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Magnetic Domains 6 Magnetic domains Magnetization reversal in thin films and some relevant experimental methods Maciej Urbaniak IFM PAN 2012 Today's plan ● Magnetic domains ● Domain walls ● Single domain particles, superparamagnetism ● Magnetic domains in bulk systems ● Magnetic domains in thin films Urbaniak Magnetization reversal in thin films and... Magnetic domains – early views original image taken from: B. D. Cullity Introduction to magnetic materials Addison-Wesley, Reading, Massachusetts 1972 Urbaniak Magnetization reversal in thin films and... Urbaniak Urbaniak image source Wikimedia Commons; authors: Zureks, Chris Vardon Magnetic domainspreliminaries - domains are magnetised "down". domain - all white domains are magnetised "up", all dark The theshow arrows direction of magnetisation in each orientation, so the domain structure is much more complex. The area outside of the grain has different crystallographic through the width ofwhole image (top half of the photo). polishing was not perfect - there is an angled scratch photographed under area 0.1sampleThe polished was mm wide. and line) of Magnetic domains in a magnetized in different directions [1,2,3] directions different in magnetized ● Real ferromagnets at zero applied field are usually divided into domains which are are which domains into divided usually are field at applied zero ferromagnets Real non-oriented electrical steel Magnetization reversal in thin films and... thin in reversal Magnetization Kerr-effect microscope single grain (outlined with a black . The The .photo shows an . The The . 10-100 of are domains magnetic the Typically with with CMOS-MagView). Visualization of meandering domains (image taken μ m size image source Wikimedia Commons; author: Matesy GmbH Magnetic domains - preliminaries ●Real ferromagnets at zero applied field are divided into domains which are magnetized in different directions [1,2,3] ●Barkhausen noise was the first confirmation of the domain concept [3]. ●Usually it can be safely assumed that magnetic moment direction within a given domain is constant in areas distant from the boundaries with other domains (domain walls) Urbaniak Magnetization reversal in thin films and... Urbaniak Urbaniak images source: http://www.ece.neu.edu/faculty/nian/mom/domains.html Magnetic domainspreliminaries - electrodynamics*: classical the within lated ● Magnetic domains are of magnetostatic origin and their shapes and sizes can be calcu- calcu- be can sizes and shapes their and origin magnetostatic of are domains Magnetic Magnetization reversal in thin films and... thin in reversal Magnetization *quantum mechanics enters through exchange, anisotropy etc. constants D. Clarke, O. A. Tretiakov, and O. Tchernyshyov, PHYSICAL REVIEW B 75, 174433 (2007) Urbaniak Urbaniak C. Grünzweig, C. David, O. Bunk, J. Kohlbrecher, E. Lehmann et al., Magnetic domainspreliminaries - J. Appl. Phys. 107, 09D308 (2010) ● ● Neutron refraction at domain walls allows the visualization of the domain structure. domain the of visualization the allows walls atdomain refraction Neutron magnetization) volume of (propagation bulk materials in visualized domains Magnetic Magnetization reversal in thin films and... thin in reversal Magnetization Origin of magnetic domains ●Consider a infinite cylinder, with 2R diameter, magnetized uniformly: ● ρ =−∇⋅⃗ The discontinuity of magnetization at magn M outer boundaries creates magnetic charges at the surface of the cylinder (see L.2) ⃗ ⋅⃗ ∇⋅ ⃗ ϕ (⃗)=∮ M ds −∫ M 3 m r ∣⃗∣ ∣⃗∣ d r' S r V r 1 ⃗ ⃗ ●Using the expression for the shape anisotropy energy (from L.5): E = V μ (N⋅M )⋅M , demag 2 0 we obtain for the energy per unit length along z: V −volume of the sample 1 1 π E = μ M 2 π R2=μ R2 M 2 demag 2 0 2 S 0 4 S + + = demag factor for cylinder N x N y N z 1 Urbaniak Magnetization reversal in thin films and... Origin of magnetic domains ●Consider now a infinite cylinder magnetized as shown below (i.e., divided into two antiparallel domains): ● ρ =−∇⋅⃗ It is assumed that both domains are of magn M equal sizes ●The division into the domains changes the distribution of magnetic y surface charges over the surface of the cylinder x ●We outline now the derivation of the demagnetization energy of the two-domain cylinder (that part is taken from A. Aharoni [2]). ●The magnetization is: +1 if y>0, i.e. , 0≤φ ≤π M =M =0, M =M ×( ) y z x s −1 if y ,0 , i.e. , π ≤φ ≤2π Urbaniak Magnetization reversal in thin films and... Origin of magnetic domains ●The step function can be expressed as a Fourier expansion: Mx ∞ [( + )ϕ ] =4 ∑ sin 2n 1 M x π M S + n=0 2n 1 3 for n =1000 φ[Rad] max 1 cosα sin β = (sin (α +β )+sin(α −β )) 2 ●The normal component of magnetization (i.e., the one creating magnetic charges) is then: ∞ [( + )ϕ ]+ [ ϕ ] = = (ϕ)=2 ∑ sin 2n 2 sin 2n 2 M n M ρ M x cos π M S + [( + )ϕ ]= [( + )ϕ+ϕ ]= = 2n 1 1 sin 2n 2 sin 2n 1 n 0 [( + )ϕ ] (ϕ )− ( ϕ ) 3 2sin 2n 1 cos sin 2n ● 1 The sum can be broken down into two sums [2]: sin [(2n+1)ϕ ]cos(ϕ )= [ sin [(2n+2)ϕ ]+sin (2 nϕ )] 2 ∞ 4 =2 ∑[ 1 + 1 ] [ ϕ ] M n π M S − + sin 2n n=1 2n 1 2n 1 the second term is null for n=0 and in the first term [( + )ϕ ] ( ϕ ) → − sin 2n 2 → sin 2n n is replaced by (n-1) so that the sum starts effectively from n=0 n n 1: + − 2n 1 2 n 1 Urbaniak Magnetization reversal in thin films and... Origin of magnetic domains ●The boundary condition, at cylinder surface, for magnetic scalar potential U is [2]: ∂ ∂ ∞ ( ϕ) ( − ) = =8 ∑ nsin 2n 1 + 1 = 4n ∂ ρ U inside ∂ ρ U outside M n π M S ( + )( − ) 2n −1 2n+1 (2n+1)(2n−1) ρ =Radius n=1 2n 1 2n 1 ●We seek a solution of the form (guessing): ∞ =∑ (ρ ) ( ϕ ) U un sin 2n n=1 ●Since magnetization is constant throughout the domains we have ∇ 2 U = 0 and: ∞ ∂2 1 ∂ 4n2 Laplacian in cylindrical coordinates: ∇ 2U =∑ sin(2nϕ)( +ρ − )u (ρ ) ∂ ρ 2 ∂ ρ ρ 2 n ∂ ∂ ∂2 ∂2 n=1 ∇ 2 =1 + 1 + f ∂ (r ∂ f ) f f r r r r2 ∂ϕ 2 ∂ z2 ●Because we are calculating demagnetization energy we need only magnetic field within the cylinder. The solutions to differential equation in the Laplacian of U are*: (ρ / R)2n if ρ ≤R u (ρ)=c ×( ) n n − (ρ / R) 2n if ρ ≥R *negative powers are introduced to have vanishing potential at infinity; n constant R is introduced to comply with magnetostatic problem - ρ is a solution too. Urbaniak Magnetization reversal in thin films and... Origin of magnetic domains ●The boundary condition, at cylinder surface, for magnetic scalar potential U is [2]: ∂ ∂ ∞ ( ϕ) ( − ) = =8 ∑ nsin 2n 1 + 1 = 4n ∂ ρ U inside ∂ ρ U outside M n π M S ( + )( − ) 2n −1 2n+1 (2n+1)(2n−1) ρ =Radius n=0 2n 1 2n 1 ●We seek a solution of the form (guessing): ∞ =∑ (ρ ) ( ϕ ) U un sin 2n n=1 ●Since magnetization is constant throughout the domains we have ∇ 2 U = 0 and: ∞ ∂2 1 ∂ 4n2 Laplacian in cylindrical coordinates: ∇ 2U =∑ sin(2nϕ)( +ρ − )u (ρ ) ∂ ρ 2 ∂ ρ ρ 2 n ∂ ∂ ∂2 ∂2 n=1 ∇ 2 =1 + 1 + f ∂ (r ∂ f ) f f r r r r2 ∂ϕ 2 ∂ z2 ●Because we are calculating demagnetization energy we need only magnetic field within the cylinder. The solutions to differential equation in the Laplacian of U are*: (ρ / R)2n if ρ ≤R u (ρ)=c ×( ) n n − (ρ / R) 2n if ρ ≥R *negative powers are introduced to have vanishing potential at infinity; n constant R is introduced to comply with magnetostatic problem - ρ is a solution too. Urbaniak Magnetization reversal in thin films and... Origin of magnetic domains ● After substitution of Uinside and Uoutside we have: ∞ n ( ϕ) 2M R ∑ (2n +2n ) ( ϕ )=8 ∑ nsin 2n ⇒ = S cn sin 2n π M S ( + )( − ) cn π ( + )( − ) n=1 R R n=1 2n 1 2n 1 2n 1 2n 1 ●And the potential inside the cylinder is: ∞ ∞ ( ϕ) ρ 2n =2 ∑ sin 2n ( ) =∑ (ρ ) ( ϕ ) U inside π R M S ( + )( − ) U un sin 2n n=1 2n 1 2n 1 R n=1 ●The field inside the cylinder is obtained from gradient of U [2]: ∂ ∂ sinϕ ∂ H inside=− U =−(cosϕ − )U x ∂ x inside ∂ ρ ρ ∂ϕ inside We need only Hx since: ● = = The demagnetizing energy is given by: M y M z 0 =−1 ∫ ⃗ ⋅⃗ =−1 μ ∫ inside E demag M B dV 0 M x H x dV 2 ρ ≤R 2 ρ ≤R ●We leave out the integration [see 2] and have for the demagnetizing energy per unit length of the cylinder: = 1 μ 2 2 E demag π 0 R M s Urbaniak Magnetization reversal in thin films and... Origin of magnetic domains π one domain 2 2 1 E =μ R M Etwo domains=μ π R2 M 2 demag 0 4 S demag 0 s ●The demagnetization energy of the cylinder with one domain is higher that that with two domains: Eone domain π 2 demag = >1 ≈2.5 two domains 4 E demag Magnetostatic interactions favor ●The result does not depend on the radius or the the subdivision of the crystal into saturation magnetization of cylinder. magnetic domains ●It may be surmised that for any ferromagnetic material the magnetostatic energy may be reduced by subdividing the crystal into at least two domains. Urbaniak Magnetization reversal in thin films and... Potential energy of magnetic charge - digression In many cases instead of calculating energy of a magnetic body in the external field from the formula: =−∫ ⃗ ⋅⃗ =−∫ ⃗⋅⃗ E magn M B dV J H dV ⃗ one can use magnetic charge method and the magnetic scalar potential (lecture 2): H =−∇ ϕ ⃗ ⃗ ⃗ ∂ ∂ ∂ E =−μ ∫ M⋅H dV =μ ∫ M⋅∇ ϕ dV =μ ∫(x̂ M + ŷ M +ẑ M )⋅(x̂ ϕ+ ŷ ϕ+ẑ ϕ )dV = magn 0 0 0 x y z ∂ x ∂ y ∂ z ∂ ∂ ∂ μ ∫(M ϕ +M ϕ+M ϕ )dV = 0 x ∂ x y ∂ y z ∂ z ∂ ∂ ∂ ∂ ∂ ∂ μ ∫[( M ϕ−ϕ M )+( M ϕ−ϕ M )+( M ϕ−ϕ M )] dV = 0 ∂ x x ∂ x x ∂ y y ∂ y y ∂ z z ∂ z z ∂ ∂ ∂ ∂ ∂ ∂ μ ∫( M ϕ + M ϕ+ M ϕ )dV −μ ∫(ϕ M +ϕ M +ϕ M )dV = 0 ∂ x x ∂ y y ∂ z z 0 ∂ x x ∂ y y ∂ z z μ ∫ ∇⋅(ϕ ⃗ ) −μ ∫ϕ ∇⋅⃗ =μ ∫ ϕ ⃗ +μ ∫ϕ ρ 0 M dV 0 M dV 0 M dS 0 magn dV surface Magnetic charge (L.2): ⃗ from Gauss divergence theorem ρ =−∇⋅ magn M =μ ∫ ϕ ⃗ +μ ∫ϕ ρ E magn 0 M dS 0 magn dV surface A.
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