Hier steht der Titel der Power Point Präsentation. · (orientation) of magnetic moments (mostly...

Professur für Anorganische Chemie II MRC07 Dr. Anna Isaeva

Dr. Anna IsaevaDresden, 8th October 2018

Basics

Recommended literature

https://www.wmi.badw.de/teaching/Lecturenotes/

Dr. Martin Valldor (IFW Dresden)Lecture series „Spins Do“ (24.10.2018, Neubau Chemie, Bergstraße 66,

Seminar room 398)http://www2.cpfs.mpg.de/~valldor/Valldor-teaching.html

https://www.itp.tu-berlin.de/fileadmin/a3233/upload/SS12/TheoFest2012/Kapitel/Chapter_8.pdf

http://www2.cpfs.mpg.de/~valldor/Valldor-teaching.html

A sneak peek into my own researchin topological insulators

https://tu-dresden.de/mn/chemie/ac/ac2/forschung/topologische-materialienThe page is in English

Semiconductor‘s surface

Possible metallic surface states

Degenerated spin states Rashba effect

Spin-resolved surface states

Local spin currents


Resilient against backscattering

Topological insulator

A bulk semiconductor (insulator) with spin-resolved (metallic) surface states

that are robust against backscattering (TR symmetry preserved)

and exhibit spin-momentum locking

3D weak TI

Spin-resolved edge states

D. Kong, Yi Cui. Nature Chem. 3, 845 (2011).

3D strong TI (Balents, Moore, 2007)


HgTe/CdTe quantum wells: 2D TI

O.A. Pankratov et al., Solid State Commun. 61, 93 (1987).

B.A. Bernevig et al., Science 314, 1757 (2006).

M. König et al. Science 318, 766 (2007).

Tetradymite-type 3D strong TIs

Bi

Te

Bi2Te3

Nature Phys. 5, 438 (2009), Science 325, 178 (2009),

PRL 103, 146401 (2009).

Helical surface states (Dirac cone)

n-doped Bulk insulator

AC II, TUD: Structure variety of new 3D WTIs (W:weak)

Bi2TeI

I. Rusinov et al. Sci. Rep. 6, 20734 (2016).

A. Zeugner et al. Chem. Mater. 29, 1321 (2017).

N. Avraham et al. arxiv.org: 1708.09062

A. Zeugner et al. Chem. Mater. 30, 5272 (2018).

B. Rasche et al. Nature Mater. 12, 422 (2013).

B. Rasche et al. Chem. Mater. 25, 2359 (2013).

B. Rasche et al. Sci. Rep. 6, 20645 (2016).

C. Pauly et al. Nature Phys. 11, 338 (2015).

Bi14Rh3I9

Dual TIGaGeTe

Optimization of crystal growth (mostly by Chemical Transport Reactions)

β-Bi4I4 Bi14Rh3I9Bi2TeI

Slide 15

Pursuit of magnetic topological insulators

Cr-doped (Bi,Sb)2Te3 below 30 mK:

C.-Z. Chang et al. Science. 340, 167 (2013).

Slide 32

0 50 100 150 200 250 3000,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

0,18

0,20

0,22

H ^ ab: m0H = 1 T, zfc

H ^ ab: m0H = 1 T, fcw

H || ab: m0H = 1 T, zfc

H || ab: m0H = 1 T, fcw

cM

ol [

em

u/m

ol]

T [K]

TN=24,1(5)K

TN=24,2(1)K

MnBi2Te4

First AFM TI

arxiv.org: 1809.07389

Spintronics

GMR effect (1988, A. Fert and P. A. Grüneberg; Nobel Prize in Physics in 2007)

• Quantum mechanical effect

• Electron scattering is dependent on the spin orientation

M

NM

M

M

NM

M

C. Chappert et al., Nature Mater. 6, 813 (2007)

Magnetic field, H

Rel. electrical

resistance, %

antipatallel

magnetisation

Historical overview

Observation and application (from the ancient times on)

Magnisia, Greece

Lodestone orMagnetite Fe3O4

Fe3O4 = (Fe3+)2Fe2+(O2–)4

A mixed valence compoundStructure type: (inverse) spinel

General formula for spinels: A3+2B

2+O4

Normal spinel: cations A in octahedral sites, B in tetrahedralInverse spinel: ½ of A in tetrahedral sites, ½ of A and B in octahedral

5μB

5μB + 4μB

μnet ~ 4μB ferrimagnetic ordering

Outline

1.1. Macroscopic level1.1.1. Classification of magnetic materials based on their susceptibility

1.2. Atomic level1.2.1. Spin-orbit coupling. Magnetocrystalline anisotropy. Coupling schemes in many-electron systems.1.2.2. Diamagnetism of valence (core) electrons and itinerant electrons1.2.3. Paramagnetism of valence (core) electrons and itinerant electrons

1.3. Interacting (magnetic) momenta1.3.1. Exchange interactions (spin-spin) or correlation effects. • Direct exchange• Superexchange, double exchange• RKKY-interactions• Anisotropic interactions (for instance, Dzyaloshinski-Moriya)

2.4. Cooperative magnetism2.4.1. Magnetically ordered states2.4.2. Magnetic frustration, spin glass, spin ice2.4.3. Band magnetism2.4.4. Metal-insulator transitions

Magnetism as interaction with an external magnetic field

Macroscopic picture

We are studying the response of a macroscopic sample introduced intoan external magnetic field

1.1. Macroscopic level


Macroscopic picture

We are studying the response of a macroscopic sample introduced intoan external magnetic field (a usual approach to explore the magneticproperties of a (unknown) sample)

Experimental parameters/variables:

• magnetic field strength H (A/m)• magnetic flux density B (T = Nm/A)• magnetization M (A/m)• susceptibility 𝜒 (dimensionless)• permeability μ (μ0μr) or relative permeability μr

• permeability of vacuum (permeability constant) μ0 = 4p·10–7 V·s/A·m

1.1. Macroscopic level

Characteristic values


• electric current• Maxwell equations• Lorentz force• magnetic dipols

Cause (H)

B = μ0H (free space)B = μ0μrH (in the solid)

Mediation (B)

• what the sample „sees“

Response (M)

or

B = μ0(H + M)M = 𝜒H

μr = 1 + 𝜒

Classification of magnetic materials

Based on susceptibility or permeability. Susceptibility is discontinuous

μr = 1 + 𝜒

χ < 0 diamagnetic material

χ = 0 vacuum

χ > 0 paramagnetic material

χ ≫ 0 magnetically ordered materials(non-linear dependence, e. g. hysteresis)

…any other complex dependence…

0 ≤ µ𝑟 < 1 diamagnetic material

µ𝑟 = 1 vacuum

µ𝑟 > 1 paramagnetic material

µ𝑟 ≫ 1 magnetically ordered material(non-linear dependence

cannot be assessed based on the absolute values of permeability or susceptibility )

𝜒 = M / H

μ = μr μ0

A note on paramagnetism

ParamagnetismMagnetization is induced and persists until theexternal magnetic field is applied. In contrast to that, ferromagnetism (or any other long-range magnetic ordering) is stable without an appliedfield (spontaneous).

Large positive experimental 𝜒 values do not suffice for

a sample to be characterized as ferro(ferri)magnetic.

Paramagnetic materials achieve saturation in high magnetic fields.

Magnetic properties are not a „permanent“ characteristic of a material at hand. They are being studied within a parameter space (magnetic fields, temperature, physical or chemical pressures are variables). Magnetic phase transitions areoften observed, in particular along the temperature scale.

Methods of synthesis/crystal growth and the associated physical shape of thesample (crystal, powder, nanoscaled product, thin film, etc.) can (strongly) influence the magnetic characteristics of a given material. Also due toconfinement effects:

• spatial confinement: nanostructures; thin films

• structure confinement: 2D, 1D and 0D structure fragments, various types ofchemical bonding between the fragments

• electronic confinement: 2DEG at interfaces, surface; heterostructures

Phase diagrams, incl. magnetic ones (H, T, p, etc.).

The range of available magnetic fields: from 50 µT on the Earth surface to

Saturation magnetization of iron ca. 2 T

Outmost weak magnetic field in a lab 10–9 T

Brain waves (Physikalisch-Technischen Bundesanstalt, Berlin)

Outmost strong (stable) magnetic field in a lab 45 T (Tallahassee, USA)

Magnetic properties in a parameter space

Magnetism as a phenomenon dealing with interactions(orientation) of magnetic moments (mostly spins)

Microscopic picture

We are looking at interactions and mutual adjustments of „elementalmagnets“ or „magnetic centres“.

Cooperative, collective (electron-based) phenomena. Electron correlations and competing ground states.

1.2. Atomic level

Magnetism as a phenomenon dealing with interactions(orientation) of magnetic moments (mostly spins)

Microscopic picture

We are looking at interactions and mutual adjustments of „elementalmagnets“ or „magnetic centres“.

Cooperative, collective (electron-based) phenomena. Electron correlations and competing ground states.

Experimental parameters/variables:

• Quantum numbers• Angular (orbital) momentum L, (total) spin momentum (S),

total angular momentum J• Bohr magneton μB = 9.27408·10–24 A·m2

1.2. Atomic level


Magnetism as a phenomenon dealing with interactions (orientation) of magnetic moments (mostly spins)

Spin angular momentum (S) = eigenrotation of an electron (core electrons anditinerant electrons). A metaphor of planetary rotation is deceiving.

Experimental confirmation: Stern-Gerlach experiment (1922)Ag [Kr]4d105s1 atoms in an inhomogeneous magnetic field

Orbital angular momentum (L) = electron is an electric charge on a circular orbit, loop current (core electrons)

Experimental confirmation: Einstein-de Haas effect (1915)A rod made of a ferromagnetic material is suspended on a string. It starts to rotatearound its axis when it is magnetized along its length.Reason: conservation of angular momentum.Orbital angular momentum is a quantum-mechanical analog of the classic angular momentum.

Einstein-de Haas experiment

Demonstrates a connection between the classic angular momentum andmacroscopic magnetization of a macroscopic body and the angular momentum ofan electron.Relates the observed magnetic moment to its angular quantum number.Rotation is a consequence of the conservation of angular momentum.

Einstein-de Haas experiment

Demonstrates a connection between the classic angular momentum andmacroscopic magnetization of a macroscopic body and the angular momentum ofan electron.Relates the observed magnetic moment to its angular quantum number.Rotation is a consequence of the conservation of angular momentum.

Magnetic moment 𝑚 is related to:• Current loops (orbital motion of electric charge)• Spin magnetic moments of elementary particles

Current loop is an orbital motion of charge and

orbital motion of a particle mass (angular momentum 𝐿)

𝑚 and 𝐿 are proportional:

𝑚 = − 𝛾 𝐿, 𝛾 – g-factor, gyromagnetic ratio

This experiment proves that the angular momentum is real.

𝑚

I

𝑚 = 𝐼 𝑑𝑆 , 𝑆 − area

Units Am2

TU Dresden, 15.10.2018 Folie 25Magnetismus

Magnetic moment

Potential energy is minimal when the moment is parallel to the external field.The relation is written by analogy with an electric dipole.The notion of magnetic dipole.Are there magnetic monopoles?

𝑚

I

𝐸𝑝𝑜𝑡 = −𝑚 ∙ 𝐵𝐵

In non-uniform magnetic field, there will be a magnetic force proportional to themagnetic field gradient, acting on a magnetic dipole:

𝐹𝑙𝑜𝑜𝑝 = ∇(𝑚 ∙ 𝐵), 𝐹𝑧 = 𝑚𝑧𝜕𝐵𝑧

𝜕𝑧

Magnetic moment

Potential energy is minimal when the moment is parallel to the external field.The relation is written by analogy with an electric dipole.The notion of magnetic dipole.Are there magnetic monopoles?

𝑚

I

𝐸𝑝𝑜𝑡 = −𝑚 ∙ 𝐵𝐵

In homogenous magnetic field, there is no force but there is torque:

𝑇 = 𝑚 × 𝐵

Angular momentum and torque lead to magnetic precession with Larmor frequency

𝛾 𝐵

Bohr magneton

One Bohr magneton corresponds to the moment of the electron in a hydrogen atom with the Bohr radius of 0.529 Å and angular quantum number l = 1.

In quantum mechanics, 𝛾 = −𝑚𝐵

ℏ, so that

𝑚 = −𝑚𝐵𝐿, 𝑚 = −𝑔𝑚𝐵Ԧ𝑆, 𝑔 − Landé-factor, 𝑔 = 2 (free electron)

For electrons in periodic solids, both core and itinerant electrons, the Landé-factor is slightly different from 2.

𝑚 = −𝑚𝐵(𝐿 + 𝑔 Ԧ𝑆) – total magnetic moment

𝑚𝐵 =𝑒ℏ

2𝑚= 9.274 ∙ 10−24

J

T= 0.579 ∙ 10−4

eV

T= 9.274 ∙ 10−21 emu


Stern-Gerlach experiment

If the spin eigenvalues were similar to the orbital ones (the l = 1 state), we wouldexpect to see 3 deflected beams. If the space quantization were due to the magneticquantum number ml, ml states were always odd (2l +1) and should produce an oddnumber of lines. ms (secondary spin quantum number) ranges from –s to +s andgenerates (2s + 1) values.Fermions (incl. electrons) take up half-integer s values.

Ag: [Kr]4d105s1

VB = -mzBFz = -(dVB/dz) = mz(dB/dz)


A History of Spin

Zeeman effect (1896)

Sodium yellow doublet (splitting of the emission lines without externalmagnetic field: a direct consequence of spin-orbit coupling)

Na atom: the 3p level splits into 2 stateswith total angular momentum (J) J = 3/2 and J = 1/2 in the presence of the internal magnetic field caused by orbital motionand spin.In the reference system of a core electron, the nucleus with a charge +Ze is rotatingaround the electron and generates a magnetic field.In this field, the electron has an additional potential energy:

𝑈𝑚 = −𝜇𝑠 ∙ 𝐵 = −𝜇𝑧𝐵𝑧 = −𝑔𝑠𝜇𝐵 𝑆𝑧𝐵𝑧 = ∓𝜇𝐵𝐵𝑧


Spin-orbit coupling I

Spin angular momentum (S) = eigenrotation of an electron (core and itinerant

electrons)

Core electrons = loop currents angular momentum (L)

Spin-orbit coupling (in a rotating reference system of an electron, a positively chargednucleus (+Ze) is rotating on a circular orbit and generating a magnetic field):

𝒋 = ℓ + 𝒔𝒋 𝟐 = ℓ 𝟐 + 𝒔 𝟐 + 𝟐ℓ ∙ 𝒔

Energy of SOC:

𝐸𝑆𝑂 = – 𝝁𝑠 ∙ 𝑩𝑜𝑟𝑏 = 𝐴 ∙ ℓ ∙ 𝒔

𝐴 — SOC constant, 𝐴~𝑍4

𝑛6,

Z – atomic number, n – principal quantum number

𝐸𝑆𝑂 lies in the range between 10 and 100 meV

R. Gross, A. Marx. Festkörperphysik. 2. Auflage. 2014


A History of Spin

Zeeman effect (1896)

Sodium yellow doublet (splitting of the emission lines without externalmagnetic field: a direct consequence of spin-orbit coupling)

Bohr-Sommerfeld atomic model (1913)

Paschen-Back effect (1921)

Stern-Gerlach experiment (1922)

S. Goudsmith, G. Uhlenbeck (1925, Universität Leiden): electron has its own orbital momentum

W. Pauli (1927): has determined the complete set (incl. spin) ofcommuting variables (CSCO), or the complete set of quantum numbersdescribing a quantum system, Pauli exclusion principle

Zeeman effects

Zeeman effect(s): Splitting of energy states in external magnetic field

Sodium atom in external field (all z-component of total angular momentum J:

Term: 2S+1LJ, selection rules


Splitting of spectral lines

Magnetic quantumnumber and spinqantum numberare decoupled

Angular momentumquantum number istaken into account, Total spin = 0


Momenta coupling in many-electron systems

How do angular ℓ𝒊 and spin 𝒔𝒊 moments of individual electrons couple in case of a many-electron system (e.g. a valence shell)?

𝒋𝒊 = ℓ𝒊 + 𝒔𝒊

jj-coupling Russell-Saunders

Russell-Saunders scheme (light elements, e.g. 3d-transition metals with L = 2, but also rare-earth elements of the 4f row L = 3): all single orbital angular momenta couple to the

total angular momentum L = σ𝑖 ℓ𝒊, all single spin momenta couple to the total spin

moment S = σ𝑖 𝒔𝒊, then L and S couple to the total total angular momentum J = L + S.

jj-scheme (for large atomic numbers Z): first pairs of individual ℓ𝒊 and 𝒔𝒊 moments coupleto 𝒋𝒊, then these couple to the total angular momentum J. This is a case of strong spin-orbit

coupling.


Spin-orbit coupling II

Energy of spin-orbit coupling: 𝐸𝑆𝑂 = – 𝝁𝑠 ∙ 𝑩𝑜𝑟𝑏 = 𝐴 ∙ ℓ ∙ 𝒔

𝐸𝑆𝑂 corresponds to the energy that is gained when spin aligns parallel to ℓ from theinitial perpendicular position to ℓ.

Since orbital angular momentum ℓ often has a „preferred“ (energetically favorable) crystallographic orientation (thanks to crystal-field effects (electrostatic potential ofsome symmetry), degeneracy of 3d-electons), the spin momentum 𝒔 will also orientparallel to this direction via SOC effect. In this spirit, spin-orbit coupling strengthensmagnetocrystalline anisotropy.

Magnetocrystalline anisotropy leads to the effect of „easy“ magnetization alongparticular crystallographic axis.

In other words, the lattice (atomic arrangement in general) can influence the spinorientation via spin-orbit effects.

Other types of magnetic anisotropy:• Shape anisotropy: easy magnetization axis in the longest direction of

measurement• Magnetoelastic anisotropy (induced anisotropy).


Further considerations…

„Quenching“ of orbital angular moment („spin-only magnetism“) in compounds oftransition metals, band magnetism

Strong SOC may „unquench“ the orbital momenta affected by electrostatic potential

Chemical bonding (differing orbital overlap between the adjacent atoms).

Electron-electon interactions (Pauli exclusion principle, Coulomb repulsion, wavefunction symmetry).

„Solid state chimera“



Magnetism as a phenomenon dealing with interactions (orientation) of magnetic moments (mostly spins)

Total magnetic moment is a sum of all contributions of all electrons in a solid. Both core and itinerant electrons cause dia- and paramagnetic contributions.


Magnetic response of core and itinerantelectrons


Delocalized electronsItinerant electrons

Localized electronsCore electrons


Magnetic response of core and itinerantelectrons

Paramagnetism Diamagnetism

Localized electrons Langevin-paramagnetism, χ = C/TContributions of spin andorbital angular momenta ofelectrons on partiallyoccupied shells

Van Vleck-paramagnetism. Spin andorbital momenta of closedshells.

Atomic or Larmor-diamagnetismContributed by the orbital angular momenta

Itinerant (quasi-free) electrons

Pauli-paramagnetismContributed by the spinmomenta

Landau-diamagnetismContributed by the orbital angular momenta

𝝌𝒗𝒂𝒏 𝑽𝒍𝒆𝒄𝒌 ≈ 𝝌𝑷𝒂𝒖𝒍𝒊 ≈ 𝝌𝑳𝒂𝒓𝒎𝒐𝒓 ≈ 𝟏𝟎−𝟔…−𝟓 ≪ 𝝌𝑳𝒂𝒏𝒈𝒆𝒗𝒊𝒏 ≈ 𝟏𝟎−𝟑…−𝟐

Diamagnetism

• Induced magnetic moments areantiparallel to the field creating them.

• Induced currents weaken the externalfield in accordance with the Lenz‘slaw.

• Note: Bohr-van Leeuwen theorem(1911 and 1919): The netmagnetization of an electronensemble in thermal equilibrium isequal to zero at finite temperaturesand finite electric or thermal fields.Consequently, electron interactionsare impossible in the classic, non-relativistic concept.

• Both in closed shells (Larmor-D.) anditinerant electrons (Landau-D.): 𝜒 ~ 10–5 — 10–6

• Universal effect• Generally temperature-independent

Examples: water, bismuth, nitrogen

https://www.wmi.badw.de/teaching/Lecturenotes/magnetismus/Kapitel-2.pdf


Diamagnetism

Diamagnetic levitation(strong fields up to 15 T)

High Field Magnet Laboratory, University of Nijmegen, 1997

Andre Geim: Ig-Nobel Prize (2000)Nobel Prize (2010)

Literature:

A.Geim, Physics Today, 9 (1998), 36-39.

http://www.ru.nl/publish/pages/682806/everyonesmagnetism.pdf

M.V. Berry, A.K.Geim, European Journal of Physics, 18 (1997), 307-313.

http://www.ru.nl/publish/pages/682806/frog-ejp.pdf

https://www.youtube.com/watch?v=KlJsVqc0ywM


Paramagnetism

• Existing magnetic moments ofelectrons align parallel to the appliedmagnetic field.

• Temperature-dependent effect(Langevin-P.) in atoms with partiallyfilled shells: 𝜒 ~ 10–2 — 10–3

• Temperature-independent, weak effect(Pauli-P.) for itinerant electrons: 𝜒 ~ 10–5 — 10–6

• No interaction between the moment, the effect vanisches when the appliedfield is cancelled.

Examples: Na (3s-electrons)alkali metals (closed shells andcompetion between Pauli-P. anddiamagnetism)Elemental copper, Cu2+ salts

Potential energy of a magnetic moment𝒎 in an external field B:

𝑈 = –𝒎 ·B

𝑈 is minimized when m is parallel to B(Zeeman energy)

https://www.wmi.badw.de/teaching/Lecturenotes/magnetismus/Kapitel-2.pdf


Langevin paramagnetism(Curie law)

Paramagnetism of core electrons (only for partially filled valence shells)

The magnetic moment of one localized magnetic centre (without strong spin-orbit coupling) :

μ2 = gJ2J(J+1)μB

2, gJ = (J(J+1) + S(S+1) – L(L+1)) / (2J(J+1)) + 1

gJ – Landé factor (characterizes coupling between orbital and spin moments)

Langevin paramagnetism follows the Curie law:

𝜒𝑚𝑜𝑙 =𝐶

𝑇mit 𝐶 = 𝜇0

𝑁𝐴µ2

3𝑘𝐵— material-specific Curie constant

Gd2(SO4)3·8H2O


Very often is accompanied by magnetic ordering (cooperative magnetism) below a critical Curie temperature (Tc)

Above the Tc the Curie-Weiss law is applied:

𝜒𝑚𝑜𝑙 =𝐶

𝑇 − Θ𝑝

ferro para ferro para

The paramagnetic Curie temperature of Weiss constant Θ𝑝 reflects the sum of all

interactions between the microscopic magnetic moments.

Curie-Weiss law


Cooperative phenomena

Antiferromagnets: MnF2, MnO, YBa2Cu3O6, LaMnO3

Ferrimagnets: Fe3O4 (Magnetit), Y3Fe5O12 (yttrium iron garnet)Helikale magnets: Tb, Dy, HoSpinglass: Cu1–xMnx, MnxGe1–x


Pauli paramagnetism

Free electrons (metals)

Why is there no temperature dependencein contrast to Langevin P. (c = C/T)?

Explanation:

Quasi-free electrons obey the Fermi statistics.

The Fermi temperature (TF) of the freeelectron gas is much higher than the roomtemperature.

Consequently, only a small portion ofelectrons can flip their spin in the narrowenergy interval near the Fermi edge (Pauli principle).

The amount of these electrons is𝑘𝐵𝑇

𝐸𝐹=

𝑇

𝑇𝐹, so

that

cPauli = 𝐶

𝑇∙𝑇

𝑇𝐹=

𝐶

𝑇𝐹(no temperature dependency)

Temperature-independent, 𝜒 ~ 10–5

Landau-D. of itinerant electrons: cLandau = –cPauli/3

Deviations from the exact ratio due tointeractions with the lattice potential (dependent from the effective mass ofitinerant electrons m*)

cPauli can increase thanks to electron-electron interactions (e.g. palladium metal)



Van-Vleck paramagnetism

Found in atoms and ions with electronconfigurations that lack one electronfor the half-filled state (e.g. 𝑑4, 𝑓6)

Temperature dependent, but onlynotable at low temperatures

Quantum effect: Contribution isconnected to field-induced electronic transition between an excited and the(non-magnetic, non-degenerate) ground state

𝜒 ~ 10–4

Examples: molecular crystals, Eu- andSm-containing compounds

Eu2O3, Eu3+ [Xe]4𝑓6


Paramagnetism

Cu [Ar]3d104s1

Resistivity: 16,8 nΩm-5 ·10-6 cm3 mol-1

Na [Ne]3s1

Resistivity: 47 nΩm16 · 10-6 cm3 mol-1

CuO: magnetic semiconductorCuS: weak Pauli paramagnetics, metallic

CuS = (Cu+)3(S2-)(S2)

2-[p]Or (Cu+)2Cu2+S2

2-S2-

Magnetic susceptibility

For localized moments:

J = 0, S = 0, L = 0: only Larmor dia

J = 0, S = L ≠ 0: (Larmor dia) + van Vleck paramagnetism

J ≠ 0: (Larmor-Dia + van Vleck para) + Langevin paramagnetism

Empirical rules to find the ground state ofa multi-electron system (Hund‘s rules):1. The total spin momentum S is

maximal.2. The total angular momentum L is

maximal.3. J = │L – S│ for less than half-filled

shell, J = L + S for more than half-filled shell.

Applies when the Russell-Saunders coupling is justified, when SOC is muchweaker than Coulomb repulsion.

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