Magnetic Properties of Solids Materials may be classified by their response to externally applied magnetic fields as diamagnetic, g paramagnetic, p g or ferromagnetic. g These magnetic g responses p differ greatly in strength. Diamagnetism is a property of all materials and opposes applied magnetic fields, but is very weak. Paramagnetism, when present, is stronger than diamagnetism and produces magnetization in the direction of the applied field, and proportional to the applied field. Ferromagnetic effects are very large; producing magnetizations sometimes orders of magnitude greater than the applied field and as such are much larger than either diamagnetic or paramagnetic effects. The magnetization of a material is expressed in terms of density of net magnetic dipole moments m in the material. We define a vector quantity called the magnetization ti ti M by b M = μtotal/V. /V Then Th th the total t t l magnetic ti field fi ld B in i th the material t i l iis given by B = B0 + μ0M where μ0 is the magnetic permeability of space and B0 is the externally applied magnetic field. When magnetic fields inside of materials are calculated using Ampere Ampere's s law or the Biot-Savart Biot Savart law, law then the μ0 in those equations is typically replaced by just μ with the definition μ = Kmμ0 where Km is called the relative permeability. If the material does not respond to the external magnetic field by producing any magnetization magnetization, then Km = 1. 1

Magnetic Properties of Solids Another commonly used magnetic quantity is the magnetic susceptibility which specifies how much the relative permeability differs from one. Magnetic susceptibility χm = Km – 1 For paramagnetic and diamagnetic materials the relative permeability is very close to 1 and the magnetic susceptibility very close to zero. For ferromagnetic materials, these quantities may be very large. Another way to deal with the magnetic fields which arise from magnetization of materials is to introduce a quantity called magnetic field strength H . It can be defined by the relationship H = B0/μ0 = B/μ0 - M and has the value of unambiguously designating the driving magnetic influence from external currents in a material, independent of the material's magnetic response. The relationship for B above can be written in the equivalent form B = μ0(H + M) H and M will have the same units, amperes/meter. Ferromagnetic materials will undergo a small mechanical change when magnetic fields are applied, either expanding or contracting slightly. This effect is called magnetostriction.

Diamagnetism The orbital motion of electrons creates tiny atomic current loops, which produce p oduce magnetic ag et c fields. e ds When e an a e external te a magnetic ag et c field e d is s app applied ed to a material, these current loops will tend to align in such a way as to oppose the applied field. This may be viewed as an atomic version of Lenz's law: induced magnetic fields tend to oppose the change which created them. Materials in which this effect is the only magnetic response are called diamagnetic. All materials are inherently diamagnetic, but if the atoms have some net magnetic moment as in paramagnetic materials, or if there is long-range ordering of atomic magnetic moments as in ferromagnetic materials, these stronger effects are always dominant. Diamagnetism is the residual magnetic behavior when materials are neither paramagnetic nor ferromagnetic. Any conductor will show a strong diamagnetic effect in the presence of changing magnetic fields because circulating currents will be generated in the conductor to oppose the magnetic field changes. A superconductor will be a perfect diamagnet since there is no resistance to the forming of the current loops loops.

Paramagnetism Some materials exhibit a magnetization g which is p proportional p to the applied pp magnetic field in which the material is placed. These materials are said to be paramagnetic and follow Curie's law:

⎛B⎞ M = C⎜ ⎟ ⎝T ⎠

; M = Magnetization B = Magnetic field C = Curie Curie' s Constant T = Temperature in Kelvins

All atoms have inherent sources of magnetism because electron spin contributes a magnetic moment and electron orbits act as current loops which produce a magnetic ti field. fi ld IIn mostt materials t i l th the magnetic ti moments t off the th electrons l t cancel, l b butt in materials which are classified as paramagnetic, the cancellation is incomplete.

Magnetostriction

Magnetostriction It is also observed that applied mechanical strain produces some magnetic anisotropy. If an iron crystal is placed under tensile stress, then the direction of the stress becomes the preferred magnetic direction and the domains will tend to line up in that direction. Ordinarily the direction of magnetization in iron is easily changed by rotating the applied magnetic field, but if there is tensile stress in the iron sample, there is some resistance to that rotation of direction. Bulk solid samples may have internal strains which influence the domain boundary movement. M Magnetostriction t t i ti can be b used d to t create t vibrators, ib t where h usually ll some llever action is used in conjunction with the magnetic deformation to increase the resultant amplitude of vibration. Magnetostriction is also used to produce ultrasonic lt i vibrations ib ti either ith as a sound d source or as ultrasonic lt i waves iin liquids which can act as a cleaning mechanism in ultrasonic cleaning devices.

Hysteresis y Curves

Properties of Permalloy thin films Ms=10/4π kG Hc=0.3 Oe Hk= 5 Oe Applications: computer memory, magnetoresistance, detector, reading Heads

Magnetic g Energies g • Exchange energy

alignment of spins, cost of energy to change direction of magnetization compensated by thermal energy ⇒ phase transition at Tc • Magnetostatic energy discontinuity of normal component across interface ⇒ demagnetizing factor f(sample shape) • Magnetocrystalline anisotropy preference of magnetization along crystallographic directions • Magnetoelastic energy change h off magnetization ti ti d due tto strain t i (magnetostriction) •Zeeman energy potential t ti l energy off magnetic ti momentt in i a field

σ exchange

2

JS ≈ 2 Na

σ anisotropy ≈ KNa i t Quantities: J = exchange integral S = spin a = atomic distance N = number of spins K = anisotropy constant

Stoner-Wohlfarth model Free energy gy in magnetic g anisotropy py

E = K1 sin ϕ 0 2

K1 = uniaxial anisotropy φ0= angle l b between t M and d easy axis i EA easy axis for energy minima HA hard axis for energy maxima

Single g Domain particles p Ferromagnetic particles sufficiently small z

z

x EA

Condition 1

x EA

h

HA M

φ0

β

EA

HA β φ0

EA

Applying external field H; E = K1 sin 2 ϕ 0 − HM cos( β − ϕ 0 ) define new parameters K K H H cos β H sin β E 2 K1 ε≡ ,h ≡ , h|| = , h⊥ = , HK ≡ MH K HK HK HK M H K = anisotropy field

ε = 12 sin 2 ϕ 0 − h|| cos ϕ 0 − h⊥ sin ϕ 0 extrema ⇒

∂ε =0 ∂ϕ 0

net torque; Λ 0 on M = 0 ∂ε = 12 sin 2ϕ 0 + h|| sin ϕ 0 − h⊥ cos ϕ 0 = 0 Λ0 = ∂ϕ 0

Condition 1

∂ 2ε ϕ 0 = answer ((stable)) onlyy if >0 ∂ϕ 02 ∂ 2ε ∂Λ 0 = = cos 2ϕ 0 + h|| cos ϕ 0 + h⊥ sin ϕ 0 = Λ1 ∂ϕ 02 ∂ϕ 0 Λ1 = uniform effective condition 1 : H || EA and condition 2 : H ⊥ EA ⇒ no hyteresis

Condition 2

Various magnetic anisotropy energies Shape anisotropy energy

a measure of the difference in the energies associated with magnetization in the shortest and longest di dimensions i off a fferromagnetic ti b body d

Magnetocrystalline anisotropy St i m g tost i tio aniostropy Strain-magnetostriction iost op M-induced uniaxial anisotropy Oblique bl incident d anisotropy

Magnetostatic Energy

Large MS energy

Smaller MS energy

Smaller MS higher wall energy

No MS energy

Closure domains: in magnetic hard directions problem: magnetostriction!

Domain wall Energy

⎛ kTc K1 ⎞ γ = 4 ( AK1 ) = 4 ⎜ ⎟ ⎝ a ⎠ a=lattice spacing, Tc=curie temperature, k = Boltzmann constant Domain wall width

A δ =π ( ) =π K1

⎛ kTc ⎞ ⎟⎟ ⎜⎜ ⎝ aK1 ⎠

Domain Wall Energy gy Energetic considerations: domain wall costs wall energy, energy but reduces magnetostatic energy More Domains = smaller spacing d Magnetostatic energy density Domain wall energy density

↑

↓

Thin films are frequently single domain, magnetization in-plane

Domain Wall Energy gy

Intrinsic magnetic properties (approximate values) of a typical hard magnetic materials (SmCo5) and a typical soft magnetic material (Fe)

Ref: R.A. McCurrie Ferromagnetic materials : structure and properties, Academic Press, 1994,Table 1.3

Domain Wall Fij = 2 S 2 J cos(θ ij ) exchange energy JS 2 σ exchange ≈ Na 2 Anisotropy energy

σ anisotropy ≈ KNa Quantities: J = exchange integral S = spin a = atomic distance N = number of spins K = anisotropy constant

Types yp of Domain Walls

Bloch and Néel Walls

in-plane

out of plane

Cross-Tie Wall

Magnetocrystalline Anisotropy

Magnetic g Films / Size z Effects single domain particles: first approximation: particle size ~ domain wall size → no domain walls walls, single domain particles more detialed: include magneto static gy energy typical rc = 3nm (Fe)

Size z effects Superparamagnetism: small particles: magnetic direction is not fixed by anisotropy or shape thermal energy might change / flip magnetic moment rsp = 20 nm each particle ferromagnet, but particles disordered => behavior like paramagnet, but higher permeability high Ms, but no Hc

Size z Effects: Summary y

Magnetism in Thin Films/Small structures

In/Out of plane magnetization

Stress and Magnetization g z I

Stress and Magnetization II

Exchange g Energy gy Coupling p g

Giant Magnetoresistance g

GMR (Fe/Cr Multilayer) y

GMR: Theory and explanation

Equivalent circuit

Spin p Valve

Link to animation

Application: pp Data storage g

Requirements q

Recording g medium

Criteria for magnetic properties

Noise

Longitudinal vs Perpendicular recording

Particulate Recording media Single domain particles Acicular particles due to shape anisotropy embedded in polymer matrix alignments by suitable deposition process or baking in magnetic ti field fi ld not for high density media (Bit length > 1μm) total magnetization reduced due to binder applications, pp , tape, p , floppy ppy disk materials: CrO2, γ-Fe2O3 Co doped γ-Fe2O3 to improve coercitivity, either alloyed or surface layer pure iron + oxidation/corrosion protection

Characteristics of Particulate media

Thin film recording media

Inductive Recording Media

Various materials for inductive recording heads

Write Head

Signal Strength in read head H x ∝ e − kx sin( kx) d

d

V = ∞ V

∫ exp(−ky)dy 0 ∞

∫ exp(−ky)dy 0

D 90 ≈ 0.37λ

= 1 − e − kd = 0.9

Magnetoresistive g head

Magnetoresistive g head Magnetoresitance = magnetic field changes electrical resistivity different mechanisms possible!

Anisotropic p magnetic g resistance ((AMR)) 2-3% % magnetic field causes oscillation of conduction electrons Giant magneto resistance (GMR) up to 100 % Magnetic field changes alignement of antiferromagnetic layers Collossal magneto resistance (CMR) several 100 % Magnetic field induces phase transition ceramic insulator® metal

GMR and spinvalves p

Summary: y requirements q read head: •Low coercivityy •low noise •high g p permeability y •low magnetostriciton •small •hard surface •inductive or magnetoresistance

write head: high Ms to magnetize recording medium up to 1 T adequate permeability at high frequency Recording medium: •should respond p to field of write head and retain information •coercivity 500 - 3000 Oe •suitable remanent magnetization •small, single domain particles (103 for a bit)