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GT-2 Magnetometry: an introduction P. Vavassori -Ikerbasque, Basque Fundation for Science and CIC nanoGUNE Consolider, San Sebastian, Spain. P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018 Introduction

  1. GT-2 Magnetometry: an introduction P. Vavassori -Ikerbasque, Basque Fundation for Science and CIC nanoGUNE Consolider, San Sebastian, Spain. P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  2. Introduction -Outline This lecture will provide an introduction to a number of important tools and methods employed in the investigation of magnetic materials. They will focus on magnetometry tools and approaches available in most laboratories: - magneto-optical Kerr effect (MOKE) magnetometry. - vibrating sample magnetometry (VSM), - superconducting quantum interference device (SQUID), - torque magnetometry, - alternating gradient magnetometry, Consideration will be also given to the special problems posed by measurements on feebly magnetic materials, like nanostructured ones, basic requirements regarding sensitivity and accuracy, and potential artifacts. P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  3. Introduction -Outline Magnetometry: what we would like to measure? • Saturation magnetization • Remnant magnetization • Coercive field • Switching field • Anisotropy symmetry and energy • Reversal process ……………. At the nanoscale P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  4. Units P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  5. Constitutive equations and units P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  6. Summary constitutive equations and units cgs SI H units Oe A/m M =  m H B units Oe T M units emu /cm 3 A/m B = m 0 ( H + M ) -> B = m H Conversions: m = m 0 (1+  m ) For H 1Oe = 10 3 / 4 p A/m = 79,58 A/m Cgs System For B 1T = 10 4 Oe B = ( H + 4 p M ) m 0 = 1 For M 1emu/cm 3 = 10 3 A/m m = (1+ 4 p  m ) Magnetic moment 1 Am 2 = 10 3 emu 1emu = 10 20 m B = 10 -3 Am 2 1 m B = 9.274 10 -24 Am 2 [J/T) P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  7. Basics: diamagnetism and paramagnetism Every material which is put in a magnetic field H , acquires a magnetic moment. In most materials M =  m H ( M magnetic dipole per unit volume,  magnetic susceptibility). m = - m B ( L + g S ) orbital and spin angular momenta In soilds m ≈ - g m B S (crystal field) paramagnetism diamagnetism Low T M M High H H H Each atom acquires a moment Each atom has a non-zero magnetic moment m  caused by the applied field H and The moments are randomly oriented (T); opposed to it H arranges these moments in its own direction. (Larmor frequency). m = 0 e.g., noble gas. E appl = - m 0 M . H temperature k b T P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  8. 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 M H remanence Limiting hysteresis curve: all the points enclosed in the loop are possible equilibrium states of the system. coercive field With an appropriate history of the H applied field one can therefore end at any point inside the limiting hysteresis loop. Fe, Co, Ni, alloys also with TM , C, and RE P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  9. Phase transition ferromagnet →paramagnet M s (T) M S Above a critical temperature called Curie temperature (T C ) all ferromagnets become regular paramagnets → M S = 0 at H = 0 T T C M S  (T C -T)  T < T C Since  = ½ mean field theory (identical average exchange field felt by all spins) This temperature for anti-ferromagnets is called Néel temperature (T N ) P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  10. Origin of hysteresis Ferromagnetic order not enough E m = - m 0 m . H Zeeman energy m ‘ Many atoms M ||H Paramagnetism m Individual atom H Ferromagnetism (exchange only) P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  11. Magneto-crystalline anisotropy: spin-orbit coupling d-orbital momentum in an atom Spin-orbit coupling tends to S induce an orbital motion as <L z > = -1, -2 H sketched…but there is the crystal field potential. H <L z > = 1, 2 S H - - - - H - - - - Orbital motion is highly hindered Orbital motion is less hindered P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  12. • 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 x2 a y2 + a y2 a z2 + a z2 a x2 ) + K 2 a x2 a y2 a z2 +…. - Uniaxial E anis = K 1 sin 2 q + K 2 sin 4 q +… ≈ -K 1 ( n . M ) 2 • 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 f 2 - 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. • 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. P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  13. Exchange+anisotropy → Hysterisis Bistable one-dimensional potential: uniaxial anisotropy M s Stoner and Wohlfarth model q E tot = E appl + E anis f H E tot = K 1 sin 2 f - m 0 M s Hcos q Easy axis E H=0 M s M ||H K 1 q 0 p H EA HA E H K 2 = K 1 1 HM s H q c m p 0 M o s P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  14. Exchange+anisotropy → Hysterisis ..... Real systems Expected EA loop (anisotropy field 282 Oe: Measured EA loop K = 48000 J/m 3 Ms 1.7 10 6 A/m ->28.2 mT) Film 1.5 1 H || Fe(100) 1.0 Easy axis 1.0 0.5 0.5 0.5 M/M sat 0.0 0 0.0 -0.5 -0.5 -0.5 -1.0 H (Oe) -1.0 H (Oe) -1 -400 -200 0 200 400 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 -750 -500 -250 0 250 500 750 Why this difference? Different reversal process: reversed domains nucleation P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  15. Magnetostatic energy. Magnetostatic energy is potential energy of magnetic moments in the field H d they have created themseves. The magnetostatic energy e m can be evaluated as: e m =  0 1 ( ) 1 ( ) ( )   m = − m  2 3 3 H r d r M r H r d r d d 0 0 2 2 all space sample If for simplicity we assume that M is uniform inside the body the integral becomes a surface integral where H d can be thought as produced by surface magnetic charges s s = M . n and the energy e m depends solely on the shape of the body . The uniformity condition can be realized only for isotropic ellipsoids and for such special cases H d = - N M , where N N is a tensor called demagnetizing tensor. Referring to the ellipsoid semi-axes the tensor becomes diagonal and the diagonal elements N x , N y , N z are called demagnetizing factors and N x + N y + N z = 1 Magnetostatic self interaction for an ellipsoid (referring to the ellipsoid semi-axes ) e m = 1/2 m 0 (N x M x2 + N y M y2 + N z M z2 ). P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  16. Magnetostatic energy: why magnetic domains form. 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. m m Energy densities In vacuum u = B 2 /2 m 0 Total energy Inside a U = ∫∫∫ ud t material All u =1/2 m 0 M s2 space P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018 ∫∫∫ t

  17. There is a cost for magnetic domains formation P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  18. Exchange+anisotropy → Hysterisis ..... Real systems Expected EA loop (anisotropy field 282 Oe: Measured EA loop K = 48000 J/m 3 Ms 1.7 10 6 A/m ->28.2 mT) Film 1.5 1 H || Fe(100) 1.0 Easy axis 1.0 0.5 0.5 0.5 M/M sat 0.0 0 0.0 -0.5 -0.5 -0.5 -1.0 H (Oe) -1.0 H (Oe) -1 -400 -200 0 200 400 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 -750 -500 -250 0 250 500 750 Why this difference? Different reversal process: reversed domains nucleation P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  19. Magnetostatic energy: Shape anisotropy favoured unfavoured Equivalent to an uniaxial anisotropy K s = 1/2 m 0 M s 2 ( N ⊥ - N || ) Osborne PRB 67, 351 (1945) P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

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