3 Magnetic hysteresis and basic magnetometry Magnetic reversal in thin films and some relevant experimental methods Maciej Urbaniak IFM PAN 2012 Today's plan ● Classification of magnetic materials ● Magnetic hysteresis ● Magnetometry Urbaniak Magnetization reversal in thin films and... Classification of magnetic materials All materials can be classified in terms of their magnetic behavior falling into one of several categories depending on their bulk magnetic susceptibility . ⃗ χ = M ⃗ In general the susceptibility is a position dependent tensor H In some materials the magnetization is 2 not a linear function of field strength. In 50000 such cases the differential susceptibility is introduced: 40000 ⃗ χ =d M ] χ d ⃗ m d H / 30000 1 2 A [ We usually talk about isothermal M χ 20000 1 susceptibility: ∂ ⃗ 10000 χ =( M ) T ⃗ ∂ H T 0 Theoreticians define magnetization as: 0 2 4 6 8 10 H[kA/m] ∂ ⃗ =−( F ) = − M ∂ ⃗ F E TS -Helmholtz free energy H T Urbaniak Magnetization reversal in thin films and... Classification of magnetic materials It is customary to define susceptibility in relation to volume, mass or mole (or spin): ⃗ ( ⃗ / ρ ) 3 ( ⃗ / ) 3 χ = M [ ] χ = M [ m ] χ = M mol [ m ] ⃗ dimensionless , ρ ⃗ , mol ⃗ H H kg H mol The general classification of materials according to their magnetic properties μ<1 <0 diamagnetic* μ>1 >0 paramagnetic** μ1 0 ferromagnetic*** *dia /daɪəmæɡˈnɛtɪk/ -Greek: “from, through, across” - repelled by magnets. We have from L2: 1 2 The force is directed antiparallel to the gradient of B2 F = V ∇ B i.e. away from the magnetized bodies 2 0 − - water is diamagnetic χ ≈ − 10 5 (see levitating frog from L2) χ ≈ −5− −3 ** para- Greek: beside, near; for most materials 10 10 [1]. ***susceptibility ranges from several hundred for steels to 100,000 for soft magnetic materials (Permalloy) Urbaniak Magnetization reversal in thin films and... Classification of magnetic materials ●Feebly magnetic material – a material generally classified as “nonmagnetic” whose maximum normal permeability is less than 4 [5]. ●Ferromagnetic materials can be classified according to the magnetic structure on atomic level: maximum permeability 1. Ferromagnets B 2. Antiferromagnets initial permeability 3. Ferrimagnets μ= B 4. Asperomagnets -random ferromagnets H μ 5. Sperimagnets – random ferrimagnets 0 H Urbaniak Magnetization reversal in thin films and... Classification of magnetic materials In general the susceptibility is frequency dependent and the magnetization depends on the preceding field values [3]: ⃗ M (t)=∫ f (t−t ' )H (t ' )dt' It is customary to introduce a complex susceptibility: χ =χ + χ i x= ( )+ ( ) real i imag e cos x i sin x Then we have: ⃗ = (χ ⃗ )= [(χ + χ ) ⃗ −i ω t ]= ⃗ (χ (ω )+χ (ω )) M R.e H R.e real i imag H 0 e H 0 real cos t imag sin t The imaginary part of susceptibility is responsible for magnetic losses. Urbaniak Magnetization reversal in thin films and... Magnetic periodic table of elements Image source: B.D. Cullity, Introduction to Magnetic Materials, Addison-Wesley 1972, p. 612 Urbaniak Magnetization reversal in thin films and... Paramagnetic materials ● We consider an ensemble of atoms with fixed magnetic moments m [2]. We neglect here and further the effect of diamagnetism which is negligible in paramagnetic and ferromagnetic materials. [2] A. Aharoni, Introduction to the Theory of Ferromagnetism, Clarendon Press, Oxford 1996 Urbaniak Magnetization reversal in thin films and... Paramagnetic materials ● We consider an ensemble of atoms with fixed magnetic moments m [2]. We neglect here and further the effect of diamagnetism which is negligible in paramagnetic and ferromagnetic materials. ● From quantum mechanics (QM) we adopt the expression for m: ⃗ = μ ⃗ μ = e h = ( )× −24 2 m g B S B π 9.27400 968 20 10 A m 4 me ● From QM again we know that Sz can assume only 2S+1discrete values ● The moments m interact with the external field but not with each other. We use the Boltzmann distribution to get the average moment of the ensemble along the field: / ∑ mz B k B T 〈 〉= mz e mz m B/ k T ∑ e z B ● Using the expression for m and remembering that S is discrete we get: S μ / ∑ μ g B n B k B T g B n e 〈 〉=n=−S mz S g μ n B/ k T ∑ e B B n=−S [2] A. Aharoni, Introduction to the Theory of Ferromagnetism, Clarendon Press, Oxford 1996 Urbaniak Magnetization reversal in thin films and... Paramagnetic materials The previous expression can be shown to give [2]: 〈S 〉 g μ S B z = ( B ) B S , where BS is a Brillouin function depending on S: S k B T 2S+1 2S+1 1 x B ( x )= coth( x)− coth( ) S 2 S 2 S 2S 2S ( )= ( ) For S=1/2 we get: B1/ 2 x tanh x 1 x For small values of x Taylor expansion leads to: coth ( x ) = + + ... and after tedious arithmetic one gets: x 3 + g μ S (S+1) ( )=S 1 → 〈 〉=( B ) BS x x S z B 3 S 3k B T = μ 〈 〉 Substituting m g B S z we get (we omit vector notation as system is isotropic): ( + ) = N S S 1 ( μ )2 〈 〉= 〈 〉 M ( ) g B M N m , where N is a number of spins 3 k T B z z B per unit volume [2] A. Aharoni, Introduction to the Theory of Ferromagnetism, Clarendon Press, Oxford 1996 Urbaniak Magnetization reversal in thin films and... Paramagnetic materials From the previous expression we get Curie Law [2]: ⃗ ( + ) χ =M =C =μ (N S S 1 )( μ )2 ⃗ with Curie constant C 0 g B H T 3 k B In paramagnets the susceptibility may depend on H but it is a single-valued function; it does not depend on the previous values of field strength (magnetic history). 1,0 ● = Accessible magnetic fields (97 T 0,8 Brillouin for S 5 as of 2011) do not allow reaching (m=5μ ) the magnetic saturation of most B 0,6 paramagnets s M ●Lowering temperature allows / 0,4 considerable decrease of M essential fields 4.2K 0,2 10K 〈S 〉 g μ S B 100K z = ( B ) BS 293K S k B T 0,0 ! 0 20 40 60 80 100 B[T] [2] A. Aharoni, Introduction to the Theory of Ferromagnetism, Clarendon Press, Oxford 1996 Urbaniak Magnetization reversal in thin films and... Paramagnetic materials χ =C Curie Law: T ●J=S=3/2, 5/2, 7/2 ●g=2 in all cases ●note high fields and very low temperatures W.E. Henry, Phys.Rev. 88 559 (1952) Urbaniak Magnetization reversal in thin films and... Ferromagnetic materials Most notable features of ferromagnetic materials: ● high initial susceptibility/permeability ● they usually retain magnetization after the removal of the external field – remanence ● the magnetization curve (B-H or M-H) is nonlinear and hysteretic ● they lose ferromagnetic properties at elevated temperatures (Curie temperature) The notable examples [4]: M [kA/m] T [K] S C @RT Fe 1714 1043 Co 1433 1403 Ni 485 630 M [Am2/kg] S @RT 217.75 161 54.39 Urbaniak Magnetization reversal in thin films and... Ferromagnetic materials Most notable features of magnetic materials: ● high initial susceptibility/permeability ● they usually retain magnetization after the removal of the external field – remanence ● the magnetization curve (B-H or M_H) is nonlinear and hysteretic ● they lose ferromagnetic properties at elevated temperatures (Curie temperature) The notable examples [4]: M [kA/m] T [K] S C @RT Fe 1714 1043 Co 1433 1403 Ni 485 630 M [Am2/kg] S @RT 217.75 161 54.39 Urbaniak Magnetization reversal in thin films and... Ferromagnetism – Weiss molecular field approximation ●in contrast to the description of paramagnets we assume that magnetic moments interact via the exchange energy ●the description is appropriate for rare earth ions (in crystal lattice) in which the uncompensated spins are in inner atomic shells ●the model is inappropriate for the description of the magnetization of metals We describe the total energy of a system in magnetic field B by [2]: =− ∑ ⃗⋅⃗ −∑ μ ⃗⋅⃗ − E J ij S i S j g B S i B J ij exchangeintegral ij ,i ≠ j i We tag one spin and replace other spins by their mean value. The energy terms in which spin i is is involved sum to: =− ∑ ⃗⋅〈 ⃗ 〉− μ ⃗⋅⃗ =−⃗⋅⃗ = ∑ 〈 ⃗ 〉+ μ ⃗ Ei 2 J ij S i S j g B S i B S i Bi ,with Bi 2 J ij S j g B B j j 1 2 3 4 1 2 3 4 1 ● ● ● 1 ● terms involving i=2 (for 4 spins) 2 ● ● ● 2 ● ● ● 3 ● ● ● 3 ● 4 ● ● ● 4 ● [2] A. Aharoni, Introduction to the Theory of Ferromagnetism, Clarendon Press, Oxford 1996 Urbaniak Magnetization reversal in thin films and... Ferromagnetism – Weiss molecular field approximation The problem of ferromagnetism is now changed into the problem of isolated spins interacting with an applied field – paramagnetism. From there we have: g μ S B S B 〈 〉= ( B )→ ( μ ⃗⋅⃗ → ⃗⋅⃗ ) 〈 〉= ( i ) S z S BS with g B S B S B S iz S BS k B T k B T Substituting Bi from previous page we obtain: S ⃗ 〈S 〉=S B ( [2∑ J 〈S 〉+g μ B]) i z S ij j B k B T j Dropping indexes (on average spins are equal) and summing only over nearest neighbors the expression can be rewritten [2]: 〈 〉 μ 2 μ= ( +α μ) μ= S z =g B S B α ( )=2 S BS h with , h , T NN J S k B T k B T transcendental equation for μ [2] A.