2022.4.6 Lecture 1

Lecture on

10:25 – 11:55

Magnetic Properties of Materials
磁性 (Magnetism)
Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Syllabus
1. Phenomenology of magnetism. Magnetization process
2. Spin magnetic moments in solids
3. Mutual interaction between spins
4. Ordered states of spins. Phase transitions
5. Magnetism in insulators
6. Magnetism of itinerant electrons
7. Some advanced topics (?)
2

How the lecture will go on?
➢ The lecture notes (in Japanese, English) will be uploaded in the site
https://kats.issp.u-tokyo.ac.jp/kats/magnetism/by the end of the lecture week.
➢ Attendance will be taken. That contributes to the achievement.
➢ Small amount of problems for your exercise at home will be given in thelast of the lecture in every two weeks. Submission deadline of thesolutions is two weeks later. In order to submit your answer, you need toregister yourself from the web page that will be prepared by the next week.
➢ In the very last of the lecture in July, the problems for your report will begiven. The deadline for the submission of the report will be notified then.
3

Contents Wed. 6th April 2022

Chapter 1 Basic Notions of Magnetism

1. Electromagnetic fields in the vacuum, andthose with materials
2. Experimental methods to measuremagnetization
3. Magnetism in classical pictures

4. Spins of electrons and their magnetic moment
4Electromagnetic fields in the vacuum
Maxwell equations for electromagnetic
fields in the vacuum
Electromagnetic induction
No magnetic monopole
Electric current can create magnetic field
𝑬: Electric field, 𝑩: Magnetic flux density (Magnetic field)

Problem of “Unit”
No 4𝜋 factor appears in the above Maxwell equations: rationalized system of units
E-B formulation, E-H formulation : Difference in the unit of magnetization!
5

2019 Redefinition of the SI base units
Redefined in 20196

Magnetic dipole: a source of magnetic field
Two ways to introduce magnetic dipole: 1. Introduction of magnetic charge

2. Magnetic dipole as the shrink limit of circularcurrentz+𝑞m𝑙/2

Introduction of magnetic charge
There is no magnetic monopole but still we can consider pairs offictitious magnetic charge with the total charge of zero.

𝒓

Magnetic potential

x
−𝑙/2
−𝑞m

y

Dipole field
Expression in polar
coordinate
7

Magnetic dipole: a source of magnetic field (2)Magnetic dipole as the shrink limit of circular currentz

Vector potential

Magnetic moment (see the next)x

Magnetic field:
A circular current can serve as a magnetic dipole.
8Magnetic moment𝑞m 𝑞 𝐻m𝑙𝜃−𝑞m 𝐻 −𝑞m

Couple of force moment:
Magnetic moment

𝑯

E-B formation𝝁2𝝁1𝒓(a) Stable(b) Unstable

Dipole-dipole interaction: the dipoles feel each other’s fields.
Potential:
9A naïve dipole model of magnetization of materials𝑙−𝑞m𝑞m −𝑞m𝑞m −𝑞m𝑞m −𝑞m𝑞m−𝑞m𝑞m −𝑞m𝑞m −𝑞m𝑞m −𝑞m𝑞m−𝑞m𝑞m −𝑞m𝑞m −𝑞m𝑞m −𝑞m𝑞m−𝑞m𝑞m −𝑞m𝑞m −𝑞m𝑞m −𝑞m𝑞m

Set of small magnets

-

-

+
+
+

+
+
+

Magnetic charges appear at the ends of the material

Density of magnets: N
Magnetization:

Surface density of magnetic charge

10Expression with “equivalent current” in materials
Magnetic moments → vector potential

Partial integration:
Equivalent current:

Introduction of magnetic field:
Maxwell equation with electric flux
density11

M-H curve
M-H relation

𝐻c : Coercive force,𝑀r : Remanent magnetization,𝑀s : Saturation magnetization

Linear: Paramagnetic materials,
diamagnetic materials

Strongly non-linear with hysteresis:
Ferromagnetic materials,
superconductors, …
12

Measurement of magnetization (1)Vibrating sample magnetometer (VSM)

Alternating-gradient magnetometer (AGM)
Piezo detector

vibration

rod

rod

samplesampleCoil for AC field

detectionpole coilpolemagnetmagnet13

Measurement of magnetization (2)Superconducting quantum interference device (SQUID) magnetometer
Superconducting loop
Critical Current Jc (arb.)

-

SQUID

+
+

-1

0

1

2

3

4

LM

5

Magnetic flux ()Wb

14

Effect of demagnetizing field
In the case of infinite plate

Magnetization causes creation ofmagnetic charges on the surface,which produces demagnetizing
field inside.

N : demagnetizing factor(depends only on shape)

-

M

+
+
+
+
+
+
+
+
+
+

Example: Permalloy (Py)
Coercive force: 0.025 Oe
Saturation magnetization field: 3860 Oe
15

Classical treatment of magnetism

Paramagnetic moment
Model: set of molecules with independent magnetic moment 𝜇 in the magnetic field withflux density B along z-axis
Moment magnetic
energy:Average on classicaldistribution:Hightemperatureapproximation:
Curie law:17

Classical paramagnetism−𝑒𝑣

Maxwell equation
Γ𝑟
𝑆

Magnetic flux 0 → 𝐵

18

Breakdown of classical magnetism
Hamiltonian
Symmetric gauge:

Dipole moment:
paramagnetic

diamagnetic

N-electron system
19

Breakdown of classical magnetism (2)Partition function:

Cancellation of paramagnetic
and diamagnetic term
Bohr- van Leeuwen theorem
20Electron spin from Dirac equation

W. Pauli

P. Dirac

R. Peierls

Dirac equation and electron spin magnetic moment
Energy-momentum relation in
Newtonian mechanics
Quantum mechanical replacement
to obtain Schroedinger equation:
Energy-momentum relation in relativity
However simple replacement is impossible: thewave equation must be the first-order in time
How to compromise (2) with (1) ?

(1)
(2)

These conditions require 𝛼𝑘 and 𝛽
to be 4 × 4 matrices.
22Pauli representation
Wave equation
Dirac hamiltonian

Pauli matrices

Pauli representation

23

Spin angular momentum

𝐿 does not commute with Hamiltonian, is thus, not a constant of motion.

4 × 4 Pauli matrices:
Then

Spin angular momentum:
24Magnetic moment of electron spin
Dirac eq. with electromagnetic field

Operation from left:
We obtain

Because25

Magnetic moment of electron spin (2)Stationary solution:

Low energy
expansion

We take first order inBohr magneton

Therefore the magnetic moment is26

Summary

1. Introduction of magnetic moment
2. Measurement of magnetization
3. Magnetism in classical interpretation and its
breakdown
4. Introduction of electron spin along Diracequation

2022.4.13 Lecture 2

Lecture on

10:25 – 11:55

Magnetic Properties of Materials
磁性 (Magnetism)

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review
Chapter1 Basic Notions of Magnetism
Classical pictures of magnetic moments in materials:

 Magnetic charges
 Circular currents

Experimental methods to measure magnetization
Paramagnetic and diamagnetic terms in classical magnetization
Breakdown of classical magnetism: cancellation of paramagnetic and diamagnetic terms(Bohr-van Lewuuen theorem)

Introduction of spin angular momentum by relativistic quantum mechanics
2

Agenda

1. Spin-orbit interaction
2. Magnetism in quantum theory
Chapter 2 Magnetism in localized systems
1. Spherical potential
2. Larmor precession
3. Magnetism of inert gas
4. LS multiplex ground state of open shell ions and Hund’s rule3

Magnetic moment of electron spin
Dirac eq. with electromagnetic field

Operation from left:
We obtain

Because4

Magnetic moment of electron spin (2)Stationary solution:

Low energy
expansion

We take first order inBohr magneton

Therefore the magnetic moment is5

Two-component separation approximation
Stationary Dirac equation
Pauli representation
When the particle sits still:𝜀particle4 × 4 matrices+ corresponds to I, - corresponds to –I in b
→ upper two laws: particle, lower two: anti-particle (?)

𝜀 = 𝑐𝑝Finite momentum p requires correction.

p
Leak to lower laws
anti-particle
6

Spin-orbit interaction
Stationary equation
Two-component approximation

Simultaneous equations

Erase of 𝜑B

Low velocity (𝑝 ≪ 𝑚𝑐) expansion

Normalization condition
7

Spin-orbit interaction (2)Introduction of magnetic field
Correction due to leakage

Corrected two-component
wavefunction
Pauli two-component approximation
ZeemanSpin-orbit interaction8Quantum Mechanical Treatment of Magnetism

Nucleus potential

g-factor

Symmetric gauge
……
……
……
9

Magnetic moment

Commutation relations

Magnetic moment

Paramagnetic

Diamagnetic

This expression does not have drastic changes other than spin magnetic moment.
However …
10

Comment: Spins of nucleonsProtons, Neutrons, Muons have spins.

MRI

J-PARC

KEK

NMR

Neutron diffraction

mSR

11

Chapter 2
Magnetism ofLocalized Electrons

Star birth12Second quantization
Number representation(index the state with number of particles occupying basis states)|0

VacuumCreation operator of j-th state(Hermitian conjugate: annihilation operator)

Fermion:anti-commutation relation

number operator
Boson: commutation relation

13

Operators in second quantization representation
Multiparticle operator
Slater determinant

Second quantization
Particle statistics
Annihilation and creation
operator (anti-)commutation
relations14Electrons in a central force potential
crystal fieldspin-orbit interaction
mutual Coulomb interaction

Localized system

single-electron (non-interaction)
Electrons in a central force (spherical) potential
Eigenfunction in polar coordinate:
Radial wavefunction
Eigen energy

15Larmor precession
B

Coulomb potential
𝜔L

Total orbital angular momentum

L

Directional quantization

Heisenberg equation
Larmor precession

In the case of spin: g-factor
16Magnetism of inert gases

rfStar birth

Total angularmomentum

Magnetic trap

Evaporation cooling17

Magnetism of inert gases
Inert gases: Closed shell structure

due to quantization!

Residual is the dielectric term:

Larmor rotation angular momentum

Moll susceptibility

18

Electronic states in magnetic ions
Open shell electronic states
Angular momentum l orbit
State of many electrons: indexed with L and S : state (L, S) degenerated in the absenceof coulomb term
(L, S) term degenerated (2L+1)(2S+1) : LS multiplex
Which state is the ground state?

20

Dominating terms

Coulomb repulsion in the same orbit

Coulomb repulsion between different orbits
Exchange term

21

Exchange integral
Spin operator
Exchange integral

22Hund’s rule
Hund’s rule

3d transition metal ions

The ground LS multiplex isdetermined by the following
1. It should have maximum S.
2. Under the condition 1., it shouldhave maximum L.

23

Summary
1. Spin-orbit interaction
2. Magnetism in quantum theory
Chapter 2 Magnetism in localized systems
1. Spherical potential
2. Larmor precession
3. Magnetism of inert gas
4. LS multiplex ground state of open shell ions and Hund’s rule

Lecture on

2022.4.20 Lecture 3
10:25 – 11:55

Magnetic Properties of Materials
磁性 (Magnetism)
Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review

1. Spin-orbit interaction
2. Magnetism in quantum theory
Chapter 2 Magnetism in localized systems
1. Spherical potential
2. Larmor precession
3. Magnetism of inert gas
4. LS multiplex ground state of open shell ions and Hund’s rule2

AgendaElectronic states in magnetic ions
LS coupling approachj-j coupling approach
Paramagnetism by magnetic ions in insulators
Curie law
Breakdown of LS coupling approach in 3d transition metals
Ligand field approach
Octahedron potential3Electronic states in magnetic ions (continued)
Periodic table of elements

Sm

4

Electronic states in magnetic ions (continued)
Ex) Lanthanoid:0
Number of 4f electrons:
Number of 4f electrons (3+ion): 0

1

3

4

5

6

7

7

9

10

11

12

13

14

15

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Lifting of degeneracy to LS multiplex
Hund’s rule to find the ground LS multiplex

j-j coupling

LS coupling (Russell-Saunders)LS multiplex

Spin-orbit
interaction

electronaccommodationJ

Spin-orbit
interaction
Coulomb
interaction

J

Coulomb
interaction
5

Spin-orbit splitting of multiplex in single-electron problem
Spin-orbit term in the Pauliapproximation:
Coulomb potential:

then

The expression tells that the SOI is more important for larger Z and orbitals closer to the nucleus.

Lanthanoid: (effect of spin-orbit interaction) > (that of crystal field)
Spin-orbit single electron
hamiltonian:𝒍, 𝒔: not constants of motion

6

Total angular momentum
𝒋 is a constant of motion

Zeeman-like terml, s : Precession around j

Eigenvalue of 𝒍 ∙ 𝒔

7Spin-orbit splitting of multiplex in single-electron problem (2)Energy eigenvalues:
j =5/2l =2
j can take values:
j =3/2spherical

spin-orbit

8

Spin-orbit interaction in the ground state of LS multiplex
Multi-electron hamiltonian:LS-coupling approach𝒔𝑖S

Hund’s ruleS

LS multiplex ground state(2𝐿 + 1)(2𝑆 + 1) degeneracy

L, S are not constant of motion.
: a constant of motion

9Spin-orbit interaction in the ground state of LS multiplex
𝒔𝑖Summation on all 𝑚𝑙 : σ 𝑙𝑖 = 0
SS

Residual part: 𝒔𝑖 and S are inverted𝒔𝑖Ground state10Electron configuration of Lanthanoid ionsSpectroscopic symbolof multi-electron state

2𝑆 + 1: number
𝐿: symbol
𝐽: number

11

Eigenfunction and second quantization representation
Eigenfunction for (𝐽, 𝑀):
Clebsch-Gordan coefficient

Second quantization representation:

Effective Coulomb potential:

12

Clebsch-Gordan calculator on Wolfram alphahttps://www.wolframalpha.com/input/?i=Clebsch-Gordan+calculator

13j-j coupling (short comment)
Pr 3+
Ground state: same as LS coupling

14

Paramagnetism by magnetic ions in insulators
Free local moment and Curie law
Due to the g-factor, the magnetization
is not parallel with the total
momentum, hence the magnetization
is not a constant of motion.

Average gives effective g-factor:

Lande g-factor

Expectation value ofmagnetization:

Partition function:
15

Free moment and Curie law

1

Brillouin function

0.8

0.6

0.4

0.2

0

2

4

6

x

8

10

12
16

Examples of experiments
Gd

7/2
II

Fe

Cr
J=3/2

5/2
3/2

Cr

FeNH4 (SO4 )2 ∙ 12H2 O
Iron Ammonium AlumW. E. Henry, PR88,
556, 1952

17

LS coupling approach for Lanthanoid (rare earth)18

3d transition metals
The discrepancy tellsthat we need to take theeffect of crystal field
into account before
going into the spinorbit interaction.

19

Magnetic ions in insulating crystals: ligands configuration
ligand

ion

(c)

(a)

K 2 NiF4(b)

Cs2 CuCl4(d)Octahedron coordination

Tetrahedral coordination20

Effect of ligand field
Color centers in insulators

Ruby red in Al2 O3
Al3+

Emerald green in Al2 O3
Cr 3+

Sapphire blue in Al2 O3
Fe2+

Hemoglobin: Fe21

Octahedron ligand field
ligandUnit: CGS

ion

Expansion:
Legendre function:

22

Octahedron ligand field (potential)
Definethen we write
for m: odd

for k: odd
23

Octahedron ligan field potential

24Summary
Electronic states in magnetic ions
LS coupling approachj-j coupling approach
Paramagnetism by magnetic ions in insulators
Curie law
Breakdown of LS coupling approach in 3d transition metals

Ligand field approach
Octahedron potential2022.4.27 Lecture 4
10:25 – 11:55

Lecture on

Magnetic Properties of Materials
磁性 (Magnetism)
Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review➢ Electronic states in magnetic ions
LS coupling approachj-j coupling approach➢ Paramagnetism by magnetic ions in insulators
Curie law
Breakdown of LS coupling approach in 3d transition metals➢ Ligand field approach
Octahedron potential
2Agenda➢ Ligand field approach to 3d orbitals in octahedral potential➢ High-spin/ Low-spin state in ligand field potential➢ Van Vleck (anomalous) paramagnetism➢ Group theoretical approach to level splitting➢ Experiments on and applications of paramagnetism

3Electronic states in magnetic ions (continued)
Periodic table of elements

4

3d transition metals
The discrepancy tellsthat we need to take theeffect of crystal field
into account before
going into the spinorbit interaction.

5

Octahedral ligand field
Potential generated by ligands at an octahedron vertices:ligandion

R
We are considering: Open shell 3d electrons
Single (3)d electron in 𝑣cb (𝒓)Diagonalization in the space of 3d wavefunctionm=0

m = ±1dm = ±2
6

Looking for eigenfunction in tetrahedral potential
Linear combination of d-orbitals

Radial part → common for 5 orbitals
Angular part → second order in (𝑥, 𝑦, 𝑧)

(1)
(2)
(3)

Possible terms: 𝑥 2 , 𝑦 2 , 𝑧 2 , 𝑦𝑧, 𝑧𝑥, 𝑥𝑦

First order in x, y, z → disappearance of off-diagonal term by integration of odd-function
Candidates:

Easily obtained by adding/subtracting (2), (3)In order for vanishing off-diagonal term of 𝑥 4 + 𝑦 4 + 𝑧 4 , we should take differences
between 𝑥 2 , 𝑦 2 , 𝑧 2 :𝑥 2 − 𝑧 2 , 𝑦 2 − 𝑧 2 orthogonalize
3𝑧 2 − 𝑟 2 , 𝑥 2 − 𝑦 2Obtained from (1) (itself), (3) (addition)7

Octahedral ligand field (2)

z
x

y

8

Energy level splitting and quenching of orbital magnetic moment

6Dq

Orbital angular momentum:
e.g.

4Dq

Neither 𝑡2𝑔 nor 𝑒𝑔 orbital does not haveangular momentum

l =2dspherical

octahedral

Explanation of quenching of orbital
angular moment
9High-spin and low-spin states
𝑛=4
low
high

𝑛=5
low
high

𝑛=6
low
high

𝑛=7
low
high

(Effect of crystal field) > (Coulomb repulsion)Ex) hemoglobin
Low spin state

4 2
No oxygen: Fe2+ (𝑡2𝑔𝑒𝑔 ) high spin
6
With oxygen: Fe2+ (𝑡2𝑔
) low spin

10

Topics in paramagnetism from 3d, 4f ions
Jahn-Teller distortion
Distortion energy = energy loweringby symmetry lowering

B.Bleaney, Proc.Phys.Soc.London A63,408(1950).

CuSiF6 6H2 O

11

Van Vleck (anomalous) paramagnetism
LS coupling approach

In the case of Eu3+ (J =0)

Excited state → finite moment
12Very short review of point group theory
Group: Set A with operator *(existence of inverse element).
Element 𝑎𝑖

𝐷(𝑎𝑖 )

square matrix
Representation of group A

projection

13Symmetry operation of point group
A set of symmetry operations around a point in space is called a point group

In crystals: requirement of (discrete) translational symmetry

32 crystal point groups

14Crystal point groups

15

Reducible/irreducible representations
R: symmetry operator

If

Symmetry operation on functionsR
then

can be a representation basis of R

If block diagonalization is possible:
reducible representation
Direct summation:

If block diagonalization is impossible:
irreducible representation: character of representation
16Symmetry operations in group O

17

Simplification to irreducible representation by character table

Simplification of representation

18

Symmetry operation and level degeneracy
Symmetry operator R
Transformation ofoperator:
Assume the system is invariant by operator R

Symmetry connected eigenfunctions: degenerated function set with eigenvalue E ofmust be irreducible
not symmetry-connected
otherwise

accidental degeneracy

19d-level splitting in various crystal fields
Simplification of representation

1
1

2
3

5-fold level

1

2
2
2

1
3

1
1

spherical
symmetry

axial

tetrahedral

octahedral

binary20

Experiments of magnetic moments on
atoms/ions and applications

21

Magnetic resonance
z

B0

𝜔L

J

Larmor precession
If we observe from rotational coordinate
with frequency 𝜔L

JPrecession stops: the effect of magnetic field isrenormalized into the rotation
22

Magnetic resonance (2)High frequency magnetic field in xy-planeJ

Two rotational
magnetic fields
On the rotational coordinate:

Ignore 2wL component: rotational wave approximation
Precession around 𝑩1
Total motion: spiral23

Summary➢ Ligand field approach to 3d orbitals in octahedral potential➢ High-spin/ Low-spin state in ligand field potential➢ Van Vleck (anomalous) paramagnetism➢ Group theoretical approach to level splitting➢ Experiments on and applications of paramagnetism

2022.5.11 Lecture 5
10:25 – 11:55

Lecture on

Magnetic Properties of Materials
磁性 (Magnetism)
Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review of last four lectures
Chapter 1 Basic Notions of Magnetism
Classical pictures of magnetic moments in materials:

 Magnetic charges
 Circular currents

Breakdown of classical magnetism: cancellation of paramagnetic and diamagnetic terms(Bohr-van Leeuwen theorem)

Quest for the sources of magnetic dipoles in materials.
Spins and spin-orbit interactions

Chapter 2 Magnetism of Localized Electrons
Spherical potential, closed shell magnetization

Electronic states of magnetic ions

 LS (j-j) coupling, Hund’s rule
 Ligand field

Representative experimental method: magnetic resonance2Outline Magnetic resonance (continued)
 Spin Hamiltonian Example of analyzing experimental data on electron paramagnetic resonance Application of paramagnetism: magnetic refrigeration

Chapter 3 Magnetism of conduction electrons
 Pauli paramagnetism
 Landau diamagnetism

3Magnetic resonance (2)High frequency magnetic field in xy-planeJ

Two rotational
magnetic fields
On the rotational coordinate:
Ignore 2wL component: rotational wave approximation
Precession around 𝑩1
Total motion: spiral
4Magnetic resonance
H0

Macroscopic magnetization M
Classical equation of motionz

M

Phenomenological introduction ofrelaxation time
𝑇1 : energy relaxation time, 𝑇2 : phase relaxation time

H1

𝑯0 : static field (z)𝑯1 : rotating field with −𝜔Then the equation of motion is given as

5Magnetic resonance (2)We introduce the coordinate system (x’, y’, z’)rotating around z-axis with freq. w.

conditions

Solution1

0.5

absorption
Original
coordinate

0large relaxation

dispersion
-0.5
-10

-5

0

5

10

6

Electron paramagnetic resonance (EPR) experimental setup
small Rlarge R𝜔0

C
L

R

Double resonator sample + cavity
Continuous wave (CW) measurement: detection of resonance dissipation
Pulse, Fourier transform measurement: detection of magnetic field due to the precession ofmagnetic moment7

Spin Hamiltonian
For the comparison of the theory with EPR experiments we need to go a little further in approximation.
Effective spin Hamiltonian:{o}Only contains spin operators, i.e. the orbital part is alreadyintegrated out.

Orbital basis:
In ket form:

diagonalizes

Energy eigenstates:{s} Spin basis for total spin S:
In ket form:
Perturbation Hamiltonian:spin-orbit

ZeemanExpand the wavefunction
with {o} and {s}as:8Spin Hamiltonian (2)Eigenenergy equation:
Second order perturbation in energy:
Orbital angular moment is quenched:
The second order term
→ reduced to second order in L:
effective magnetic field for L

The effective spin Hamiltonian:where L is a tensor given by

Expansion:
9

Spin Hamiltonian (3)The third term is small, does not contribute to level splitting → Drop
The first term: extension of Zeeman
energy with effective g-tensor:
The second term is written as
principal axes: x, y, z
D: axial fine structure parameter

E: rhombic fine structure parameter
The form frequently used forthe analysis of experiments
10Weak crystal field approximation
𝑑4, 𝑑6

𝑑, 𝑑6
2

5

𝐷, 𝐷

−4𝐷𝑞 (𝑇2 )

5

𝐷, 2𝐷

𝑑3, 𝑑83

2𝐷𝑞 (𝑇2 )4𝐹, 𝐹−6𝐷𝑞 (𝑇1 )−6𝐷𝑞 (𝐸)

a

b

c

𝐻

6𝐷𝑞 (𝑇1 )12𝐷𝑞 (𝐴2 )𝑑2, 𝑑7

4𝐷𝑞 (𝑇2 )ESR signal

6𝐷𝑞 (𝐸)−2𝐷𝑞 (𝑇2 )

𝐹, 3𝐹

4

±3/2
−12𝐷𝑞 (𝐴2 )2𝐷octahedral 𝑆 = ±1/2

ℎ𝜈

ℎ𝜈

𝑧

ℎ𝜈
11

EPR signal from Cr3+ and Fe3+ ions in BaTiO3

3000−5/2 ↔ −3/2

𝐻 ⊥c-axis−3/2 ↔ −1/2

−3/2 ↔ −1/20 ↔10 ↔ −1

Cr3+

2000

c

1/2 ↔ −1/2b

1/2 ↔ 3/2

1/2 ↔ 3/2

Fe3+
5/2 ↔3/2

EPR signal intensity (arb.)a

H// c-axis

H (Oe) 4000

hexagonal

tetragonal

𝐸=0

𝐸≠0

5000
Bairavarasu et al. SPIE Proc. 6698-05
12

EPR signal from Cr3+ and Fe3+ ions in BaTiO3

hexagonal

Boettcher et al. JPCM 17, 2763 (2005)

cubic

tetragonal

𝐸=0

𝐸≠0

Bairavarasu et al. SPIE Proc. 6698-05
13

Hyperfine structureselectron-nuclear spin exchange interaction:
the same form as spin-orbit interaction

N 𝜇N

e 𝜇e

NV center

Rama et al., PRB 94, 060101 (`16).

14

ESR detector/analyzer

Commercial machines

EPR public software
EPR-WinSim
https://www.niehs.nih.gov/research/
resources/software/tox-pharm/tools/

Easyspinhttps://www.easyspin.org/

Works on MATLAB

GUI front: cwEPR etc.
Announcement ofEasyspin for Octave
https://octave.discourse.group/t/easyspin-for-octave/1177
15

Magnetic refrigeration
Many attempts for commercial use

Entropy S

E
isothermal

adiabatic𝑇l𝑇hTemperature T𝑇hisothermal𝑇hadiabatic

𝑇l

t
16

Magnetic refrigeration (2)Entropy of a free spin system
1.8
1.6
1.4

for g =2, T =1 K

1.2

B in T

1
0.8
0.6
0.4
0.2

Cooling material
0

1

2

a/2

3

4

Maxwell relation:
17

Active magnetic refrigeration
coolant pump
heat
exhaustion

heat absorption

refrigeration cellmagnetmagnetic
materialmotor

magnet yokemagnetrotation axis
18

Chapter 3 Magnetism of Conduction Electrons

Lise Meitner
Niels Bohr

Werner Heisenberg

George Gamow
Wolfgang Pauli

Lev Landau

Hans Kramers

Photo: heisenbergfamily.org 1930 conference
https://medium.com/lantern-theater-company-searchlight/the-fractured-physics-community-639742855629
19

Chapter 3 Magnetism of Conduction Electrons
3.1 Pauli paramagnetism

3.2 Landau diamagnetism
3.3 An example of orbital diamagnetism

https://sci-toys.com/scitoys/scitoys/magnets/pyrolytic
20

Spin paramagnetism in free electrons
Hamiltonian: kinetic
energy + spin
spin variable:k

How to count k in metals?
Periodic boundary conditionin L-cube
Constant E sphere radius:Pauli paramagnetism
Expectation value ofmagnetic moment:

Fermi distributionfunction:
Chemical potential 𝜇 is determined from

Magnetization:
1
E

Pauli paramagnetic susceptibility

22

Landau quantization
Hamiltonian free electron +magnetic field
Landau gauge:
Schrödinger
equation:
Homogeneous for y and z→ Functional form assumption:
Differential equation for x
Harmonic oscillator at: Cyclotron frequency

Landau quantization
23

Landau quantization: forms of wavefunctions

Diagonalize X

Diagonalize 𝑋 2 + 𝑌 2← Symmetric gauge

Orbital diamagnetism
How to count density of states?
Periodic boundary condition in a cube with side length L

Number of 𝑘𝑧 below 𝐸𝑧z-direction
y-direction
x-direction

Landau level degeneracy (in xy-plane) is
the number of states
below the total energy:

Free energy:
25Orbital diamagnetism (2)Partial integration

26

Orbital diamagnetism (3)To calculate

We use an asymptotic expansion
which can be obtained by applying Euler-Maclaurin formula to

The free energy:
Landau orbital diamagnetism:
Total susceptibility of freeelectrons:
27Summary Magnetic resonance (continued)
 Spin Hamiltonian Example of analyzing experimental data on electron paramagnetic resonance Application of paramagnetism: magnetic refrigeration

Chapter 3 Magnetism of conduction electrons
 Pauli paramagnetism
 Landau diamagnetism

2022.5.18 Lecture 6
10:25 – 11:55

Lecture on

Magnetic Properties of Materials
磁性 (Magnetism)

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review Magnetic resonance (continued)
 Spin Hamiltonian Example of analyzing experimental data on electron paramagnetic resonance Application of paramagnetism: magnetic refrigeration

Chapter 3 Magnetism of conduction electrons
 Pauli paramagnetism

 Landau quantization (→ diamagnetism)2

Outline

1. Landau diamagnetism
2. de Haas-van Alphen effect
3. Orbital diamagnetism of graphene, graphite
Chapter 4 Interaction between spins

1. Exchange interaction from Heitler-London approximation

3

Landau quantization
Hamiltonian free electron +magnetic field
Landau gauge:

Schrödinger
equation:
Homogeneous for y and z→ Functional form assumption:
Differential equation for x
Harmonic oscillator at: Cyclotron frequency

Landau quantization
4

Orbital diamagnetism
How to count density of states?
Periodic boundary condition in a cube with side length L

Number of 𝑘𝑧 below 𝐸𝑧z-direction
y-direction
x-direction

Landau level degeneracy (in xy-plane) is
the number of states
below the total energy:

Free energy:
5Orbital diamagnetism (2)Partial integration

6

Orbital diamagnetism (3)To calculate

We use an asymptotic expansion
which can be obtained by applying Euler-Maclaurin formula to

The free energy:
Landau orbital diamagnetism:
Total susceptibility of freeelectrons:7

de Haas-van Alphen effect: orbital magnetization at high magnetic fieldDensity of states

Free energy expression0

n =0 1

2

3

4

Rapid change in the free energy at
Motion in z-direction: Density of states in one-dimensional system

1

2

3

4

5

Then the density of states is given by

Magnetization formula fora spherical Fermi surface
varies slowly compared with the rapidly oscillating sine term other than at around 𝑘𝑧 = 0
8

de Haas-van Alphen effect in Tl2Ba2CuO6+d
Rourke et al., New J. Phys.
12, 105009 (2010).

9

Experimental data on Tl2Ba2CuO6+dTorque measurement todetect the oscillations inmagnetization

10

Graphite and Graphene
Graphite

Graphenevan der Waals

covalent

11Graphene lattice structure and a simple thought on the band structure
Honeycomb lattice

Atomic orbitals

Dirac corn

Dirac point

Simple thought on band gap opening𝜀0𝜉Triangular neighborhood

𝜀0

𝜀0
1d nearest neighbor

𝜉
𝜀0

2𝜉

𝜀0

𝜉

𝜀0
𝜉

𝜀0

𝜀0

𝜉
𝜉

Degeneracy remains

Graphene band structure: Dirac points in k -space

A Dirac point
Graphene magnetic susceptibility

Why this happens?
No acceleration
Remember classical diamagnetism𝑘𝑥
𝑘𝑦Infinite acceleration

Measurement of graphene diamagnetic susceptibility
Graphene magnetic susceptibility

𝜒McClure, Phys. Rev. 104, 666 (1956).

Bustamante et al., Science 374, 6573 (`21).

𝐸𝐹
0
Dirac point

14

Magnetic field screening, repulsion in graphenez (a)Graphene𝑞m(b)−𝛼𝑔 𝑞m

(c)

0

𝛼𝑔 𝑞m
Magnetic charge magnetic
force line

Virtual charge to express
induction field in the region𝑧 < 0.

Mirror charge to express
induction field in the region𝑧 > 0.

Magnetic field
induced current andmagnetization

0.16 g/cm2
15

Multi-layer graphene
N=2M+1 layers

1: Dirac point
M: zero gap+ gapped

16

Magnetic levitation of graphite

17

Chapter 4

Chiririn Goma

小泉製作所

Exchange interaction
Classical dipole interaction:
This cannot explain ferromagnetism

Quantum mechanical origin of spin-spin interaction: Symmetry of wavefunction
Fermion wavefunction is anti-symmetric: (Orbital part: anti-symmetric) → (Spin part: Symmetric)

If the anti-symmetric orbital part is energetically favorable, this should work as ferromagnetic coupling.

Heitler-London approximation: A two-atom system without hopping

Atomic orbitals: 𝜑𝑎 , 𝜑𝑏
Spin up (a):Spin down (b):Spin states: 𝜒𝑎 , 𝜒𝑏

Wavefunction Slater determinantHeitler-London approximation
Pauli exclusion:
Basis:
Example of interaction
Hamiltonian calculation:

𝐾𝑎𝑏
𝐽𝑎𝑏Matrix elements:

Exchange integral

20Spin Hamiltonian

Eigenstates:

Spin triplet

Spin singlet

Spin operators: 𝒔𝑎 , 𝒔𝑏

21

Heisenberg Hamiltonian
Direct exchange interaction

Effective Hamiltonian:

Heisenberg Hamiltonian:

Exchange interaction

Exchange integral:
Positive → Ferromagnetic
interaction
Weak hopping correction. Superposition ofElectron hopping:
Energy gain

22

Summary
1. Landau diamagnetism
2. de Haas-van Alphen effect
3. Orbital diamagnetism of graphene, graphite
Chapter 4 Interaction between spins
1. Exchange interaction from Heitler-London approximation

2022.5.25 Lecture 7
10:25 – 11:55

Lecture on

Magnetic Properties of Materials
磁性 (Magnetism)
Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review

1. Landau diamagnetism
2. de Haas-van Alphen effect
3. Orbital diamagnetism of graphene, graphite
Chapter 4 Interaction between spins
1. Exchange interaction from Heitler-London approximation2

Outline

Spin Hamiltonian and quantum entanglement
Hubbard Hamiltonian

Superexchange interaction
RKKY interaction
Double exchange interaction

Theory of Magnetic insulators

Molecular field approximation3

Spin Hamiltonian and quantum entanglement
Spin Hamiltonian for EPR analysis:
Obtained by integrating out the orbital
part in the second order perturbation.
Direct exchange interaction:
This gives the same matrix elements forthe basis of relevant levels.

Quantum entanglement
Two systems, freedoms

bases

statesNot entangled:
The state is written as a direct product.

Maximally entangled state:

Two states are unseparable.
4Quantum entanglement and effective Hamiltonian
Maximally entangled state:
Another maximally entangled state:
Let us consider the case the basis is limited toConsider a Hamiltonian working on {|1ۧ, |2ۧ}Consider a Hamiltonian working on {|𝑝ۧ, |𝑞ۧ}Thoughandare completely different, as long as we limit the basis todistinguishand.

we cannot
5Quantum measurement and entanglement
In inductive measurement the EPR needs
𝑁𝑚𝑖𝑛 ~1010
How you make this to one?
What is measurement?
System to be measured:
Degree of freedom which human can distinguish:
Measurement is to create amaximally entangled state
between them.
Schrödinger’s cat problem is a problem of measurement.

6

Coulomb blockade in quantum dots
mN+1

Constant interaction: U
EFElectron number: N
Interaction energy

EF
mN

Chemical potential

mN-1

Coulomb oscillation
Vg

𝐽
EF

m N+2

EF

m N+1
mN
m N -1

𝑉g
7

Pauli blockade
dot1 dot2
When both dot1 and dot2
are occupied by up (down)spin, the conduction isblocked.

K. Ono et al., Science 297, 1313 (2002)8

Spin quantum bit
F. H. Koppens et al. Nature 442, 766 (2006)

Ec
Even electron QD
No spinVL

VL

Odd electron QD
Spin 1/2

Quantum dot isolated

Pauli blockade

Larmor precession
Readout

rf lineGate voltaget9CNOT gate for electron spins
Zajac et al. Science 359, 439 (2018).

10Detection of Larmor precession

11

Spin-charge entanglement and detection of spin-spin interaction12

Hubbard modelAnti-ferromagnetic exchange interaction by electron transfer(𝑖, 𝑗)

Two site model:
Hopping operator:ttSecond perturbation energy gain

Intermediate state
Enhancement U

In Hamiltonian form:
Two-site Hubbard Hamiltonian13

Effective Hamiltonian for 2-site Hubbard Hamiltonian
Possible 6-states:
Good quantum number
operators

Anti-ferromagnetic
14

Superexchange interaction
Fe
K

Compounds of magnetic ions and closed shellnegative ions often have anti-ferromagnetism or
ferromagnetism.

F
What is the mechanism (spin-spin interaction?) of
magnetism?

Magnetic ion Negative ion Magnetic ion

Superexchange mechanism
Small amount of electrons on a negative ionsmoves to a neighboring magnetic ion. Then
spin appears on the negative ion which haveexchange interaction with anotherneighboring magnetic ion.
15

Goodenough-Kanamori rules

(a)

(b)

Angles, orbitals, electrons numbers determine ferromagnetic, anti-ferromagnetic
and the strength16s-d exchange interaction
Scattering of electrons s by a local magnetic ion S at the origin.

This works as if a delta-function magnetic field:
Fourier transformation:
Magnetic moment spatial distribution
and susceptibility in frequency space.
Perturbation of

on plane waves

17

RKKY interaction
2

1

0

1

2

18

RKKY interaction (2)5 × 10−3

Second magnetic ion at R
0

−5 × 10−3
0

10

20

30

19

Double exchange interaction

LaMnO3
LaMnO3 are anti-ferromagnetic insulator (Mott insulator). But when some La is replacedwith Ca, Mn4+ ions appear. The system becomes metallic and at the same time thismaterial shows ferromagnetism.
: Kind of kinetic exchange interaction.

20

Chapter 5
W. D. Coolidge

Y. Kato

Theories of Magnetic Insulators
Fehttp://teetokue.air-nifty.com/blog/2008/04/59_9f47.html

MnOMolecular field approximation
Ferromagnetic Heisenberg Hamiltonian:
Average field approximation:(molecular field approximation)

Remember

replacethenBrillouin function is expanded asMolecular field approximation (2)
then

with
The first order term drops atwhich gives the Curie temperature.

Curie-Weiss law

23Summary

Spin Hamiltonian and quantum entanglement
Hubbard Hamiltonian
Superexchange interaction
RKKY interaction

Double exchange interaction

Theory of Magnetic insulators
Molecular field approximation

2022.6.01 Lecture 8

Lecture on

10:25 – 11:55

Magnetic Properties of Materials
磁性 (Magnetism)
Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto
Kanra Park Gunma Prefecture

Review• Spin Hamiltonian and quantum entanglement➢ Hubbard Hamiltonian➢ Superexchange interaction➢ RKKY interaction➢ Double exchange interaction
Ch. 5 Theory of Magnetic insulators➢ Molecular field approximation2

Outline➢ Molecular field approximation➢ Phenomenology of phase transition GL theory• Free energy• Spontaneous Symmetry Breaking➢ Critical Exponent➢ Theoretical models of magnetic materials• XY model• Ising model➢ Antiferromagnetic Heisenberg model3

Molecular-field approximation on ferromagnetic Heisenberg model
Ferromagnetic (J >0) Heisenberg model:
nearest neighbor

Mean field (molecular-field) approximation:

Replace the neighboring spins with averaged one

i+1
i

j

i+4

i+2

i

The averaged spins work as an effective field:

i+3

Remember paramagnetic representation ofmagnetization:

Replacement:then4Curie-Weiss law
Brillouin function is expanded as

𝜒

M

then
where
0

1

: the Curie law

The first order term drops atwhich gives the Curie temperature and the Curie-Weiss law

There exists non-zero solution for M,
which has lower energy than M =0.

Solving the above we get

5Ginzburg-Landau Theory(1)The Curie-Weiss law
involves that the establishment of spontaneous magnetization is the result of a cooperative phenomenon.
Phenomenology: discuss the physical properties that do not depend on details of models.

The Ginzburg-Landau theory was developed for phenomenology of superconductivity.
Consider a symmetry of the Heisenberg model at B =0.
A symmetry operation:

On the other hand

hence

Expansion to the series of power should be
To obtain stable (minimum) points

6Ginzburg-Landau Theory (2)
a
0

a>0

a<0

M

M
(a)

(b)

M

(c)

Continuous variation in free energy: Second
order phase transition
T : relevant parameter

Magnetic equation of state7

Spontaneous Symmetry Breaking
Spontaneous magnetization
a<0

The symmetry of the system (Hamiltonian)is kept unchanged.
M
However the symmetry of the state is
broken.

MH curve

Spontaneous Symmetry Breaking One of the central concepts in physics.
Phase transition, mass appearance, big bang, ⋯Associated with appearance of Nambu-Goldstone mode8

Critical exponent
C

In the presence of spontaneous magnetization, the free energy aroundthe stable point is

Then the specific heat is obtained by
TC

0

T

Small B

Physical quantity that appears
at the critical pointat the critical point

Shift of a relevant parameter from the critical point
9

Critical Exponent and Universality Class
Universality Class: Classification of the systems by symmetry, range of interaction, etc.
Each system which belongs to a universality class has the same set of critical exponents.
In the case of mean field approximation:
One of the key features in analyzing phase transitions.

Wikipedia

10Models of magnetic systems (spin systems)
XY model:Spins are confined in a two-dimensional plane.
𝜙𝑖 : Angle of each spin

Two-dimensional XY model:

No long range order (Mermin-Wagner theorem)
Berezinskii-Kosterlitz-Thouless (BKT) transition
Quasi long range order (power decay)
Realization of XY model: Josephson array𝜙𝑖𝜙𝑗Josephson energy

11Berezinskii-Kosterlitz-Thouless Transition

12

Ising model
Directions of spins are limited to z

Solution: 1d Ising, 2d Onsager

https://www.youtube.com/watch?v=kjwKgpQ-l1s
13Antiferromagnetic Heisenberg model: Néel order and lattice partitioning
Antiferromagnetic
Heisenberg Hamiltonian

J<0

Néel order in 2D square lattice

Partial Lattice A

Partial Lattice B
14

Antiferromagnetic Heisenberg model(2)Molecular-field effective
Hamiltonian

Averaged moments

Vector Brillouin function

Self-consistent equation

Uniform susceptibility
Alternative susceptibility
15

Antiferromagnetic Heisenberg model
Around stable points

Self-consistent equation for 𝑀𝑆

0

T

16

Summary➢ Molecular field approximation➢ Phenomenology of phase transition GL theory• Free energy• Spontaneous Symmetry Breaking➢ Critical Exponent➢ Theoretical models of magnetic materials• XY model• Ising model➢ Antiferromagnetic Heisenberg model

2022.6.8 Lecture 9

Lecture on

10:25 – 11:55

Magnetic Properties of Materials
磁性 (Magnetism)

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review➢ Molecular field approximation➢ Phenomenology of phase transition GL theory• Free energy• Spontaneous Symmetry Breaking➢ Critical Exponent➢ Theoretical models of magnetic materials• XY model• Ising model➢ Antiferromagnetic Heisenberg model2

Outline➢ Anti-ferromagnetic Heisenberg model : parallel fieldsusceptibility➢ Spin flop and metamagnetic transition➢ Ferrimagnetism➢ Molecular-field approximation➢ Helimagnetism➢ Spin wave3

Antiferromagnetic Heisenberg model
Antiferromagnetic
Heisenberg Hamiltonian
J<0
Sublattice magnetic field

spontaneous sublattice magnetization

Self-consistent equation

Around stable points
Self-consistent equation for 𝑀𝑆

0

T

4

Antiferromagnetic Heisenberg model: parallel field
Consider sublattice-dependent effective magnetic fieldSet of self-consistent equations for
sublattice-dependent magnetic field

This set of equations should be solved numerically.
Then the parallel susceptibility is given by
𝑇→0
0
T

then

On the other hand, at 𝑇 = 𝑇N
5

Examples of spin configuration in metal-oxide antiferromagnets
MnO

magnetic
unit celllatticeunit cell

NiO

MnF2

FeO
6

Temperature dependence of susceptibility

High temperature side (𝑇 > 𝑇N )

𝜃: Weiss temperature7

Typical anti-ferromagnets Néel and Weiss temperatures

8

Spin phase transition at higher fields in antiferromagnets
Consider a material with susceptibility 𝜒MmetaWith applying magnetic field B the energy lowering in the material isspin flop0

B

States under vertical magnetic field is more stable

However crystals often have magnetic anisotropy. Let K be the anisotropic energy.
The anisotropic energy is overcome by the Zeeman energy atSpin flop transition

Meta magnetism transition: antiferromagnetic interaction is overcome by the Zeeman energy
9Spin flop transition, metamagnetic transition
Metamagnetic transition

Sm0.5Ca0.25Sr0.25Mn3

Helical magnet

Ho single crystal

Remember in magnetic refrigeration, to have a good efficiency
Spin flop transition in polymer
anti-ferromagnet

should be large.
Because the metamagnetic transition is temperature sensitive, very
high efficiency may be attainable.

10Ferrimagnetism
Ferrimagnetism

Magnetism in ferrites

Anti-ferromagnetic exchange interaction
between the two sublattices: the same as
anti-ferromagnetism
However the amplitudes ofmagnetization (the magnetic moments)
are not the same.

Spontaneous magnetizations do notcancel out.
Molecular field approximation
Molecular fields: intrasublattice interaction is
included
Self-consistent set of equations

spinel ferrite

11Compensated ferrimagnetism
𝑀

Mn1.5V0.5FeAl
Cubic Heusler
L21
𝐹𝑚3ത 𝑚𝑀A
𝑀A − 𝑀B
TX
𝐹 4ത 3𝑚

Stinshoff et al.
PRB 95, 060410(R)
(`17)

𝑀B

Hasegawa et al. AIP conf. proc. 24, 110 (`75).

12

Helimagnetism
Heisenberg model (again!)in the first place

Remember the agenda of molecular field approximation.
1.
2.
3.

Find classical ground stateConsider the field configuration to stabilize the classical ground state
Write down the self consistent equation

𝑸

Look for a stable state.
Fourier expansion

Then
The expectation value of the
Hamiltonian is written as

𝒗

𝒖

where
13

Helimagnetism (2)In
the summation on i in the right handside can be carried out as
Then
Let ±𝑸 be wavenumbers at which 𝐽𝒒 take the maxima

this should be a constraint.
𝑸 = 0: Ferromagnetism
𝑸 = 𝑲 − 𝑸: Antiferromagnetism

Then we assume
The equation on the
top isFrom the constraint

14

Helimagnetism (3)𝑸

Then we can write with taking u and v
as orthogonal unit vectors as
Then the ground state spin configuration
is given byThis represents the helical structure.
Molecular field approximation
Stabilization field𝒖𝒗

Molecular field
Effective Hamiltonian
Self consistent equation

Helical susceptibility
Helical order temperature
15

Lorentz transmission microscope
Overview

Effect of defocusing

Li-cong et al. Ch. Phys. B 27, 066802 (`18)

16

Spatially localized magnetic structures
Li-cong et al. Ch. Phys. B 27, 066802 (`18)

17

Real space observations of spin structures by Lorentz microscope
Uchida et al., Science 311, 359 (`06)

Fe1-xCoxSi

𝜀-FeSi B20-type cubic non-centrosymmetric lattice

Dzyalosinsky-Moriya interaction
causes helimagnetism

Helical structure can be detectedin the Fresnel mode of Lorentz
microscope.

18

Observation of skyrmions by Lorentz microscope
Cu2OSeO3
The DM
interaction
causeshelimagnetism.
Seki et al.
Science 336, 198 (`12)

19

Spin wave (ferromagnetic)
Ferromagnetic Heisenberg model

Heisenberg equation of motion
can be re-written as a torque equation

Fourier transformed equation of motion
has much larger value than others.
Then we can approximate
20

Spin wave (ferromagnetic) (2)These equation represents precession aroundz-axis (in Fourier space)

Hence we write
to obtain the excitation energy
Holstein-Primakoff transformation

21

Summary➢ Anti-ferromagnetic Heisenberg model : parallel fieldsusceptibility➢ Spin flop and metamagnetic transition➢ Ferrimagnetism➢ Molecular-field approximation➢ Helimagnetism➢ Spin wave

2022.06.15 Lecture 10

Lecture on

10:25 – 11:55

Magnetic Properties of Materials
磁性 (Magnetism)

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review➢ Anti-ferromagnetic Heisenberg model : parallel fieldsusceptibility➢ Spin flop and metamagnetic transition➢ Ferrimagnetism➢ Molecular-field approximation➢ Helimagnetism➢ Spin wave2

Outline➢ Spin wave (classical) in a ferromagnet➢ Quantization of spin wave (magnon)➢ Magnons in an anti-ferromagnet➢ Magnon approximation for weak excitations in ferro- andanti-ferromagnets➢ Nambu-Goldstone mode in phase transition➢ Experiments on magnons3

Spin wave from phase shift of spin precessions
https://www.youtube.com/watch?v=pWQ3r-2Xjeoxsnapshotcf. Bloch electrons → Magnons can be described in magnetic Brillouin zone.
Total spin

Heisenberg equation:

Such a motion of macroscopic magnetic
moment can be confirmed by
Ferromagnetic resonance (FMR)

Phase shifts of precessions with sites:
Then a snapshot should be expressed in a Fourier form.
However for 𝜔0 we need to consider the spin-spin interaction.

4

Equations of motion in the momentum space
Fourier transform, inverse Fouriertransform:
Heisenberg Hamiltonian, equation of
motion:,Substituting the above Fouriertransforms into equation of motion,
we obtain a set of equations of
motion in the momentum space as:

Fourier transform of interaction J:(i can be taken somewhere)

Nearest neighbor approximation:
Small angle approximation, i.e.,
replace 𝑆𝑗𝑧 with 𝑆:

5

Spin wave (ferromagnetic)
These are the equation we obtained in
the last lecture but B:

These equation represents precession aroundz-axis (in Fourier space)

Hence we write
to obtain the excitation energy
Remember the message:

However for 𝜔0 we need to consider the spin-spin interaction.
6

Holstein-Primakoff transformation
Let us consider the quantization of the spin wave.
Spin operator:: eigenfunction of 𝑆𝑧 with eigenvalue of 𝑚.

We define up/down operator as:
vacuumThen from the properties of spin operator:

This is as if we are treating number states |𝑛ۧ.
Let us introduce boson creation/annihilation operators:“Vacuum” of the boson:n-boson state:
Then as is for ordinary boson operators, we obtain:

With number operator

we can write

Holstein-Primakofftransformation
7

Non-interacting spin wave approximation
In Holstein-Primakoff transformation we have nonlinear terms. → Interaction between the bosons.

Expand the square roots in Holstein-Primakoff transformation

Then the Hamiltonian is expanded as

Take up to quadratic terms

8Ferromagnetic spin wave: ferromagnetic magnon
Fourier transform ofcreation/annihilation
operators:

Magnon Hamiltonian

Substitute these to the approximated Hamiltonian

Total magnetization:with Bose distribution function:

Magnon dispersion

Then we obtain9Why we have introduced the concept: Magnon?
Low temperature Magnetic
moment:
The internal energy is:

Then low temperature specificheat is obtained by
As above, by considering magnons we can calculate low energy excitations and obtain important
quantities.
Magnons: Low temperature model of ferro (anti-ferro) magnets.

10

Spin wave modeling of anti-ferromagnets
Anti-ferromagnet → Decompose into A, B sublattices
A sublattice: we can consider magnon model
B sublattice:

Magnetization is reversed

Take the vacuum as
Boson creation/annihilation operators
vacuumThen the Holstein-Primakoff transform is

Quadratic Hamiltonian :11

Spin wave modeling of anti-ferromagnets (2)Fourier transformation ofcreation/annihilation operators

Momentum representation of
the Hamiltonian
with nearest neighbor summation

But the above Hamiltonian is still not diagonalized. (Néel ordered state is not true ground state.)
Bogoluibov transformation

Bosonic commutation relations

12

Spin wave modeling of anti-ferromagnets (3)Condition for diagonalization:
Diagonalized Hamiltonian:
Ground state energy:
Néel orderedstate energy

Energy lowering due tohybridization of 𝑆, 𝑆 − 1, ⋯ , −𝑆
statesis shorter than S

How?
13

Spin wave modeling of anti-ferromagnets (4)Magnon dispersion

Simple cubic case

Asymptotic form 𝑞 → 0
Internal energy

Results

14

Specific heat of an organic anti-ferromagnet
Anti-ferromagnetic TN : 2.5 K

Normal metal specific heat:

Superconducting Tc : 1.1 K

Fukuoka et al., PRB 93, 245136 (2016)15

Spontaneous Symmetry Breaking and Nambu-Goldstone mode
Spontaneous symmetry breaking
Nambu-Goldstone theorem
M

M

When a spontaneous symmetry
breaking takes place, a mode with zeroenergy at long wavelength limit
appears.

00 massless

Nambu-Goldston mode
(Nambu-Goldston boson)

Magnons in the case of ferromagnets(type-B) and anti-ferromagnets (type-A).
16

Generalization of Nambu-Goldstone theorem (column)
Nambu-Goldstone mode, Higgs mode → Birth of particle mass; Standard theory of elementary particles
The theories based on the principle prevail all over the physics.
However, there still have been many open questions!
An example: According to the primitive statement, the number of NG mode should be the sameas that of broken symmetries.
Broken symmetry
N-G theorem

Number of NG modes

Number of broken
symmetryxy=x

CrystalTranslational symmetry

3

3

3D Ferromagnet

Rotational symmetry

1

2

Spinor BEC

Rotational symmetry

2

3

Skymion crystal

Translational symmetry

1

2

Extended theorem (2012) :Watanabe, Murayama PRL 108, 25162 (2012); Hidaka PRL 110, 091601 (2013); Hidaka, Minami PTEP 2020, 033A01
17

Magnon dispersion relation measurement in MnF2
Neutron time-of-flight (TOF) measurement
neutron magnetic field
electron moment
neutron inelastic
scattering

momentum shiftenergy loss

TOF measuerment
magnon dispersion

MnF2

Calculated dispersions of anti-ferromagnetic magnons in MnF2 and FeF2.
(Taken from Fundamentals of Magnonics.)

Low et al., JAP 35, 998 (1964).

18

Magnon dispersions in metallic ferromagnets

19

Bose-Einstein condensation of magnons
Magnons: not completely bosons (para statistics) however can be treated as“hard core” bosons
Sharp
TlCuCl3
enhancement ofmagnetization
BEC in cold Rb atom ensemble
Sharp increase in particle density
TheoryNikuni et al.,PRL 84, 5868
(2000)
20

Magnon (spin wave) resonance in thin films
CoFe Nano disk

Kajiwara et al., Nature 464, 262 (2010)

Spin wave resonance in a thin film

Kittel Phys. Rev. 110, 1295 (1958)

Dobrovolskiy et al., Nanoscale 12, 21207 (2020).

21

Real space imaging of magnons
YIG

YIG

NV center ESR detection of magnon fields

Gruszecki et al. Sci. Rep. 6, 22367 (2016)

Bertelli et al., Sci. Adv. 6, eabd3556 (2020).

22

Summary➢ Spin wave (classical) in a ferromagnet➢ Quantization of spin wave (magnon)➢ Magnons in an anti-ferromagnet➢ Magnon approximation for weakexcitations in ferro- and anti-ferromagnets➢ Nambu-Goldstone mode in phasetransition➢ Experiments on magnons

2022.6.22 Lecture 11

Lecture on

10:25 – 11:55

Magnetic Properties of Materials
磁性 (Magnetism)

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review➢ Spin wave (classical) in a ferromagnet➢ Quantization of spin wave (magnon)➢ Magnons in an anti-ferromagnet➢ Magnon approximation for weak excitations in ferro- andanti-ferromagnets➢ Nambu-Goldstone mode in phase transition➢ Experiments on magnons2Outline➢ Ferromagnetic and Antiferromagnetic resonance➢ Spin wave resonance➢ Experiments on magnons➢ Scaling relations➢ Renormalization group➢ Derivation of scaling ansatz

3

BEC of quasi-equilibrium magnons at room temperature
Demokritov et al., Nature 443, 431 (2006).

4

Ferromagnetic resonanceFree energy:
In case of ferromagneticallyordered state:
Anisotropic field

ZeemanDemagnetizing field

Effective field other than H :Kinetic equation of macroscopic
moment:𝛾: gyromagnetic ratio

When the anisotropy is uniaxial,
the shape of the sample, themagnetic field are on line:5

Antiferromagnetic resonance
Effective field for two magnetic sublattices (1,2) :In the case of antiferromagnet:

No demagnetizing effect!

Anisotropy tensor:
Assumption: Anisotropy energy

is uniaxial.

Anisotropy tensor:
Resonance frequencies:

Critical field of spin-flop transition:
When the anisotropy is small:6

Spin wave (Magnon) resonanceFerromagnetic spin wave dispersion relation:
Standing wave condition:

L

Seavey, Tannenwald, PRL 1, 168 (1958).
7

Magnon (spin wave) resonance in thin films
CoFe Nano disk

Kajiwara et al., Nature 464, 262 (2010)

Spin wave resonance in a thin film

Kittel Phys. Rev. 110, 1295 (1958)

Dobrovolskiy et al., Nanoscale 12, 21207 (2020).

8

Real space imaging of magnons
YIG

YIG

NV center ESR detection of magnon fields

Gruszecki et al. Sci. Rep. 6, 22367 (2016)

Bertelli et al., Sci. Adv. 6, eabd3556 (2020).

9

Section 5.10

Renormalization group and scaling theory of phase transition

Kenneth Wilson

Jacques Friedel

Jun Kondo

1936 - 2013

1921 - 2014

1930 - 2022

1982 Nobel prize

Correlation function
Magnetic moment (local) density :Free energy density:

Free energy functional:
Partition function:: functional integral
Probability density of realization of m(r) :Expectation value of physical quantity A
Temperature dependence assumption:

11

Correlation function (2)Correlation function of orderparameter fluctuation:
The second term = 0 for 𝑇 > 𝑇C
Fourier representation:

BecauseAnd from translational invariance:
Free energy:
Ignore 4th order
Weight function:
Sum over independent k
12

Correlation function (3) Scaling relations
Finally
Yukawa type function
Exponential decay + (1/r)

Correlation
lengthdiffers from g(r)Appearance of long range order
Scaling relations

temperaturemagnetic field hCritical exponents:

13

Scaling relationsd : dimensionality
GL theory gives

Scaling relations:

Scaling ansatz: Critical behavior is described by a single relevant parameter.

Δ: Gap exponent

Free energy expression:

14

Scaling ansatz and scaling relations

Then we assume the asymptotic form:
From the scaling relations:
For m to be finite at 𝑡 → 0:Compare with

thenHyperscaling relation:

15

Ising model
Directions of spins are limited to z

Solution: 1d Ising, 2d Onsager

https://www.youtube.com/watch?v=kjwKgpQ-l1s
16Renormalization group
2D square lattice Ising model

Renormalization grouptransformation of scaling factor x as

Four spins are averaged and coarse-grained to a single spin.
Ordinary no inverse element: Semigroup
Renormalization group transformation of scaling
factor 2.
17

Flow diagramA

x: continuous variable
Renormalization group transform → changes system parameters

B

Continuous movement of the system in the parameter space

Flow diagram
Flow diagram

Complete order, complete disorder: the parameters do not change
Stable fixed points
Just on the critical point: Unstable fixed point

18

Derivation of scaling ansatz
𝑡: Temperature, h : Magnetic field
Renormalization group transform with scaling factor x

Expansion around the unstable fixed point t = h = 0
Symmetry difference between t and h. h is reversed by reversing the magnetization but t is not.

Then they cannot have linear relations:This should hold for any x >1, natural number nThenshould be power functions of x.

19

Derivation of scaling ansatz (2)Hence we write

Let us consider the case of starting at ( t , h ).
n-times operation of RGT with SF x

System temperature

is far from the critical point (assume)

Correlation length
Remember

In a d-dimensional system, 𝑓(𝑡, ℎ) becomes 𝑥 𝑑 times of the original.

Hence by some function 𝑓± (𝑥) we can write

20Chapter 6

Magnetism of
Itinerant Electron
Systems

Magnetic Puzzle

Summary➢ Ferromagnetic and Antiferromagnetic resonance➢ Spin wave resonance➢ Experiments on magnons➢ Scaling relations➢ Renormalization group➢ Derivation of scaling ansatz

2022.6.29 Lecture 12

Lecture on

Magnetic Properties of Materials

10:25 – 11:55

磁性 (Magnetism)

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review Magnon condensates, Magnonics
 Scaling, Renormalization group
Coarse graining

A
System transition in a parameter space
Flow diagram

B

Stable fixed point: Extreme conditions
Unstable fixed point: critical point

Renormalization group transform with scaling factor x

Renormalization group equation2

Outline

Chapter 6 Magnetism of Itinerant Electron Systems
 Ferromagnetism in Electron gas• Hartree-Fock approximation• Diffusion Monte-Carlo calculation

 Hubbard model: mean field theory• Hartree-Fock approximation: Stoner criterion• Magnetic susceptibility

 Magnetism in 3d transition metals• Slater-Pauling’s curve• Density of states by APW method3

Chapter 6

Magnetism of
Itinerant Electron
Systems

Magnetic Puzzle

Periodic table of the elements

5

Hartree-Fock approximation for ferromagnetism in electron gas
Hartree-Fock approximation: A way to treat electron-electron interaction (correlation) in mean field theory.

Let us consider an N-particle system

Single-particle wavefunctions

Many-particle wavefunction fulfilling
particle exchange statisticsSlater determinant:Assumption: Hamiltonian =single body + two-body:𝑥𝑖 : all freedoms of a single particle

Expectation value of the total energy:
Hartree-Fock approximation = minimize

in variational method on

6

Hartree-Fock approximation (2)Constraint – Orthonormal basis:(1) Direct integraland (2) Exchange integral:
We apply the method of Lagrange multipliers.

(1)

(2)

Consider the quantity:

Extremals condition for the variation of

Density matrix (definition):

We further define
Then the extremal condition is7

Hartree-Fock approximation (3)We takefor an eigenfunction of operator

Then taking N of eigenstates with the lowest eigen energies, and make the Slater determinant from them.
Hartree-Fock ground state
Operator

depends on

Self-consistent equation

Hartree-Fock equation

8

Magnetism in jellium model+e-eElectrons in a
lattice potential

-e

-e
+e
-e

+e -e +e

-e
+e
-e

+e

-e

+e
+e
-e -e
-e
-e
+e -e +e+e -e +e

-e

-e
-e

-e
-e

-e
-e

-e
-e
-e

Jellium model
-e

-e
-e
-e

Electrons in a uniformbackground

Jellium model ground state of noninteracting electrons

Hamiltonian with interaction:

System parameter: Averaged particle
distance measured by Bohr magneton:
9Magnetism in jellium model (2)In the jellium model, plane waves are already the self-consistent equation. Then the plane wave states thatminimize the energy is the solution of HF approximation.
Remember

Kinetic energy per an electron:
No direct integral term (Hartree) due to the charge neutral condition in the case of jellium model.
Exchange energy per an electron:

Integration gives

Hartree-Fock energy is given by
10

Magnetism in jellium model (3)Magnetic polarization:

Ferromagnetic state is the ground state

Overestimation of
the stability offerromagnetism

Taken from Ashcroft-Mermin
Solid State Physics

11Correlation energy
In a realistic electron gas, the electrons keep away from each other lowering the Coulomb energy
even between ones with the opposite spin directions.
Difference from the HF interaction energy:

Correlation energy
Phase diagram by diffusion Monte-Carlo
method2.5
2.0

Polarized Fermi fluid
1.5

Huge deviation from 3d metals

Metastable
Bose fluid

1.0

UnpolarizedFermi fluid

0.5

Wigner crystal
0

40

80

120

160

200

Ceperly, Adler, PRL 45, 566 (1980).

Hubbard model
3d transition metals: 3d4s open shell(1) 3d: Tendency to localize(2) 4s: Delocalize, light mass → screen long range Coulomb interaction

Two-site Hubbard Hamiltonian

General Hubbard Hamiltonian(1) → Hopping

On-site Coulomb ← (2)Fermion commutation relation:
In the present case, this Hamiltonian only acts on d-electrons explicitly.

13

Hubbard model (2)Hubbard Hamiltonian
Fourier expansion

Tendency to localize but still itinerant

Itinerant electron system14

HF approximation in Hubbard model
Local magnetic moment, electron
number (per site)

Fluctuation term:
dropped in HF
approximation
Moving in the averaged fieldof opposite spin

15

HF approximation in Hubbard model (2)averaging

Single electron energy shift:
Total energy:

Energy shift by magnetization
Spin-dependence ofDifference in the numbers of ↑ electrons and ↓ electrons
should be consistent with m
16

HF approximation in Hubbard model (3)for non-zero m

Self-consistent equation
Density of states
Increase in the kinetic energy by
spontaneous magnetization

Decrease in interaction energy
by spontaneous magnetization
Stoner condition

For ferromagnetism to take place, the Coulomb energy should be larger than the band width.

(Still has a problem of overestimating the Coulomb effect in the case of anti-parallel spins.)

17

Magnetism in 3d transition metals

Elementary ferromagnetic metals

From D. Coey in Materials for Spin Electronics, Springer 2008
18

Magnetism of 3d transition metals: Slater-Pauling’s curve
Balke et al., Sci. Technol. Adv.
Mater. 9, 014102 (2008).

Slater-Pauling’s curve
Experimental data are in line.

The gradient is ±1 !
Abrupt change around FeAPW method to calculate DOS

Muffin-tin potential:

Hartree:Exchange:

Variational wavefunction:

Iteration for convergence for each k20

Density of states in Ni and Fe
Major

Ni
Fe
full
hole

hole
Major
hole

21

Fermi energy “locking” around a valley in density of states𝐸𝐹

Enhance particle number

Decrease particle number

𝐸𝐹

𝐸𝐹
Needs large kinetic
energyEnhance inoccupied states

22Summary
Chapter 6 Magnetism of Itinerant Electron Systems
 Ferromagnetism in Electron gas• Hartree-Fock approximation• Diffusion Monte-Carlo calculation

 Hubbard model: mean field theory

 Magnetism in 3d transition metals• Slater-Pauling’s curve• Density of states by APW method

2022.7.6 Lecture 13

Lecture on

10:25 – 11:55

Magnetic Properties of Materials
磁性 (Magnetism)

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review

Chapter 6 Magnetism of Itinerant Electron Systems➢ Ferromagnetism in Electron gas• Hartree-Fock approximation• Diffusion Monte-Carlo calculation➢ Hubbard model: mean field theory• Hartree-Fock approximation: Stoner criterion• Magnetic susceptibility➢ Magnetism in 3d transition metals• Slater-Pauling’s curve• Density of states by APW method2Outline➢ Magnetism in 3d transition metals• Slater-Pauling’s curve• Density of states by APW method➢ Kubo formula (linear response)➢ Fluctuation dissipation theorem➢ Random phase approximation (RPA)3

3d and 4s electrons in isolated transition metal atoms
3d

4s
4s
3d
K

Ca

Sc

Ti

Radial wavefunction

4

3d and 4s electrons in isolated transition metal atoms
bcc a-Fe3d2.86 × 10−10 m = 5.4aB
4s
4s

e

3d
K

Ca

Sc

s-electrons

Ti
Minor

Radial wavefunction

Major
d-electrons5Magnetism of 3d transition metals: Slater-Pauling’s curve
Balke et al., Sci. Technol. Adv.
Mater. 9, 014102 (2008).

Slater-Pauling’s curve
Experimental data are in line.

The gradient is ±1 !
Abrupt change around FeAPW method to calculate DOS

Muffin-tin potential:

Hartree:Muffin-tin potential

Exchange:

Variational wavefunction:

Iteration for convergence for each k7

Density of states in Ni and Fe
Major

Ni
Fe
full
hole

hole
Major
hole

8

Explanation of Slater-Pauling’s curve (1)Case of Ni
3𝑑 ↑ : full 5 electrons
3𝑑 ↓ : 4.4 electrons
Increase of electrons → filling up the holes andthe magnetic moment decreases
Decrease of electrons → opening holes in 3𝑑 ↓and the magnetic moment increases
9Explanation of Slater-Pauling’s curve (2)Case of Fe 3𝑑 ↓ : 2.5 electrons

3𝑑 ↑ : 4.7 electrons (not full, hole exists)
Increase of electrons → filling up the holes in ↑and the magnetic moment increases
After complete filling up of ↑ → filling up theholes in ↓ and the magnetic moment decreases
10Magnetic susceptibility in HF approximation
Magnetic moment:
Magnetic susceptibility per atom:
Electron energy in magnetic field:
where

with

This should be positive for the appearance of ferromagnetism. (remember GL theory).
Then minimization of 𝐸𝐵 should give 𝑛− as

We finally obtain
Stoner factor

11Temperature dependence of susceptibility in HF approximation
By using the identity for degenerated
Fermi gas:

we writeBy defining

Susceptibility with temperature correction:

This is not Curie-Weiss observed in experiments.

12

Kubo formula
Dynamical response to magnetic field

Time-dependent perturbation

Heisenberg equation of motion
Single body density matrix:
Initial condition:
Unperturbed system partition function:
Then the density matrix should satisfy (see lecture note for the calculation)

where
13

Kubo formula (2)For linear response we can replace

made of eigenstates of unperturbed Hamiltonian

Then we can write

External field
Expectation value of general physicalquantity Q
where
is a pure imaginary.
Field with frequency w

Definition of susceptibility 𝜒(𝜔)We can equalize this with

14

Kubo formula (3)Definition of susceptibility 𝜒(𝜔)We can equalize this with

The first equation →The second equation →Kubo formula15

Fluctuation-dissipation theorem
Green’s function(+: retarded, -: advanced)

Heaviside function

Kubo formula is re-written as:

Fourier transform to w-space

Correlation function
Fluctuation-dissipation theorem:

See the lecture note for the calculation

16

Example of fluctuation-dissipation theorem
Johnson-Nyquist noise
Thermal noise

Low pass 5 kHz

100 kHz17

Random phase approximation (RPA)
External magnetic field:
Hubbard model:

Local magnetization (inunit):
Perturbation Hamiltonian:Magnetization in q-space18

RPA: susceptibility
Kubo formula
Correspondence
Susceptibility

To calculate above, let us consider a
Green’s function

19

RPA: susceptibility (2)Hubbard Hamiltonian

Mean field approximation

Random phase approximation (RPA)20RPA: susceptibility (3)In paramagnetic state:

Fourier transformation:
Fermi distributionfunction
Summation on k

Susceptibility of noninteracting system:

unit

21

Susceptibility of non-interacting system
Wavenumber unit: 𝑘𝐹

w

Energy unit: 𝐸𝐹

Mathematical identity:

2

I

II

1

III

2

IV

0
1

q

Boundary of Kohn anomaly:
22

Kohn anomaly, Stoner condition, SDW
Re[𝜒 0 𝑞, 0 ]In region III

1
𝜌
2

→

0

1

2

3

(𝑞 → 0)4qStoner condition

𝜌(𝜖𝐹 )
2

Magnetic order
Spin density wave (SDW)23

Summary➢ Magnetism in 3d transition metals• Slater-Pauling’s curve• Density of states by APW method➢ Kubo formula (linear response)➢ Fluctuation dissipation theorem➢ Random phase approximation (RPA)

2022.7.13 Lecture 14
10:25 – 11:55

Lecture on

Magnetic Properties of Materials

磁性 (Magnetism)
Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Notification

Deadline for exercise 0629 is now 21st July.

Problems for the final report will be uploaded in the eveningof 14 July.
The deadline for the submission of report is 2nd August.

2

Review Magnetism in 3d transition metals• Slater-Pauling’s curve• Density of states by APW method Kubo formula (linear response) Fluctuation dissipation theorem Random phase approximation (RPA)3

Outline Paramagnon theory
 Self-consistent renormalization spin-fluctuation
theory4Why and how we consider magnons in marginally paramagnetic metals?
Itinerant electron magnetism• Hartree-Fock approximation for jellium model• HFA for Hubbard Hamiltonian overestimation of stability of ferromagnetism

 Some successes: Explanation of Slater-Pauling curve Still has the overestimation problem• Dynamic mean field approximation by random phase approximation

 Curie-Weiss law cannot be reproduced
 Finding of spin-density-wave (SDW) i.e. existence of spin
fluctuation (magnon)
Hypothesis to improve the approximation: Spin fluctuations exist in thermal equilibrium and lower theenergy of marginally paramagnetic states
Agenda: Hellmann-Feynman theorem to treat the effect of fluctuation, fluctuation-dissipation theorem
5Paramagnons in “nearly ferromagnetic” materials
Resonant inelastic x-ray scatteringb-FeSe iron-based superconductor
TC 100 K in monolayer(?) anti-ferromagnetic order at 90 K

Rhan et at., PRB 99, 014505 (2019).

6

Hellmann-Feynman theorem
Hamiltonian with parameter p
Normalized eigenstateswith eigenenergy

Variation in an eigenstate |𝑝, 𝑛 caused by a small
variation 𝛿𝑝 in p is expressed as a linear
combination of |𝑝, 𝑚
Linear approximation

Then taking the inner productTherefore 𝑐𝑛 = 0 from the normalization condition. Hence 𝐶𝑛 = 0 within the linear approximation in 𝛿𝑝.
Within linear in 𝛿𝑝(Other contribution should be in the second order of 𝛿𝑝.)7

Hellmann-Feynman theorem (2)Then the shift in the eigenenergy is given by

Hellmann-Feynman theorem
Free energy of the system under consideration: 𝐹(𝑝)8Paramagnon theory
Hamiltonian

Interaction Hamiltonian with interaction parameter I

Introduction of interaction 𝐼′ : 0 → 𝐼

We consider paramagnon (spin-fluctuation) contribution to specific heat
Hubbard Hamiltonian
Fourier expansion

Interaction parameter
Up/down operators
9

Paramagnon theory (2)The interaction Hamiltonian canbe developed as

Fermioncommutation relation

Change of summation
representation

10

Paramagnon theory (3)does not change by spin inversion in paramagnetic state

Then can be written in a form
anti-commutation relation
Variation of free energy

Remember retarded Green’s function
By writing a parallel expression for anadvanced Green’s function, we can obtain11

Paramagnon theory (4)Fluctuation-dissipation
Linear response
Let |𝑛 be a many-body eigenstate with eigenenergy 𝐸𝑛

Boltzmann factor
See Appendix 14A for the derivation of the above equation

Multiply both sides withand integrate with 𝜔

12Paramagnon theory (5)Then from

RPA expression

13

Paramagnon theory (6)In order for calculation of specific heat we pick up temperature-dependent part from the free energy variation.
w

Paramagnon zero-point motion and ignored

2

I

II

Contribution from region-III is the largest, where we can expand

1

III
0

2

1

IV

q

Wavenumber unit: 𝑘𝐹
Cutoff 𝑞𝑐 ~ 1

Energy unit: 𝐸𝐹
14

Paramagnon theory (7)Low temperature approximation

Because the free energy is proportional to 𝑇 2
we can write

Free electron expression

Logarithmic divergence for the Stoner condition15

Self-consistent renormalization spin fluctuation theory
In paramagnon theory, we take the effect of spin-fluctuation (magnon) into account.
However, the effect of magnons should be reflected back to magnons and they should be selfconsistent.
Otherwise, we cannot treat ferromagnetic cases, in which spontaneous magnetization appears.
Free energy in the presence of magnetization

Non-interacting

Magnetic equation of state

Zeeman→ determines spontaneous magnetization

HF approximation is expressedin these terms

16

SCR-SF theory (2)Non-interacting starting point derivative:

where

We apply RPA to 𝜒±−

17SCR-SF theory (3)To obtain magnetic equation of state, we take differentiation by 𝑚 = 𝑀/𝑁Magnetic equation of state for non-interacting systemthen

In paramagnetic caseSquare of spin-fluctuation
Ignored18

SCR-SF theory (4)Coupling constantApplication of RPA → Breakdown of self-consistency
𝑇=0
Correction of overestimation of stability in ferromagnetic state
Ignore magnon zero-point
motionExpansion around (0,0)

Self-consistent
determination of 𝜒
19

Improvements by SCR-SF theory

Molecular field approximation (×1/5)

Temperature dependence of susceptibility

Critical temperature vs interaction parameter20

Problems in SCR-SF theory

Makoshi & Moriya, JPSJ 38, 10 (1975).

Takeuchi & Masuda, JPSJ 46, 468 (1979).

21

Enhanced thermopower due to spin fluctuation
Weak ferromagnet
Spin fluctuation enhancement around TC
Enhancement of entropy
Magnon drag effectTsuji et al. Science Advances 5, eaat5935 (2019).

22

Lecture review
Chapter 1 Basic Notions of Magnetism
Breakdown of classical magnetism: cancellation of paramagnetic and diamagnetic terms(Bohr-van Leeuwen theorem)
Quest for the sources of magnetic dipoles in materials.
Spins and spin-orbit interactions

Chapter 2 Magnetism of Localized Electrons
Electronic states of magnetic ions

 LS (j-j) coupling, Hund’s rule
 Ligand field

Magnetic resonance Spin Hamiltonian23

Lecture review
Chapter 3 Magnetism of conduction electrons
Pauli paramagnetism
Landau diamagnetism

Chapter 4 Interaction between spins
Exchange interaction

 Heisenberg Hamiltonian Hubbard modelSuperexchange interaction

 Spin Hamiltonian

Chapter 5 Theory of magnetic insulators

Chapter 6 Magnetism of itinerant electrons24Thank you

Have a nice summer vacation

Notification

Deadline for exercise 0629 is now 21st July.

Problems for the final report will be uploaded in the eveningof 14 July.
The deadline for the submission of report is 2nd August.

26