Atomic Magnetic Moment Virginie Simonet Virginie.Simonet@Neel.Cnrs.Fr

Home , Electron magnetic moment, Magnetic moment, Van Vleck paramagnetism

Atomic Magnetic Moment Virginie Simonet [email protected]

Institut Néel, CNRS & Université Grenoble Alpes, Grenoble, France Fédération Française de Diffusion Neutronique

Introduction One-electron magnetic moment at the atomic scale Classical to Quantum Many-electron: Hund’s rules and spin-orbit coupling Non interacting moments under magnetic field Diamagnetism and paramagnetism Localized versus itinerant electrons Conclusion

ESM 2019, Brno 1 Introduction

A bit of history Lodestone-magnetite Fe3O4 known in antic Greece and ancient China (spoon-shape compass) Described by Lucrecia in de natura rerum

Medieval times to seventeenth century: Pierre de Maricourt (1269), B. E. W. Gilbert (1600), R. Descartes (≈1600)… Gilbert Properties of south/north poles, earth is a magnet, compass, perpetual motion

ted experiment (1820 - Copenhagen)

Modern developments: H. C. Oersted, A. M. Ampère, M. Faraday, J. C. Maxwell, H. A. Lorentz… Unification of magnetism and electricity, field and forces description Oersted 20th century: P. Curie, P. Weiss, L. Néel, N. Bohr, W. Heisenberg, W. Pauli, P. Dirac… Para-ferro-antiferro-magnetism, molecular field, domains, (relativistic) quantum theory, spin… Dirac

ESM 2019, Brno 2 Introduction

Magnetism:

science of cooperative effects of orbital and spin moments in matter

Wide subject expanding over physics, chemistry, geophysics, life science.

At fundamental level: Inspiring or verifying lots of model systems, especially in theory of phase transition and concept of symmetry breaking (ex. Ising model)

Large variety of behaviors: dia/para/ferro/antiferro/ferrimagnetism, phase transitions, spin liquid, spin glass, spin ice, skyrmions, magnetostriction, magnetoresistivity, magnetocaloric, magnetoelectric effects, multiferroism, exchange bias… in different materials: metals, insulators, semi-conductors, oxides, molecular magnets,.., films, nanoparticles, bulk...

Magnetism is a quantum phenomenon but phenomenological models are commonly used to treat classically matter as a continuum

Many applications in everyday life

ESM 2019, Brno 3 Introduction

Magnetic materials all around us : the earth, cars, audio, video, telecommunication, electric motors, medical imaging, computer technology…

Flat Disk Rotary Motor

Hard Disk Drive

Write Head Voice Coil Linear Motor

Read Head

Discrete Components : Transformer Filter Inductor

ESM 2019, Brno 4 Introduction

Topical research fields in magnetism

• Magnetic frustration: complex magnetic (dis)ordered ground states • Molecular magnetism: photo-switchable, quantum tunneling • Mesoscopic scale (from quantum to classical)  quantum computer • Quantum phase transition (at T=0)

• Low dimensional systems: Haldane, Bose-Einstein condensate, Luttinger liquids de • Magnetic topological matter • Multiferroism: coexisting ferroic orders (magnetic, electric…) • Magnetism and superconductivity • Nanomaterials: thin films, multilayers, nanoparticles 1.0 90 nm • Spintronics: use of the spin of the electrons in electronic devices Magnetoscience … • Skyrmionics: new media for encoding information • Magnetic fluids: ferrofluids • Magnetoscience: magnetic field effects on physics, chemistry, biology … !"#$"%#&

ESM 2019, Brno 5 Atomic magnetic moment: classical

✔ An electric current is the source of a magnetic field B µ0 Id~l ~r −7 ~ = × 0 4 = 10 B 2 µ / π 4⇡ ZC r r ~ ✔ Magnetic moment/magnetic field generated by a single-turn coil ~ dB dl B

Magnetic moment µ0 (~m.~r)~r ~m B~ = m=[3 − ] with ~m = IS~n 4⇡ r5 r3

ESM 2019, Brno 6 Atomic magnetic moment: classical

✔ Orbiting electron is equivalent to a magnetic moment

⇥ ⇥ −ev 2 −evr µ⇥ ` = I.S = πr ⇥n = ⇥n Nucleus Ze 2πr 2 e- orbiting around the nucleus

ESM 2019, Brno 7 Atomic magnetic moment: classical

✔ Orbiting electron is equivalent to a magnetic moment

⇥ ⇥ −ev 2 −evr µ⇥ ` = I.S = πr ⇥n = ⇥n Nucleus Ze 2πr 2 ✔ The magnetic moment is related to the angular momentum e- orbiting around the nucleus L! = !r × p! = !r × m!v −e µ⇥ ` = L⇥ = γL⇥ 2m Orbital magnetic moment

Gyromagnetic ratio gyroscope

https://en.wikipedia.org/ Wiki/Angular_momentum

ESM 2019, Brno 8 Atomic magnetic moment: classical

✔ Orbiting electron is equivalent to a magnetic moment

⇥ ⇥ −ev 2 −evr µ⇥ ` = I.S = πr ⇥n = ⇥n 2πr 2 ✔ The magnetic moment is related to the angular momentum L! = !r × p! = !r × m!v −e µ⇥ ` = L⇥ = γL⇥ 2m

https://fr.wikipedia.org/wiki/Effet_Einstein-de_Haas Einstein-de Haas effect (1915): suspended ferromagnetic rod magnetized by magnetic field  rotation of rod to conserve total angular momentum

ESM 2019, Brno 9 Atomic magnetic moment: classical

✔ The magnetic moment is related to the angular momentum: consequence Larmor precession. 1.1 Magnetic moments 3 induced by rotation. BothEnergy phenomen Ea =demonstrat−~µ . B~ e that magnetic moments are associated with angular momentum. Torque applied to the moment G~ = ~µ × B~

1.1.2 Precession Equation of motion  We now consider a magnetiVariationc momen of thet magneticu in a magnetimomentc (hyp. field no B dissipation) as shown in Fig. 1.3. The energy E of the magnetic moment is given by d~µ = γ~µ × B~ Fig. 1.3 A magnetic moment u in a magnetic dt field B has an energy equal to —u . B = (see Appendix B) so that the energy is minimized when the magnetic moment —uB cos 0. = lies along the magnetic✔ field The. magneticThere wil momentl be a torqu precessese G o nabout the magneti the fieldc at momen the Larmort frequency ωL |γ|B given by 1 For an electric dipole p, in an electric field £, the energy is £ = — p . E and the t o r q u e (see Appendix B) which, if the magnetic moment were noESMt associate 2019, Brnod with is G = 10 p x E. A stationary electric dipole any angular momentum, would tend to turn the magnetic moment towards the moment is j u s t two separated stationary elec- magnetic field.1 tric charges; it is not associated with any angular momentum, so if £ is not aligned However, since the magnetic moment is associated with the angular mo- with p, the torque G will tend to turn p mentum L by eqn 1.3, and because torque is equal to rate of change of angular towards E. A stationary magnetic moment momentum, eqn 1.5 can be rewritten as is associated with angular m o m e n t u m a n d so behaves differently.

2Imagine a top spinning w i t h its axis inclined This means that the change in u is perpendicular to both u and to B. Rather to the vertical. The weight of the top, acting than turning u towards B, the magnetic field causes the direction of u to downwards, exerts a (horizontal) torque on the top. If it were not spinning it would just precess around B. Equation 1.6 also implies that \u\ is time-independent. Note fall over. But because it is spinning, it has that this situation is exactly analogous to the spinning of a gyroscope or a angular momentum parallel to its spinning spinning top.2 axis, and the torque causes the axis of the In the following example, eqn 1.6 will be solved in detail for a particular spinning top to move parallel to the torque, in a horizontal plane. The spinning top pre- case. cesses.

Example 1.1 Consider the case in which B is along the z direction and u is initially at an angle of 6 to B and in the xz plane (see Fig. 1.4). Then

so that uz is constant with time and ux and uy both oscillate. Solving these Fig. 1.4 A magnetic moment u in a magnetic differential equations leads to field B precesses around the magnetic field at the Larmor precession frequency, y B, where y is the gyromagnetic ratio. The magnetic field B lies along the z-axis and the m a g n e t i c moment is initially in the xz-plane at an angle 0 to B. The magnetic moment precesses around a cone of semi-angle 0. where is called the Larmor precession frequency. Joseph Larmor (1857-1942) Atomic magnetic moment: classical to quantum

Consequences:

✔ Orbital motion, magnetic moment and angular momentum are antiparallel

✔ Calculations with magnetic moment using formalism of angular momentum

No work produced by a magnetic field on a moving e- ~ − × ~ hence a magnetic field cannot modify its energy f = e(~v B) and cannot produce a magnetic moment.

ESM 2019, Brno 11 Atomic magnetic moment: classical to quantum

Consequences:

✔ Magnetic moment and angular momentum are antiparallel

✔ Calculations with magnetic moment using formalism of angular momentum

In a classical system, there is no thermal equilibrium magnetization! (Bohr-van Leeuwen theorem)

 Need of quantum mechanics QUANTUM MECHANICS THE KEY TO UNDERSTANDING MAGNETISM Nobel Lecture, 8 December, 1977

J.H. VAN VLECK Harvard University, Cambridge, Massachusetts, USA

ESM 2019, Brno 12

L(x) :x, Atomic magnetic moment: classical to quantum

Reminder of Quantum Mechanics

ˆ ˆ ˆ ∗ ˆ Wavefunction ψ and operator A Aφi = aiφi hAi = dτψ Aψ 2 ∗ Z |ψ| = ψ ψ = ˆ 2 ψ X ciφi hAi = X |ci| ai i i

Commutator [A,ˆ Bˆ]=AˆBˆ − BˆAˆ ˆ dψ ˆ Schrödinger equation Hψ = i~ Hφi = Eiφi dt ˆ ˆ ˆ ˆ Angular momentum operator ~L~ = ~r ⇥ ~p = −i~~r ⇥r ˆ 2 ˆ |hφi|V |φki| Perturbation theory E ⇡ Ek + hφk|V |φki + X Ek − Ei i6=k

ESM 2019, Brno 13 Atomic magnetic moment: quantum

Magnetism in quantum mechanics:

Distribution of electrons on atomic orbitals, which minimizes the energy  Building of atomic magnetic moments Ψ The electronic wavefunction n`m ` is characterized by 3 quantum numbers (spin ignored) ~2 Ze2 `l=0 H = − r2 − Eigenstate 2me 4⇡✏0r HΨi = EiΨi `l=1 m go Ψn`m` (~r )=Rn`(r).Y` (✓, φ) ` Radial part Spherical l=2 harmonics

ESM 2019, Brno 14 Atomic magnetic moment: quantum

Magnetism in quantum mechanics:

: principal quantum number (electronic shell) : orbital angular momentum quantum number

: magnetic quantum number

ESM 2019, Brno 15 Atomic magnetic moment: quantum

Magnetism in quantum mechanics: quantized orbital angular momentum

~ ` is the angular momentum operator 2 Electronic orbitals are eigenstates of ` and `z

`2 ~2` ` Ψn`m` = ( + 1)Ψn`m` ` = ~m Ψ z ` n`m`

The magnitude of the orbital momentum is

The component of the orbital angular momentum along the z axis is

ESM 2019, Brno 16 Atomic magnetic moment: quantum

Magnetism in quantum mechanics: quantized orbital angular momentum ` ~ Ex. vector model for l=2 ` is the angular momentum operator 2 Electronic orbitals are eigenstates of ` and `z

`2 ~2` ` Ψn`m` = ( + 1)Ψn`m` ` = ~m Ψ z ` n`m`

The magnitude of the orbital momentum is

The component of the orbital angular momentum along the z axis is

Degeneracy 2 ` +1 , can be lifted by magnetic field (Zeeman effect)

ESM 2019, Brno 17 Atomic magnetic moment: quantum

Magnetism in quantum mechanics: spin angular momentum of pure quantum origin ~s

2 2 s Ψs = ~ s(s + 1)Ψs Nucleus Ze szΨs = ~msΨs

With the quantum numbers :

The magnitude of the spin angular momentum is

The component of the spin angular momentum along the z axis is

ESM 2019, Brno 18 Atomic magnetic moment: quantum

Magnetism in quantum mechanics: spin angular momentum of pure quantum origin ~s

z s2 ~2s s Ψs = ( + 1)Ψs M S H - 1/2 s = 1/2 -1/2 -ħ/2 szΨs = ~msΨs 2 H µ0µB g√[s(s+1)]ħ2 - 1/2 ħ/2 With the quantum numbers : 1/2

The magnitude of the spin angular momentum is

The component of the spin angular momentum along the z axis is

Degeneracy 2 s +1=2 , can be lifted by magnetic field

ESM 2019, Brno 19 Atomic magnetic moment: quantum

Magnetism in quantum mechanics:

Magnetic moment associated to 1 electron in the atom Two contributions: spin and orbit

Nucleus Ze Magnetic moments

~µ ` = −g`µB~l ~µ s = −gsµB~s

) −3 With and =2+O(10 )

~e −24 −1 and the Bohr magneton µB = =9.27.10 J.T 2me

ESM 2019, Brno 20 Atomic magnetic moment: quantum

Magnetism in quantum mechanics: several e- in an atom What are the ground state values of J, L and S ? (How to fill the atomic shells 1 règle: Distributions des électrons sur tous les états d'orbit

m ? 2 ml -3 -2 -1 0 1 2 3 Ex : 4f S m szs + principe de Pauli projection maximale de spi Semi-empirical rules: Pauli principle, minimize Coulomb interaction and finally spin-

ESM 2019, Brno 21 Atomic magnetic moment: quantum

Magnetism in quantum mechanics: several e- in an atom

~ ~ ~ L = X ` S = X ~s

ne− ne−

Combination of the orbital and spin angular momenta of the different electrons: related to the filling of the electronic shells in order to minimize the electrostatic energy and fulfill the Pauli exclusion principle (one e- at most in quantum state)

ESM 2019, Brno 22 Atomic magnetic moment: quantum

Magnetism in quantum mechanics: several e- in an atom

~ ~ ~ L = X ` S = X ~s

ne− ne−

Spin-orbit coupling: λL.~ S~

Total angular momentum J~ = L~ + S~

A given atomic shell (multiplet) is defined by 4 quantum numbers : L, S, J, M J with −J

ESM 2019, Brno 23 Atomic magnetic moment: quantum

Origin of spin-orbit coupling: change of rest frame + special relativity B~ Beffnuc ! ! +Ze ! +Ze r !v S -e -e ! me v Ch E~ × ~v ~r dV (r) B~ = with E~ = − lectron feels an effective magneticnuc c2 r dr 1 dV (r) ~ dV (r) Hso = µBS.~ B~ nuc = µB S~(~r . × ~v )=µB S.~ L~ c2r dr mc2r dr ~ ~ Spin-orbit Hamiltonian Hso = λS.L

λ increases with atomic number Z4 Consequence: m ` and m s are no longer good quantum numbers ( L, S, J, M J instead)

ESM 2019, Brno 24 Atomic magnetic moment: quantum

Magnetism in quantum mechanics: several e- in an atom

Hund’s rules for the ground state

1st rule maximum

2nd rule maximum in agreement with the 1st rule

~ ~ 3rd rule from spin-orbit coupling Hso = λS.L

for less than ½ filled shell for more than ½ filled shell

ESM 2019, Brno 25 Atomic magnetic moment: quantum

Magnetism in quantum mechanics: several e- in an atom

Hund’s rules for the ground state

1st rule maximum

2nd rule maximum in agreement with the 1st rule

~ ~ Atomic term 3rd rule from spin-orbit coupling Hso = λS.L labeling the ground state: 2S+1 LJ

for less than ½ filled shell for more than ½ filled shell L = S, P, D... With Degeneracy 2J+1, can be lifted by a magnetic field

ESM 2019, Brno 26 Atomic magnetic moment: quantum

Application of Hund’s rule : L and S are zero for filled shells

Ex. Lu3+ is 4f14, 14 electron to put in 14 boxes ( = 3)

↓ ↓ ↓ ↓ ↓ ↓

↓ ↓ ↓ ↓ ↓ ↓ ↓

so L = 0 and S = 0, J = 0

ESM 2019, Brno 27 Atomic magnetic moment: quantum

Application of Hund’s rule : L and S are zero for filled shells

Example of unfilled shell

Ce3+ is 4f1, 1 electron to put in 14 boxes ( = 3)

so L = 3 and S = 1/2

The spin-orbit coupling applies for less than ½ filled shell so J = 5/2 and -5/2 < MJ < 5/2 E = LS LS = λ L~ .S~ L~ .S~ so h |Hso| i h i i− h i i  k k # The ground state is 6-fold degenerate Xi,up i,downX

ESM 2019, Brno 28 Atomic magnetic moment: quantum

Application of Hund’s rule : L and S are zero for filled shells

Example of unfilled shell

Tb3+ is 4f8, 8 electrons to put in 14 boxes ( = 3)

so L = 3 and S = 3

The spin-orbit coupling applies for more than ½ filled shell so J = 6 and -6 < MJ < 6 E = LS LS = λ L~ .S~ L~ .S~ so h |Hso| i h i i− h i i  k k # The ground state is 13-fold degenerate Xi,up i,downX

ESM 2019, Brno 29 Atomic magnetic moment: quantum

The Zeeman interaction:

Hyp. H Z << H so Wigner-Eckart theorem (projection theorem), 1st order perturbation theory

~µ = −µB(L~ +2S~)

Hz = µB(L~ +2S~).B~ ≈ µBgJ J.~ B~ In the J multiplet basis

J(J + 1) + S(S + 1) − L(L + 1) With the Landé g -factor gJ =1+ J 2J(J + 1)

Hence the Zeeman energy is E z = g J µ B M J B with M J ∈ {−J, J} and the level separation with ∆ M J = ± 1 is gJ µBB of the order of 1 K ≈ 0.1 meV

ESM 2019, Brno 30 Atomic magnetic moment: quantum

Summary: Total magnetic moment of the unfilled shell

~µ = −µB(L~ +2S~)

µ = gJ µBpJ(J + 1)

~µ J = −gJ µBJ~

3+ Example Tb , J=6, gJ= 3/2 so that the magnetic moment is 9 µB

ESM 2019, Brno 31 Atomic magnetic moment: quantum

Summary of atomic magnetism

 Electrons in central potential (nucleus + central part of e-e interactions)  fundamental electronic configuration, degeneracy 2n2, energies labeled by n ΔE≈108 K  Add non-central part of e-e interactions separates the energies in different terms labeled by (L, S), degeneracy (2L+1)(2S+1) ΔE≈104 K  Add spin-orbit coupling  each term decomposed in multiplets characterized by J (and L, S), degeneracy 2J+1 ΔE≈100-1000 K for 3d ions, 1000-10000 K for 4f ions  Add magnetic field: lift multiplet’s degeneracy, ΔE≈1 K

ESM 2019, Brno 32 Atomic magnetic moment: quantum

Summary of atomic magnetism Ex. free ion Co2+ 3d7, S=3/2, L=3, J=9/2

= 2, 10 boxes to fill

Ground state

↓ ↓

ESM 2019, Brno 33 Atomic magnetic moment: quantum

Summary of atomic magnetism Ex. free ion Co2+ 3d7, S=3/2, L=3, J=9/2 Fundamental L = 1 S = M configuration J free J = L − S = − H 1st, 2nd Hund’s. rules ee((eV) − (2S+1)(2L+1)

L = 3 S = 3rd Hund’s rules Ground state − Hso (10 meV)

J = L + S = (2J+1) ↓ ↓ − HZ (0.1 meV)

ESM 2019, Brno 34 Atomic magnetic moment in matter

Magnetism is a property of unfilled electronic shells: Most atoms (bold) are concerned but ≈22 are magnetic in condensed matter Magnetic Periodic Table

ESM 2019, Brno 35 Atomic magnetic moment in matter

Atom in matter:

✔ Chemical bonding  filled e- shells : no magnetic moments

H H H H

- e e-

Magnetic Non Magnetic

ESM 2019, Brno 36 Atomic magnetic moment in matter

Atom in matter:

✔ Chemical bonding  filled e- shells : no magnetic moments, exceptions:

Rare-earth element Transition-metal element

4f electrons: inner shell (localized moment) 3d electrons: outer shell (more delocalized, less screened)

ESM 2019, Brno 37 Atomic magnetic moment in matter

Validity of empirical Hund’s rules:

L-S (Russel-Saunders) coupling scheme assumes spin-orbit coupling << electrostatic interactions: L and S combined separately, then apply spin-orbit coupling.

No more valid for high Z (large spin-orbit coupling) j-j coupling scheme: s and ` coupled first for each e-, then couple each electronic j.

ESM 2019, Brno 38 34 Isolated magnetic moments

Table 2.2 Magnetic ground states for 4f ions using Hund's rules. For each ion, the shell configuration and the predicted values of S, L and J for the ground state are listed. Also shown is the = 1 2 calculated value of p = ueff /uB 8J [ J(J + 1)] / using these Hund's rules predictions. The next column lists the experimental value pexp and shows very good agreement, except for Sm and Eu. The experimental values are obtained from measurements of the susceptibility of paramagnetic salts at temperatures kBT » ECEF a where ECEF is crystal f i e l d energy.

ion shell S L J term p Pexp

Ce3+ 4f 1 5 2 2.54 2.51 2 3 F5/2 Pr3+ 2 3 4f 1 5 4 H4 3.58 3.56

3 3 3 9 4 Nd + 4f 6 I9/2 3.62 3.3-3.7 3 4 5 Pm + 4f 2 6 4 I4 2.68 - Sm3+ 5 5 5 6 1.74 4f 5 1 I5/2 0.85 6 3 4f 7 Eu + 3 3 0 F0 0.0 3.4 3 7 7 7 8 Gd + 4f 0 S7/2 7.94 7.98 3 8 7 Tb + 4f 3 3 6 F6 9.72 9.77 3 9 5 15/2 6 Dy + 4f 5 H15/2 10.63 10.63 10 3 4f 5 Ho + 2 6 8 I8 10.60 10.4 3 ll 3 Er + 4f 6 4 9.59 9.5 ¥ Il5/2 3 l2 3 Tm + 4f 1 5 6 H6 7.57 7.61 3 13 7 Yb + 4f 2 1/2 3 F7/2 4.53 4.5 l4 3 4f 1 ! Lu + 0 0 0 S0 0 0 Atomic magnetic moment in matter 13 The present chapter deals only with free the 3d ions). S rises and becomes a maximum i n the middle of each group. L atoms or ions. Things will change when the and 7 have maxima at roughly t h eValidity quarter an ofd empirical three-quarte Hund’sr p o s i t i o rules: n s , a l t h ogood u g h for 4f atoms are put in a crystalline environment. for J there is an asymmetry between these maxima which reflects the differing The changes are quite large for 3d ions, as may be seen in chapter 3. rules for being in a shell which is less than or more than half full. Bilan de l'a peff = gJ pJ(J + 1)µB ions: ique

Fig. 2.14 S, L and J for 3d and 4f ions total according to Hund's rules. In these graphs « is the number of electrons in the subshell (3d és or 4f). J = L+S La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu3+ J = L+S-1 From eqn 2.44 we have found that a measurement of the susceptibility allows one to deduce the effective moment. This effective moment can bexcepte forEu, Sm: contribution from higher (L, S) levels expressed in units of the Bohr magneton uB as

ESM 2019, Brno 39 34 Isolated magnetic moments

value pexp and shows very good agreement, except for Sm and Eu. The experimental values are obtained from measurements of the

susceptibility of paramagnetic salts at temperatures kBT » ECEF a where ECEF is crystal f i e l d energy.

ion shell S L J term p Pexp

Ce3+ 4f 1 5 2 2.54 2.51 2 3 F5/2 Pr3+ 2 3 4f 1 5 4 H4 3.58 3.56

3 3 3 9 4 Nd + 4f 6 I9/2 3.62 3.3-3.7 3 4 5 Pm + 4f 2 6 4 I4 2.68 - Sm3+ 5 5 5 6 1.74 4f 5 1 I5/2 0.85 6 3 4f 7 Eu + 3 3 0 F0 0.0 3.4 3 7 7 7 8 Gd + 4f 0 S7/2 7.94 7.98 3 8 7 Tb + 4f 3 3 6 F6 9.72 9.77 3 9 5 15/2 6 Dy + 4f 5 H15/2 10.63 10.63 10 3 4f 5 Ho + 2 6 8 I8 10.60 10.4 3 ll 3 Er + 4f 6 4 9.59 9.5 ¥ Il5/2 3 l2 3 Tm + 4f 1 5 6 H6 7.57 7.61 3 13 7 Yb + 4f 2 1/2 3 F7/2 4.53 4.5 l4 3 4f 1 Lu + 0 0 0 S0 0 0 Atomic magnetic moment in matter 13 The present chapter deals only with free the 3d ions). S rises and becomes a maximum i n the middle of each group. L atoms or ions. Things will change when the Validity ofan empiricald 7 have maxim Hund’sa arules:t roughl goody t h e for quarte 4f butr an lessd three-quarte good for 3dr (duep o s i tto i o ncrystal s , a l t h ofield) u g h atoms are put in a crystalline environment. for J there is an asymmetry between these maxima which reflects the differing The changes are quite large for 3d ions, as may be seen in chapter 3. rules for being in a shell which is less than or more than half full.

For 3d ions works better if J is replaced by S (influence of crystal field)

Fig. 2.14 S, L and J for 3d and 4f ions according to Hund's rules. In these graphs « is the number of electrons in the subshell (3d or 4f). Sc Ti V Cr Mn Fe Co Ni Cu Zn2+ From eqn 2.44 we have found that a measurement of the susceptibility allows one to deduce the effective moment. This effective moment can be expressed in units of the Bohr magneton uB as

ESM 2019, Brno 40 Atomic magnetic moment in matter

Summary

ESM 2019, Brno 41 Assembly of non-interacting magnetic moments

Measurable quantities:

Magnetization : magnetic moment per unit volume (A/m) derivative of the free energy w. r. t. the magnetic field

∂F M = −

∂B

Susceptibility: derivative of magnetization w. r. t. magnetic field, alternatively, ratio of the magnetization on the field in the linear regime (unitless) ∂M M χ = µ0 ≈ µ0! " ∂B B lin −7 µ0 =4π10

ESM 2019, Brno 42 Assembly of non-interacting magnetic moments

N atomic moments in a magnetic field B !" !" B~ =0 =0K B 0K B

a field B, how will the : saturated magnetiz Non-interacting At T=0 K

ESM 2019, Brno 43 Assembly of non-interacting magnetic moments

N atomic moments in a magnetic field B !" !" B~ =0 =0K B 0K B

a field B, how will the : saturated magnetiz Non-interacting At T=0 K

Calculation of magnetization and susceptibility Thermal average (Boltzmann statistics) + perturbation theory

with β =1/kBT

ESM 2019, Brno 44 Assembly of non-interacting magnetic moments

One atomic moment in a magnetic field B

Z 2 (~p i − eA~(~r i)) H = + Vi(ri) + gµBB.~ S~ B~ = r⇥A~ 2me Xi=1 ✓ ◆ ~ ~ B × ~r With the magnetic vector potential (Coulomb gauge) A(~r )= 2 2 2 pi e 2 H = + Vi(ri) + µB(L~ +2S~).B~ + (B~ × ~r i) 2me 8me Xi ✓ ◆ Xi

Zeeman hamiltonian: coupling of total magnetic moment with the magnetic field

Diamagnetic hamiltonian: induced orbital moment by the external magnetic field

ESM 2019, Brno 45 22 Isolated magnetic moments

Assembly of non-interacting magnetic moments

2 e 2 Energy: HEBB = µB(L~ +2S~).B~ + X(B~ × ~r i) 8me i

Fig. 2.2 The measured diamagnetic molar Diamagnetic term for N atoms:susceptibilitie s Xm of various ions plotted 2 against Zeffr , where Zeff is the number of electrons in the ion and r is a measured ionic radius. 2 N e 2 χ = − µ0 V 4me perpendicular to the field due to the induced moment by the magnetic field

 Larmor diamagnetism  negative weak susceptibility, concerns all e- of the atom, T-independent Fig. 2.3 (a) Naphthalene consists of two  Large anisotropic diamagnetismfused benzene rings found. (b) Graphit ine consist planars systems of sheets of hexagonal layers. The carbon atoms are shown as black blobs. The carbon with delocalized e- (ex. graphite,atoms are in registry ibenzene)n alternate, not adjacen t planes (as shown by the vertical dotted lines).

ESM 2019, Brno 46 The effective ring diameter is several times larger than an atomic diameter and so the effect is large. This is also true for graphite which consists of loosely bound sheets of hexagonal layers (Fig. 2.3(b)). The diamagnetic susceptibility is much larger if the magnetic field is applied perpendicular to the layers than if it is applied in the parallel direction. Diamagnetism is present in all materials, but it is a weak effect which can either be ignored or is a small correction to a larger effect. Assembly of non-interacting magnetic moments

2 e 2 Energy: HEBB = µB(L~ +2S~).B~ + X(B~ × ~r i) 8me i

Paramagnetic term for N atoms :

with

and the Brillouin function:

ESM 2019, Brno 47 Assembly of non-interacting magnetic moments

Paramagnetic term

Brillouin functions for different J values,

Classical limit

Limit x >> 1 i.e. B >> kBT N https://fr.wikipedia.org/wiki/ Saturation magnetization Ms = gJ JµB V Fichier:Brillouin_Function.svg

ESM 2019, Brno 48 Assembly of non-interacting magnetic moments

Paramagnetic term

(J + 1)x 3 Limit x << 1 i.e. kBT >> B BJ (x)= + O(x ) 3J 2 2 N (µBgJ ) J(J + 1) N peff C Curie law: χ = = = V 3kBT V 3kBT T

1/χ with C the Curie constant χ and the effective moment

peff = gJ pJ(J + 1)µB

It works well for magnetic moments without interactions and negligible CEF: 3+ 3+ 2+ ex. Gd , Fe , Mn (L=0) T (K)

ESM 2019, Brno 49 Assembly of non-interacting magnetic moments

Summary of magnetic field response of non-interacting atomic moments

versus magnetic field versus temperature

M Paramagnetic χ Paramagnetic – Curie law

B T

Diamagnetic Diamagnetic – independent of temperature

Rmq: Another source of paramagnetism (2nd order perturbation theory, mixing with excited states) Van Vleck paramagnetism  weak positive and temperature independent

ESM 2019, Brno 50 Assembly of non-interacting magnetic moments

Summary20 Isolated of magneticmagnetic moments field response of non-interacting atomic moments

paramagnetic diamagnetic

Fig. 2.1 The mass susceptibility of the first 60 elements in the periodic table at room temperature, plotted as a function of the atomic number. Fe, Co and Ni are ferromagnetic so that they have a spontaneous magnetization with no applied magnetic field. ESM 2019, Brno 51 2.3 Diamagnetism

3 The prefix dia means 'against' or 'across' All materials show some degree of diamagnetism, a weak, negative mag- (and leads to words like diagonal and diame- netic susceptibility. For a diamagnetic substance, a magnetic field induces a ter). magnetic moment which opposes the applied magnetic field that caused it. This effect is often discussed from a classical viewpoint: the action of a 4 electromotive force magnetic field on the orbital motion of an electron causes a back e.m.f., which by Lenz's law opposes the magnetic field which causes it. However, the Bohr- van Leeuwen theorem described in the previous chapter should make us wary of such approaches which attempt to show that the application of a magnetic 5 See the further reading. field to a classical system can induce a magnetic moment. The phenomenon of diamagnetism is entirely quantum mechanical and should be treated as such. We can easily illustrate the effect using the quantum mechanical approach. Consider the case of an atom with no unfilled electronic shells, so that the paramagnetic term in eqn 2.8 can be ignored. If B is parallel to the z axis, then B x ri = B(-yi,xi,0)and

so that the first-order shift in the ground state energy due to the diamagnetic term is 2.6 Adiabatic demagnetization 37

the system probabilistically, we use the expression:

Alternatively, the equation for the entropy can be generated by computing the Helmholtz free energy, F, via F = — N k B T In Z and then using S — -(0F/0T)B. Let us now explore the consequences of eqn 2.59. In the absence of an applied magnetic field, or at high temperatures, the system is completely disordered and all values of mJ are equally likely with probability p(mJ ) = 1/(2J + 1) so that the entropy 5 reduces to Assembly of non-interacting magnetic moments

in agreement with eqn 2.57. As the temperature is reduced, states with low energy become increasingly probable; the degree of alignment of the Adiabatic demagnetization:magnetic moment coolings parallel to a na applie sampled magnetic downfield (the magnetization to mK) increases and the entropy falls. At low temperatures, all the magnetic moments will align with the magnetic field to save energy. In this case there is only one way of arranging the system (with all spins aligned) so W = 1 and The entropy is a monotonicallyS = 0. decreasing function of B/T The principle of magnetically cooling a sample is as follows. The param- Two steps: agnet is first cooled to a low starting temperature using liquid helium. The a-b isothermal magnetizationmagnetic coolin gby then applyingproceeds via tw oa step magnetics (see also Fig. 2.15)field.  reduces the entropy b-c Removing the magnetic field adiabatically (at constant entropy) lower the temperature

Fig. 2.15 The entropy of a paramagnetic salt as a function of temperature for several different applied magnetic fields between zero and some maximum value which we will call Bb. Magnetic cooling of a paramagnetic salt from temperature Ti to Tf is accomplished as indicated in two steps: first, isothermal magnetization from a to b by increasing the magnetic field from 0 to Bb at constant temperature Ti; second, adiabatic demagnetization from b to c. The S(T) curves have been calculated assuming J = 1 / 2 ( s e e eqn 2.76). A term oc T 3 has been added to these curves to simulate the entropy of the lattice vibrations. The curve for B = 0 is actually for B small but non-zero to simulate the effect of a small residual field.

ESM 2019, Brno 52 Itinerant electrons

Magnetism in metals

Starting point: the free electron model, properties of Fermi surface, Fermi-Dirac statistics, electronic band structure

e e Energy, Energy, Energy, Energy,

Non-interacting electron waves confined in a box

k y Fermi wavevector 2 1 kF =(3π n) 3 kF Density of states at kx Fermi level (T=0) 3n D↑,↓(EF )= 4EF Density of states, D(e) k-space: For a non-magnetic metal: Each points is a possible state same number of spins ↑ and ↓ for one spin up and down electrons at Fermi level

ESM 2019, Brno 53 Itinerant electrons

Magnetism in metals

Applying a magnetic field Spin-split bands (at T=0) by magnetic field

Ez = gµBBms  magnetization e Energy, Energy,

−4 ∆E =2µBB ≈ 10 eV EF

gµB(n ↑−n ↓) M = 2 2 M = µBD(EF )B 2 B χP = µ0µBD(EF )

Density of states, D(e) Pauli paramagnetism (effect associated to spin of e-) Temperature independent > 0, weak effect. Small correction at finite temperature ∝ T 2

ESM 2019, Brno 54 Itinerant electrons

Magnetism in metals

Applying a magnetic field

Pauli paramagnetism Orbital response of e- gas to magnetic field associated to spin of electrons 2 χP = µ0µBD(EF ) kz

e The applied magnetic field results

in Landau tubes of electronic states Energy, B = 0 n = 3 n = 2  Landau diamagnetism, n = 1 n = 0 Temperature independent < 0 Wavevector, k 2 z 1 me χL = − χP 3 ∗ ✓m ◆  Oscillations of the magnetization (de Haas-van Alphen effect)

ESM 2019, Brno 55 Conclusion

Summary

 Magnetism is a quantum phenomenon  Magnetic moments are associated to angular momenta  Orbital and Spin magnetic moments can be coupled (spin-orbit coupling) yielding the total magnetic moment (Hund’s rules)  Magnetic moment in 3d and 4f atoms have different behaviors  Various responses of non-interacting magnetic moments in applied magnetic field, different for localized or delocalized electrons: Curie-law/Pauli paramagnetism, Larmor/Landau diamagnetism

But… does not explain spontaneous magnetization/magnetic order in absence of magnetic field, 3d ions magnetism and anisotropic behaviors…  Next lectures will introduce missing ingredients: magnetic interactions and influence of the environment (crystal field)

ESM 2019, Brno 56 Further reading

• Material borrowed from presentations of D. Givord, L. Ranno, Y. Gallais, Thanks to them!

• “Magnetism in Condensed Matter” by Stephen Blundell, Oxford University press (2003)

• “Introduction to magnetism” by Laurent Ranno, collection SFN 13, 01001 (2014), EDP Sciences, editors V. Simonet, B. Canals, J. Robert, S. Petit, H. Mutka, free access DOI: http://dx.doi.org/10.1051/sfn/20141301001

• “Magnetism and Magnetic Materials” by J.M.D. Coey, Cambridge Univ. Press (2009)

• Lectures of Yann Gallais Website: www.mpq.univ-paris-diderot.fr/spip.php?rubrique260

• Any questions: [email protected]

ESM 2019, Brno 57