy The direction of the magnetization inside each domain is NOT arbitrary. For instance, the crystal structure is not isotropic so it is expected that along certain crystallographic directions it is easier to magnetize the crystal, along others it is harder (confirmed by experiments). The exchange energy term introduced so far (Heisemberg) is isotropic. We have to introduce a phenomenological expression for this additional term E anis. There are several types of anisotropy, the most common of which is the magnetocrystalline anisotropy caused by the spin-orbit interaction (the electron orbits are linked to the crystallographic structure and by their interaction with the spins they make the latter prefer to align along well-defined crystallographic axes. In this case E anis will be a power series expansions that take into account the crystal symmetry. nano@nanogune.eu
Anisotropy sources (a) Magnetocrystalline anisotropy: dependence of internal energy on the direction of sposntaneous magnetization respect to crystal axis. It is due to anisotropy of spin-orbit coupling energy and dipolar energy. Examples: - Cubic E anis = K 1 (a x 2a 2 y + a y 2a 2 z + a z 2a x 2) + K 2 a x 2a 2 y a 2 z +. - Uniaxial E anis = K 1 sin 2 q + K 2 sin 4 q + -K 1 (n. M) 2 [K] = J/m 3 Surface and interface anisotropy: due to broken translation symmetry at surfaces and intefaces. The surface energy density can be written: - E surf = K p a 2 f - K s a n 2; where a n and a f are the director cosines respect to the film normal and the in plane hard-axis. nano@nanogune.eu
Anisotropy sources (b) Strain anysotropy: strain distorts the shape of crystal (or surface) and, thus can give rise to an uniaxial term in the magnetic anisotropy. E s = 3/2 ls sin 2 q; where l is the magnetostriction coefficient (positive or negative) along the direction of the applied stress s and q is the angle between the magnetization and the stress direction. Growth induced anisotropy: preferential magnetization directions can be induced by oblique deposition or by application of an external magnetic field during deposotion. nano@nanogune.eu
Summary of energy contributions E tot = E appl + E ex + E anis + E m E appl is the Zeeman energy related to the spin alignment in the external magnetic film H. E appl = - m 0 M. H E ex is the interatomic exchange interaction favoring parallel atomic moments alignment (short range). E ex = -(1/V) ij Jij Si. Sj ( ij is over the nearest neighbors and V is the unit cell volume) E anis is the magnetic anisotropy energy associated to preferential magnetization directions. For a preferential axes n : E anis = -K1(n. M) 2 E m is the magnetostatic self-interaction due to the long-range magnetic dipolar coupling. Responsible for domain formation in bulk- and film-like specimens E m = -1/2 m 0 H d. M nano@nanogune.eu
Anisotropy and domain structures. a) Magnetostatic energy is not the only ingredient to determine the actual domain structure. Anisotropy energy plays a role: b) y? structure a) is expected with cubic anisotropy; structure b) with uniaxial anisotropy with EA along x; structure c) with uniaxial anisotropy with EA along y. x But still, what decides the number of subdivisions, for c) instance? Somwhere ther is also Exchange energy stored. Where? nano@nanogune.eu
Domain boundaries Bloch domain wall Neél domain wall (thin films) So to set up a domain structure and reduce the magnetostatic energy there is a price to pay: an excess of anisotropy and exchange energy has to be stored in the boundaried between domains, the domain walls (there is also some magnetostatic energy). [A] = J/m Domain wall energy, per unit of surface: s w AK 1 with A=nJS 2 /a, the exchange stiffness constant, where n is the number of sites in the unit cell, J is the average exchange integral value, S is the spin number and a is the unit cell edge. Domain wall width: w w A K 1 nano@nanogune.eu
Nèel Energies and widths of domain boundaries Threshold between Bloch and Nèel walls in a Fe film Bloch nm nano@nanogune.eu
Energies and widths of domain boundaries (e) Threshold between Bloch and Nèel walls in a typical soft film (sligthly anysotropic) Nèel Bloch nm nano@nanogune.eu
Magnetization reversal and domains H ext = 0 H ext H ext Ideally, reversal through domain walls motion does not cost energy because the wall energy necessary at the new position is released at the previous position (reversible). The annihilation of DWs costs energy, of course. In the case of domain wall pinning at local defects (non-magnetic impurities, voids, grain boundaries ) some activation energy is necessary to release the domain wall from the pinning centre (abrupt displacement, Barkhausen jumps, viscosity due to Lenz law, energy dissipation -> irreversible process -> hysteresis). nano@nanogune.eu
Let s see a real example Domain wall motion is the preferred way of changing the magnetization at low fields. With increasing field strength, first domain walls will move and increase the size of domains with a magnetization component parallel to the field (with the magnetization in every domain being parallel to an easy axis). Therefore some misalignment with the applied field remains if the field is not aligned with one of the easy axes. domain walls displacement (low fields) rotation (high fields) Barkhausen jumps At high fields the domain walls are removed and the magnetization is rotated coherently towards the field direction. nano@nanogune.eu
Hysteresis due to magnetization rotation Stoner and Wohlfarth model: coherent rotation of an uniaxial particle uniformly magnetized. E = K 1 sin 2 - mm s Hcosq Free energy for unit volume M s q H Easy axis Ho 2K1 mo M s nano@nanogune.eu
Ferromagnetic nano-structures Ferromagnetic nano-structures offer a unique opportunity to investigate properties at length scales previously unattainable. Because intrinsic magnetic length-scales (e.g., exchange length or the domain wall thickness) are comparable to the sample size, novel physical properties can be expected, respect to bulk- and film-like materials. Surface effects become relevant (dominant). Novel physical properties can be expected with respect to bulk- and film-like materials (e.g., completely different magnetization reversal mechanism). nano@nanogune.eu
Applications This field has attracted much attention because of its close ties to potential technological applications. Nonvolatile Magnetic Random Access Memories (MRAM). Periodic bi-dimensional arrays of magnetic dots for future high density magnetic storage media (1 Tbit/in 2 ). Magnetoelectronic devices with new functionalities (sensors). A key issue for such applications is to understand and control the magnetic switching of small magnetic elements. nano@nanogune.eu
Scale length parameters The relative strength of the anisotropy and magnetostatic (1/2m M s2 ) energies (per unit volume) can be expressed by the dimensionless parameter: k = 2K 1 /m M s2. k provides a quantitative definition of the conventional distinction between soft (k <<1, i.e., dipolar effects dominate over anisotropy ones) and hard (k >1) materials. The competition between exchange and dipolar energy is expressed in terms of the exchange 2A m0m s length: l ex = 2 The comparison between exchange and anisotropy may be expressed through the length: l w = DW width A K 1 l ex k A (J/m) M s (A/m) K (J/m 3 ) l ex (nm) l w (nm) k (adim.) Fe 21 10-12 1.7 10 6 48 10 3 3.4 20.9 2.6 10-2 Co 30 10-12 1.42 10 6 520 10 3 4.9 7.6 4.2 10-1 Ni 9 10-12 0.49 10 6-5.7 10 3 7.5 39.7 3.7 10-2 Py 13 10-12 0.86 10 6 0 5.3 0 nano@nanogune.eu
From 3D to 2D (infinite) E tot = E appl + E exc + E anis + E m unfavoured favoured + + + + M n M > 0 - - - - n M < 0 M H d = 0 H d = - M Real 2D systems have a lateral finite size more favoured nano@nanogune.eu
Magnetostatic effects due to shape. For uniformly magnetized bodies M = 0 (M is uniform everywhere) so that magnetostatic energy depends only on surface magnetic charges n M 0 shape of the body) H d = 0 favoured H d = - M unfavoured H d,z = - N z M H d,x = - N x M N z < N x Aspect ratio H d,z < H d,x favoured unfavoured Elongated particles nano@nanogune.eu
Demagnetizing tensor : shape anisotropy. The uniformity condition can be realized ONLY for isotropic ellipsoids and for such special cases H d = -N M, where N is a tensor called demagnetizing tensor. Referring to the ellipsoid semi-axes the tensor becomes diagonal and Nx, Ny, Nz are called demagnetizing factors and Nx + Ny + Nz = 1. Magnetostatic self interaction for an ellipsoid (referring to the ellipsoid semi-axes ) E d = 1/2 m (N x M x 2 + N y M y 2 + N z M z 2 ). y M q Uniaxial anisotropy E anis = K 1 sin 2 q z K 1 = 1/2 m M s2 (N y - N z ) nano@nanogune.eu
Lateral confinement and single domain state Upon sufficient lateral confinement the size of a ferromagnet can become so small that not even a single domain wall can be accomodated inside the system. The magnetization configuration inside the ferromagnet can become a single domain (this concept will be defined more precisely later on). Size Multi-domains structure Single domain nano@nanogune.eu
Single domain particles: the spherical case Domains are favourable from the point of view of magnetostatics (minimize dipolar energy) but they cost domain wall energy For a sphere of radius R of material with A and K1, by equating the energy of the single domain state with that of two domains, one gets that the critical radius below which the single domain state is energetically stable is: s w AK 1 R SD = 36 m AK 2 0M s 1 For spherical particles one finds that R SD for Co is 70 nm, whereas for Fe is 15 nm, and for Ni 55 nm [ hard magnetic particles (Co) are more stable than soft magnetic ones (Fe, Ni, Py)]. nano@nanogune.eu
Too small!! If the size becomes too small (as for R SD for Co, Fe, and Ni) the magnetic moment of the single domain ferromagnet can fluctuate due to thermal energy: superparamagnetic limit. Size Single domain Superparamagnet nano@nanogune.eu
Thermal stability of the remanent state: superparamagnetism Competition between thermal and magnetic (anisotrpy) energy: thermal activated reversal. This leads to a relaxation time at zero field (for uniaxial particle): t K V 10 10 e 1 k B T K 1 is the anisotropy constant V is the particle volume, k B the boltzmann s constant and T the absolute temperature. t depends on the particle volume V: if the particle is too small it becomes unstable at room temperature -> superparamagnetic limit. E k B T 0 K 1 V p q nano@nanogune.eu
Retarding the onset of superparamagnetism - Increase K 1 as much as possible. - Larger aspect ratio retards the onset of superparamagnetism by adding shape anisotropy to the material intrinsic anisotropy. K 1 = 1/2 m M s2 (N y - N z ) Shape anisotropy hard soft Material effect (K 1 ) nano@nanogune.eu
Lateral confinement and remanent state: summary Size effects Closure domains Single domain Superparamagnetic Shape effects Closure domains Single domain Aspect ratio effects Closure domains Single domain nano@nanogune.eu
A caveat about single domain state in confined systems The uniformity condition can be realized ONLY for isotropic ellipsoids. Real particles are never ellipsoids..deviations from uniformity close to the edges. Flower state Leaf state In order to avoid these deviations from uniformity the size of the particle (cubic) should be: d ~ l ex which is of a few nm for Co,Fe and Ni. nano@nanogune.eu
An interesting case: very soft material Fe 20 Ni 80 Eg., a particle 10 nm thick and with lateral size below 700 nm could not accomodate even a domain wall. Nèel Bloch nano@nanogune.eu
Soft nano-scale disk and rings: vortex state Magnetization configuration determined by magnetostatic and exchange energies (no anisotropy). Vortex core The energy is almost all due to exchange. Residual magnetostatic energy is confined in the core. The core is necessary to avoid the singularity in the in-plane magnetization curling. Systems developing vortes state configurations are interesting because of their reduced sensitivity to edge effects. Stabilization of the vortex state by removing the high energy core in a ring structure. Magnetic force microscopy image nano@nanogune.eu
Incoherent vs. coherent reversal For single domain particles the reversal process can be still incoherent, in a way different from doman wall displacement: curling mode. reversal 2K 1 m o M s H N 2K 1 m o M s nano@nanogune.eu
Magnetization revrsal is a dynamic process Coherent rotation accomplishes magnetization reversal much faster than inhomogeneous and domain walls displacement mechanisms. Magnetization rotation : 100 ps to 1 ns Domain walls displacement : 100 ns up to 100 ms E = - m B dj G dt G = m B dm dt m B m= -m B g J = - J M motion is damped (a) Larmour precession ( c = B) damping Low damping m y Magnetization reversal through coherent rotation takes place with magnetization precession (N.B., always counter-clockwise looking from +z). If the precession is effectively dumped the reversal can be very fast. torque High damping m y m x m x nano@nanogune.eu
Damping and field dependance of the reversal of a magnetic moment t t m H Damping a 0.05 m 1700 emu x (10nm) 3 H 5 koe: reversal time 2121 ps H 500 Oe: reversal time 20481 ps t = 0 Damping a 0.2 m 1700 emu x (10nm) 3 H 5kOe : reversal time 646 ps H 500 Oe: reversal time 5950 ps Damping a 1.0 m 1700 emu x (10nm) 3 H 5 koe: reversal time 185 ps H 500 Oe: reversal time 1565 ps My 1.0 0.5 0.0-0.5-1.0-1.0-0.5 0.0 0.5 1.0 Mz 1.0 1.0 0.5 0.5 0.0 0.0-0.5-0.5-1.0-1.0-1.0-1.0-1.0-0.5 0.0 0.5 1.0-1.0-0.5 0.0 0.5 1.0-1.0 Mx My Mz Mx My Mz Mx nano@nanogune.eu
References and further reading S. Blundell, Magnetism in condensed matter, Oxford University Press, 2006. A. Aharoni, Introduction to the theory of Ferromagnetism, University Press, Oxford 2000. W. F. Brown, Micromagnetics, Wiley, New York 1963. S. Chikazumi, Physics of Magnetism, Wiley, New York 1964. M. Prutton, Thin ferromagnetic films, Butterworth &Co. (Publishers) Ltd., London 1964. nano@nanogune.eu