Order 1252715: Condensed Matter physics

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Lect-16-Magnetism-1.pdf

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26/10/2018

Magnetism Not only permanent ferromagnets,

many applications

motors, transformers, imaging,

data storage (probably just as

important as semiconductors for

modern computers)

FeCo/Pt superlattices high saturation

magnetism promising for magnetic

data storage

• How do solids react to an external field? • What is the cause of spontaneous magnetic ordering?

Magnetism

Magnetism is an extremely active area of research with many

still unanswered questions

Condensed matter physics uses magnetism as a testing ground

for understanding complex quantum and statistical physics

Most magnetic phenomena caused by quantum mechanical

behaviour of the electrons

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• Magnetic moments in atoms • Weak magnetism in solids • Magnetic ordering

Magnetic properties

Weak magnetism in solids can largely be understood by atomic properties

magnetic ordering cannot - cannot describe it as

ordering of totally localised moments on atoms because these have

to “talk” to each other, otherwise there is no ordering in the first place

Ashcroft and Mermin Ch 31, 32; Oxford Basics Ch 19

Macroscopic description of magnetism:

Fundamental quantities In vacuum we have:

magnetic field intensity

magnetic induction

When a material medium is placed in a magnetic field,

the medium is magnetized. This is described by the

magnetization vector M – the dipole moment per unit volume

we interpret as the “external field”

permeability of free space 4πx107 (SI units: N.A-2)

Magnetization induced by the field assume M is proportional to H

- magnetic susceptibility of the medium

(Real crystals anisotropic, and susceptibility is a second-rank tensor (ignore such effects)

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Units

• Both, and are measured in Tesla (T) • 1 T is a strong field. The magnetic field of the earth is only

of the order of 10-5 T.

potential energy of one dipole in the external field:

Classification of materials

All magnetic materials may be grouped into three magnetic classes

depending on the magnetic ordering and the sign and magnitude of the

magnetic susceptibility:

(more later)

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Diamagnetism

• Diamagnetism is caused by “currents” induced by the external field. According to Lenz’ law, these currents always

lead to a field opposing the external field.

Potential energy U

increase in potential energy for higher field, unfavourable.

Paramagnetism

Potential energy U

the potential energy is lowered when moving the magnetized bodies to a

higher field strength

. • Paramagnetism is caused by aligning some dipoles, which

are already present, with the magnetic field

Ferromagnetism

A ferromagnet is a material where M can be nonzero

even in the absence of an applied magnetic field

Magnetism is said to be Spontaneous when it occurs even in the

absence of an externally applied magnetic field, as in the

case of a ferromagnet

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Ferromagnetism

d-electron states/orbitals

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f-electron states/orbitals

Isolated Atoms

• The magnetic moment of a free atom has three main sources

1. the spin of the electrons

2. the electron orbital angular momentum about the nucleus

3. the change in orbital moment induced by an applied

magnetic field

• In classical picture, electrons orbit around nucleus • Each orbit like a loop of electric current • A loop of current produces a magnetic field, so electrons in

an atom generates a magnetic field

• Quantum numbers n,l,m l and ms label the electrons in an atom (alternatively called n, l, lz, σz )12

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Hund’s Rules for Isolated Atoms

Set of Rules – Hund’s Rules that determines how electrons fill orbitals

Recall from QM, an electron in an atomic orbital can be labelled

by four quantum numbers:

Principle quantum number

Angular momentum

z-component of ang. mom.

z-component of spin

Sometimes the angular momentum shells

are known as and can accommodate

electrons, respectively.

Start with some fundamentals of electrons in isolated atoms

Hund’s Rules for Isolated Atoms

lz = 2

lz= 1

lz= 0

lz=-1

lz=-2

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Hund’s Rules for Isolated Atoms

Hund’s 0th Rule – (Aufbau Principle) shells filled starting with lowest available energy state. An entire shell filled

before another started

Madelung Rule): energy ordering is from lowest value of n+l to largest when two shells have the same n+l, fill one with

smallest n first

Two examples:

Nitrogen (N) 7 electrons, filled 1s shell with 2

electrons spin-up and spin-down, 2 electrons in

the 2s shell with 2 electrons spin-up and spin-down,

3 electrons in the 2p shell

Praseodymium (Pr) 59 electrons

or as

Hund’s Rules for Isolated Atoms

The shell filling sequence is the rule which defines the overall

structure of the periodic table.

When shells are partially filled need to describe which of the orbitals

are filled in these shells and which spin states are filled

Hund’s Rules

(1) Electrons try to align their spins, i.e. the electrons

should occupy the orbitals such that the maximum

possible value of the total spin S is realized.

Consider Pr as example - the 3 valence electrons will

have spins that point in same direction giving S=3/2

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Hund’s Rules for Isolated Atoms

(2) The electrons should occupy the orbitals such that

the maximum of L, consistent with S, is realized.

For Pr, this means giving

so we have and

(3) The total angular momentum J is calculated

• If the sub-shell is less than half-full J=L-S • If the sub-shell is more than half full J=L+S • If the sub-shell is half full, L=0 and J=S

For Pr, since shell less than half-full we use J=L-S = 6 - 3/2 = 9/2

Preference for spins to align comes from the Coulomb interaction

energy between the electron and nucleus

For spins anti-aligned, electrons

are closer and the nucleus is

partially screened by the

negative charge of the other

electron.

For spins aligned the electrons

repel each other and see the full

positive charge of the nucleus.

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• The first of Hund’s rules requires S=3/2. • The possible l z values for the 3d shell are -2,-1,0,1,2.

Hund’s second rule requires to choose the largest possible

value of L, i.e. to choose l z =0,1,2, so L=3.

• Since the sub-band is less than half filled, J=L-S=3-3/2=3/2.

Another example: Cr 3+

Cr3+ has three electrons in the 3d sub-shell

20 Ashcroft & Mermin Ch31

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(2S+1) X J

Coupling of electrons in atoms to an external

field

Seen how electron orbital and spin can align with each other

Now consider how electrons couple to an external magnetic field

A is the vector potential

particle in electromagnetic field

change the momentum (operator)

First recall -

for electrons q=-e

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Coupling of electrons in atoms to an external

field

In absence of a magnetic

field the Hamiltonian for an

electron in an atom is:

V is electrostatic

potential from the

nucleus

In presence of a magnetic field:

where is the electron spin, g is the electron g-factor (about 2)

and the Bohr magneton is

Zeeman term

Zeeman effect

(l z)

Splitting of spectral lines when atom is placed in an external magnetic field

Predicted by Lorentz, first observed by Zeeman

Energy level splitting in the normal Zeeman effect for singlet levels l=2 and

l=1

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Coupling of electrons in atoms to an external

field

For a uniform magnetic field, we can take and

and so

can be written as:

First two terms just Hamiltonian in absence of field,

Can rewrite 3rd term as:

where is the angular momentum of the electron

is the Bohr magneton

Coupling of electrons in atoms to an external

field

With 3rd term as

Can combine with 5th term of below

To obtain final expression:

paramagnetic

term Coupling of field to total

magnetic moment of electron

diamagnetic

term

These two terms are

responsible for the paramagnetic

and diamagnetic response of atoms

to external magnetic fields

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Coupling of electrons in atoms to an external

field Free spin (Curie or Langevin) Paramagnetism

Consider the paramagnetic term in previous equation – generalize to multiple

electrons in the atom:

L and S, orbital and spin components

of all electrons, and

Can write as:

where (see Oxford Solid

State Basics for

derivation or

Ashcroft and Mermin

Appendix P)

The partition function is

And the corresponding free energy is

It describes the reorientation of free spins in an atom

Coupling of electrons in atoms to an external

field

Free spin (Curie or Langevin) Paramagnetism

Given the free energy

The magnetic moment per spin is

Assuming a density n of these atoms it can be shown that the susceptibility is:

Curie Law

Called “Curie paramagnetism or Langevin paramagnetism”

Curie constant

Ashcroft & Mermin

Ch31

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Coupling of electrons in atoms to an external field

Curie paramagnetism dominant when

So can only observe diamagnetism when

For example, filled shell configurations like

noble gases

[Or if J=0, but L and S not equal to zero. This

occurs when shell has one electron fewer than

being half full ]

Larmor Diamagnetism

The expectation of the diamagnetic term for B in the z direction is

The atom is rotationally symmetric:

Consider now diamagnetic term – coupling of the orbital motion to the

magnetic field

Coupling of electrons in atoms to an external field

Larmor Diamagnetism

The atom is rotationally symmetric:

So we have

and the magnetic moment per electron is:

Assume density of electrons, can write:

Larmor Diamagnetism

(recall M=χ H = χ B/ μ0)

= χ B/ μ0

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Will be some amount of diamagnetism in all materials, above good description

for core electrons – but for conduction electrons in a metal we have

Landau-diamagnetism:

where is the susceptibility of the free Fermi gas 33

Magnetism of atoms in solids

Diamagnetism in solids

In above, Larmor diamagnetism applied to isolated atoms with

At low temperatures, noble gas atoms form weak bonds in crystal and

description still applies, where density of electrons is put equal to the

atomic number Z times the density of atoms, n. the radius r is the

atomic radius

Magnetism of atoms in solids Curie Paramagnetism

Recall: Curie paramagnetism describes the reorientation of free spins in an atom

Does it occur in solids?

Yes, possible,

e.g. through “crystal field splitting” where atoms are no longer in a rotationally

symmetric environment

Also, the number of electrons on an atom can become modified in a material,

e.g. Pr, we had 3 free electrons in valence (4f) shell (J=9/2), but in many

compounds Pr donates two of its 6s electrons and one f electron (J=4).

Paramagnets can have many different effective values of J – need to know

microscopic details of bonding in system!

(in this case, L=5, S=2/2 and J=L-S = 5 -1=4)

e.g Fe iron

d 6

4μB