Olivier FRUCHART Univ. Grenoble Alpes / CEA / CNRS, SPINTEC, France - - PowerPoint PPT Presentation

Olivier FRUCHART

• Univ. Grenoble Alpes / CEA / CNRS, SPINTEC, France

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Dear Institute, I've always had a fascination with electromagnetism, and have pondered the theories of gravity. One thing I've come across in preliminary research is that the current theories largely fail to include human element in, as if we're just baseless objects trapped here without a role in the ultimate reason. (...) Humans are magnets, too, as we possess iron. (…) If you take two magnets, they stick together when proper polars are placed near each other. What causes humans to act as the 2nd magnet in gravity is the iron found in humans. Earth, obviously the big magnet with the most iron, is able to control humans, the far smaller magnet with less iron. (…) Ultimately there is one controlling magnet for the entire universe somewhere in space holding it all together, like Galileo said. Calculations of Earth's maximum gravitation pull could be made by testing individual boosters on humans and converting the thrust needed into some kind of formula which returns Earth's magnetic energical pull. (…) While it doesn't conclude why other things on Earth are in the same situation as us, it is also based on magnetism and humans have to have their own role in the matter. Further research into it needs to be done as these are very preliminary original thoughts. Regards, XXX YYY.

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

What is a quantity? What is a unit ?

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Quantity

Example: speed Dimension:

𝐰 = 𝜀ℓ/𝜀𝑢 dim(𝐰) = L ∙ T−1 Units

Why? Provide a measure Universality: share with others Possible formalism:

𝑌 = 𝑌𝛽 𝑌 𝛽 𝑀 = 50 𝑀 SI = 5000 𝑀 cgs 𝑀 SI = meter = 100 𝑀 cgs SPEED

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Facts: interaction between charges Modeling by the Physicist

Electric field Charges are scalar sources of electric field

+𝑟1 𝐆1→2 = 𝑟1𝑟2 4𝜌𝜗0𝑠

12 2 𝐯12

+𝑟2 +𝑟1 − 𝑟2 𝐅1→2 𝐅(𝐬) = 𝑟 4𝜌𝜗0𝑠2 𝐯 𝐆1→2 = q2𝐅1→2

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Microscopic level: Maxwell equation Macroscopic level: Gauss theroem

Ostogradski theorem

׮

𝒲 𝛂 ⋅ 𝐅 d𝒲 =װ 𝜖𝒲 𝐅 ⋅ 𝐨 d𝒯

𝛂 ⋅ 𝐅 = 𝜍 𝜗0 𝜍 = 𝜀𝑅 𝜀𝒲 Volume density

• f electric charge

𝑅 𝜗0 = ׮ 𝒲 𝜍 𝜗0 d𝒲 =װ 𝜖𝒲 𝐅 ⋅ 𝐨 d𝒯

𝛂 ⋅ 𝐅 = 𝜖𝐹𝑦 𝜖𝑦 + … = 𝐹𝑦 𝑦 + 𝜀𝑦 − 𝐹𝑦 𝑦 𝜀𝑦 + ⋯ Link

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Century-old facts Œrsted experiment in 1820

Magnetic materials (rocks)

Bir Birth of

• f

ele lect ctromagneti tism

Magnetic field of the earth

Magnetite Light-struck

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Facts: interaction between charge currents Modeling by the Physicist

Magnetic induction field: Biot & Savart law

+𝐽1 𝜀𝐆1→2 = 𝜈0 𝐽1𝐽2 𝜀ℓ2 × (𝜀ℓ1 × 𝐯12) 4𝜌𝑠

12 2

−𝐽2 Note: former definition of the Ampère: The force between two infinitely wires 1m apart with current 1A is 2 × 10−7 N/m 𝜀𝐂 = 𝜈0𝐽𝜀ℓ × 𝐯 4𝜌𝑠2 +𝐽

Retrieve the force (Laplace)

𝜀𝐆2 = 𝐽2𝜀ℓ × 𝐂(𝐬2) 𝐆 = 𝑟 𝐰 × 𝐂 Magnetic in induction fie ield ld defi fined ed through Lo Lorentz Force

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Microscopic level: Maxwell equation Macroscopic level: Ampere theorem

Stokes theorem

𝛂 × 𝐂 = 𝜈0𝐤 j: Volume density of current (A/m2) J J is the vectorial source of curl of B ׭

𝒯

𝛂 × 𝐂 ⋅ 𝐨 d𝒯 = ׯ

𝝐𝒯 𝐂 ⋅ dℓ

𝐽 = 𝜈0 ׭

𝒯 (𝐤 ⋅ 𝐨) d𝒯 = ׯ 𝝐𝒯 𝐂 ⋅ dℓ

𝛂 × 𝐂 = … … 𝜖𝐶𝑧 𝜖𝑦 − 𝜖𝐶𝑦 𝜖𝑧 = … … 𝐶𝑧 𝑦 + 𝜀𝑦 − 𝐶𝑧 𝑦 𝜀𝑦 − 𝐶𝑦 𝑧 + 𝜀𝑧 − 𝐶𝑦(𝑧) 𝜖𝑧 Link 𝐕𝐨𝐣𝐮 𝐠𝐩𝐬 𝐂: tesla (T)

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

𝛂 × 𝐂 = 𝜈0 𝐤 + 𝜗0 𝜖𝐅 𝜖𝑢 𝛂 ⋅ 𝐅 = 𝜍 𝜗0 𝛂 × 𝐅 = − 𝜖𝐂 𝜖𝑢 𝛂 ⋅ 𝐂 = 0

Gauss theorem Ampère theorem Faraday law of induction B is divergence free (no magnetic poles)

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Ampere theorem and Œrsted field 𝐶𝜄 = 𝜈0𝐽 2𝜌𝑠 𝐽 Not

• te: 1/r dec

ecay The magnetic point dipole 𝐂 = 𝜈0 4𝜌𝑠3 3 𝑠2 𝛎 ⋅ 𝐬 𝐬 − 𝛎 𝛎 = 𝐽𝒯 𝐨 𝐂 = 𝜈0 4𝜌𝑠3 2𝜈 cos 𝜄 𝐯𝑠 + 𝜈 sin 𝜄 𝐯𝜄 𝜀𝐂 = 𝜈0𝐽𝜀ℓ × 𝐯 4𝜌𝑠2 Biot and Savart Not Note: 1/r2 dec ecay Not

• te: 1/r3 dec

ecay 𝐕𝐨𝐣𝐮: A ⋅ m2 𝛎 = 1 2 ම

𝒲

𝐬 × 𝐤 𝐬 d𝒲

General definition Simple loop

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Torque 𝚫 = ර 𝐬 × 𝐽 dℓ × 𝐂 = 𝛎 × 𝐂 Energy

Zeeman energy

(𝐾)

Demonstration Work to compensate Lenz law during rise of B Integrate torque from Laplace force while flipping dipole in B

Force 𝐆 = 𝛎 ⋅ (𝛂𝐂)

Valid only for fixed dipole No force in uniform magnetic induction field Inducing precession of dipole around the field It is energy-conservative, as expected from Laplace (Lorentz) force

ℰ = −𝛎 ⋅ 𝐂

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Energy ℰ = − 𝜈0 4𝜌𝑠3 3 𝑠2 𝛎1 ⋅ 𝐬 𝛎2 ⋅ 𝐬 − 𝛎𝟐 ⋅ 𝛎2 ℰ = + 𝜈0𝜈1𝜈2 4𝜌𝑠3 ℰ = − 𝜈0𝜈1𝜈2 4𝜌𝑠3 ℰ = +2 𝜈0𝜈1𝜈2 4𝜌𝑠3 ℰ = 0 ℰ = −2 𝜈0𝜈1𝜈2 4𝜌𝑠3 Examples

The dipole-dipole interaction is anisotropic

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Definition 𝐍 = 𝜀𝛎 𝜀𝒲

Volume density of magnetic point dipoles

A/m

Total magnetic moment of a body

𝓝 = න

𝒲

𝐍 d𝒲 A ⋅ m2 Appli lies es to:

• : fer

erromagnets, paramagnets, dia iamagnets etc. c. Must be be defin ined ed at at a len ength scale much la larger th than atom

• ms

Is Is the basis for th the micromagnetic th theo eory Equivalence with surface currents

Name: Amperian description of magnetism Surface current equals magnetization A/m

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Back to Maxwell equations

Disregard fast time dependence: magnetostatics

The magnetic field H A/m 𝛂 × 𝐂 = 𝜈0 𝐤 + 𝜗0 𝜖𝐅 𝜖𝑢

Consider separately real charge current, from fictitious currents of magnetic dipoles

𝛂 × 𝐂 = 𝜈0 𝐤c + 𝐤m

𝐤c 𝐤m

𝛂 × 𝐍 = 𝐤m

One can show:

𝐍 × 𝐨 = 𝐤m,s 𝛂 × 𝐈 = 𝐤c

By definition:

𝐈 = 𝐂 𝜈0 − 𝐍 𝛂 × 𝐂 𝜈0 − 𝐍 = 𝐤c

One has:

B versus H : definition of the system

M: local (infinitesimal) part in of the system defined when considering a magnetic material H: The remaining of B coming from outside , liable to interact with the system

𝜀𝒲 𝜀𝒲 A/m A/m2 Outs tsid ide e matter, and c coin

• incid

ide and have e exactly th the e same mea eaning.

𝐂

𝜈0𝐈

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

The dipolar field Hd

By definition: the contribution to H not related to free currents (possible to split as Maxwell equations are linear)

𝛂 × 𝐈d = 0 𝐈d = −𝛼𝜚d Derive the dipolar field 𝛂 ⋅ 𝐈d = −𝛂 ⋅ 𝐍 𝐈d 𝐬 = −𝑁s ම

𝒲′

𝛂 ⋅ 𝐧 𝐬′ (𝐬 − 𝐬′) 4𝜌 𝐬 − 𝐬′ 3 d𝒲′ 𝐈d 𝐬 = ම 𝜍 𝐬′ 𝐬 − 𝐬′ 4𝜌 𝐬 − 𝐬′ 3 d𝒲′ + ඾ 𝜏 𝐬′ 𝐬 − 𝐬′ 4𝜌 𝐬 − 𝐬′ 3 d𝒯′

Maxwell equation → To lift the singularity that may arise at boundaries, a volume integration around the boundaries yields:

Analogy with electrostatics 𝛂 × 𝐅 = 0 𝐅 = −𝛼𝜚 𝛂 ⋅ 𝐂 = 𝟏 𝐈 = 𝐈d + 𝐈𝐛𝐪𝐪 𝜍(r)= − 𝑁s 𝛂 ⋅ 𝐧(𝐬) 𝜏(r)=𝑁s 𝐧 𝐬 ⋅ 𝐨(𝐬)

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Vocabulary

Generic names Magnetostatic field Dipolar field Inside material Demagnetizing field Oustide material Stray field

𝐈d 𝐬 = ඾ 𝜏 𝐬′ 𝐬 − 𝐬′ 4𝜌 𝐬 − 𝐬′ 3 d𝒯′ Example

Permanent magnet (uniformly-magnetized) Surface charges Dipolar field

𝜏(r)=𝑁s𝐧 𝐬 ⋅ 𝐨(𝐬)

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

From: M. Coey’s book

Pseudo-charges source of Hd Fictitious currents source of B

Coulombian Amperian 𝛂 ⋅ 𝐂 = 𝟏

No magnetic monopole

𝛂 × 𝐈 = 𝟏

No closed lines

Δ𝐈 ⋅ 𝐨 = 𝜏 Δ𝐼∥ = 0 Δ𝐶⊥ = 0 Δ𝐂 = 𝜈0𝐤 × 𝐨

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Examples of magnetic charges

Note for infinite cylinder: no charge ℰd = 0 Charges on side surfaces Surface and volume charges

Dipolar energy favors alignment of magnetization with longest direction of sample Take-away message

The long cylinder

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Dipolar energy ℰd = − 1 2 𝜈0 ම

𝒲

𝐍 ⋅ 𝐈d d𝒲 ℰd = 1 2 𝜈0 ම

𝒲

𝐈d

𝟑 d𝒲

Zeeman energy of microscopic volume

𝜀ℰZ = −𝜈0𝐍𝜀𝒲 ⋅ 𝐈ext

Elementary volume

• f

a macroscopic system creating its own dipolar field

𝐹d = 𝜀ℰd/𝜀𝒲 = − 1 2 𝜈0𝐍 ⋅ 𝐈d

Total energy of macroscopic body

Alw lways pos

• sit

itiv

• ive. Zer

ero mea eans min inimum 𝐈d 𝐬 = Volume + ඾ 𝜏 𝐬′ 𝐬 − 𝐬′ 4𝜌 𝐬 − 𝐬′ 3 d𝒯′ Size considerations

Unchanged if all lengths are scaled: homothetic. Check that the following is a solid angle:

d𝛁 = 𝐬 − 𝐬′ d𝒯′ 𝐬 − 𝐬′ 3 Hd doe

• es not
• t dep

epen end on

• n th

the siz ize of

• f th

the bod

• dy

Neit either doe

• es th

the volu

• lume den

ensity ity of

• f en

ener ergy Said id to to be be a lon long-range in inter eraction

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Range

Upper bound of dipolar field

𝐈d(𝐬) ≤ 𝑁s𝑢 න 2𝜌𝑠 𝑠3 dr Non-homogeneity 𝐈d(𝐒) ≤ Cste + 𝒫(1/𝑆)

Example: flat strip with aspect ratio 0.0125 Average (de demag coe

• efficient)

True profile

Lateral position (a.u.) Dip Dipola lar field fields are short-ranged in in lo low dim imensionali lity Dip Dipola lar field fields are high ighly non

• n-homogeneous in

in lar large aspec ect rati tio systems Con Conse sequences es: non

• n-uniform magn

gneti tization swit itching, exci citation modes etc.

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Dipolar energy for uniform magnetization 𝐈d 𝐬 = ඾

𝜖𝒲

𝐍 𝐬′ ⋅ 𝐨 𝐬′ 𝐬 − 𝐬′ 4𝜌 𝐬 − 𝐬′ 3 d𝒯′ = 𝑁s𝑛𝑗 ඾

𝜖𝒲

n𝑗 𝐬′ 𝐬 − 𝐬′ 4𝜌 𝐬 − 𝐬′ 3 d𝒯′ 𝜍(r)= − 𝑁s 𝛂 ⋅ 𝐧 𝐬 = 0

No volume charges:

ℰd = − 1 2 𝜈0 ම

𝒲

𝐍 𝐬 ⋅ 𝐈d 𝐬 d𝒲 = − 1 2 𝜈0𝑁s

2𝑛𝑗 ම 𝒲

d𝒲 ඾

𝜖𝒲

𝑜𝑗 𝐬′ 𝐧 ⋅ 𝐬 − 𝐬′ 4𝜌 𝐬 − 𝐬′ 3 d𝒯′ 𝐍 𝐬 = 𝐍 = 𝑁s(𝑛𝑦ො 𝐲 + 𝑛𝑧 ො 𝐳 + 𝑛𝑨ො 𝐴) ℰd = −𝐿d𝑛𝑗𝑛𝑘 ම

𝒲

d𝒲 ඾

𝜖𝒲

𝑜𝑗 𝐬′ 𝑠

𝑘 − rj ′

4𝜌 𝐬 − 𝐬′ 3 d𝒯′

Dipolar field: Dipolar energy:

ℰd = 𝐿d𝑊 𝐧 ⋅ ന 𝐎 ⋅ 𝐧 𝐈𝐞(𝐬) = −𝑁s ന 𝐎 ⋅ 𝐧

See more detailed approach: M. Beleggia et al., JMMM 263, L1-9 (2003)

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

For any shape of body ℰd = 𝐿d𝑊 𝐧 ⋅ ന 𝐎 ⋅ 𝐧 𝐈𝐞(𝐬) = −𝑁s ന 𝐎 ⋅ 𝐧 For ellipsoids etc.

Condition: boundary is a polynomial of the coordinates, with degree at most two Slabs (thin films), cylinders, ellipsoids demagnetizing tensor. Always positive, and can be diagonalized. ന 𝐎

𝑂𝑦 + 𝑂𝑧 + 𝑂𝑨 = 1

Along main directions

𝐼d,𝑗(𝐬) = −𝑂𝑗𝑁s ℰd = 𝐿d𝑊 𝑂𝑦𝑛𝑦

2 + 𝑂𝑧𝑛𝑧 2 + 𝑂𝑨𝑛𝑨 2

Dipolar anisotropy is always of second order Hypothesis uniform M may be too strong Remember: dipolar field is NOT uniform

ℰd = 𝐿d𝑊 𝐧 ⋅ ന 𝐎 ⋅ 𝐧 𝐈𝐞 = −𝑁s ന 𝐎 ⋅ 𝐧

Along main directions

𝐼d,𝑗 = −𝑂𝑗𝑁s

M and H may not be colinear along non- main directions

𝑨2 = 𝑢 2

2

𝑦 𝑏

2

+ 𝑧 𝑐

2

= 1 𝑦 𝑏

2

+ 𝑧 𝑐

2

+ 𝑨 𝑑

2

= 1

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Reminder about plane symmetry Magnetic fields are pseudo-vectors

Points Vectors Example: electric field 𝛂 ⋅ 𝐅 = 𝜍

𝜗0 𝝉 𝐅 𝜀𝐂 = 𝜈0𝐽𝜀ℓ × 𝐯12 4𝜌𝑠2

Curl is a chiral operator

𝐉 𝐉 B is antisymmetric What use? Example: Ampere theorem and Œrsted field 𝐉

Symmetry of I with plane containing I Antisymmetry of B: is azimuthalℓ

𝐶𝜄 = 𝜈0𝐽 2𝜌𝑠

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Time inversion symmetry of Maxwell equations Example 𝐉 −𝐉 𝛂 × 𝐂 = 𝜈0 𝐤 + 𝜗0 𝜖𝐅 𝜖𝑢 𝛂 ⋅ 𝐅 = 𝜍 𝜗0 𝛂 × 𝐅 = − 𝜖𝐂 𝜖𝑢 𝛂 ⋅ 𝐂 = 0

What happens with operation 𝑢 → −𝑢

Maxw xwell eq equati tions rem emains vali lid Soluti tions must comply with th time- reversal symmetry What use? Magneto-crystalline anisotropy 𝐹 𝜄 = 𝐿10 cos 𝜄 + 𝐿01 sin 𝜄 + 𝐿11 cos 𝜄 sin 𝜄 + 𝐿20 cos2 𝜄 + 𝐿02 sin2 𝜄 + 𝐿30 cos3 𝜄 + 𝐿03 sin3 𝜄 + 𝐿21cos2𝜄 sin 𝜄 + 𝐿12 cos 𝜄 sin2 𝜄 + ⋯ Odd ter erms are for

• rbid

idden

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Problems with cgs-Gauss

The quantity for charge current is missing No check for homogeneity Mix of units in spintronics Inconsistent definition of H Dimensionless quantities are effected: demag factors, susceptibility etc.

More in in th the practical on

• n units

its

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Define quantities Fixed values

Times Length Mass Electric charge Speed of light -> Define meter Planck constant -> Defines kg Charge of the electron

To be measured

Magnetic permeability of vacuum

𝜈0 ≠ 4𝜌 × 10−7 S. I. 𝜈0 = 4𝜌 1 + 2.0 2.3 ⋅ 10−10 × 10−7 S. I.

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