Order 1252715: Condensed Matter physics

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Magnetism 2

where is the g-factor. The spin is and the Bohr

magneton is

Then the energy of an electron with spin up (same direction as

Magnetic field is:

And that with spin down is

where

Magnetic Spin Susceptibility – Pauli

Paramagnetism Consider the response of free electrons to an externally applied

magnetic field. The electron’s motion can be curved due to the

Lorentz force, but also the spins can flip. Looking at latter effect.

The Hamiltonian becomes:

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The spin magnetization of the system (magnetic moment per

unit volume) in direction of the field is:

With applied magnetic field the energy is lower when the spins

point down, so more of them will point down and a magnetisation

develops in the direction of applied field – known as

Pauli paramagnetism (spin magnetization of free electron gas)

Magnetic Spin Susceptibility – Pauli

Paramagnetism

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Magnetic Spin Susceptibility – Pauli

Paramagnetism Find Pauli paramagnetism for T=0

For no magnetic field, electrons

are filled up to the Fermi energy with With magnetic field, up electrons

more energetically unfavorable by

therefore will have

fewer spin up electrons

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Magnetic Spin Susceptibility – Pauli

Paramagnetism That is, with the magnetic field that states with up and down spin are

shifted in energy by and , respectively.

Hence, spin up electrons that are pushed above the Fermi energy

can lower their energies by flipping their spins to become spin down

electrons. The total number of spins that flip (the area of the

approximately rectangular shape) is roughly

Then from:

we obtain:

and

so

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Spontaneous magnetic order in solids

Heisenberg Hamiltonian

Model description of how spins align – assume an interaction between

neighbouring spins – so-called “exchange interaction”

Assume an insulator, so electrons don’t hop from site to site.

Model Hamiltonian is:

is the spin on site i and B is the magnetic field experience by the spins

is the interaction energy. Neglecting the magnetic field, and

assume each spin coupled to its neighbour with the same strength, can

drop i,j

is the interaction energy

Factor of ½ avoids over-counting in sum

If lower energy when spins aligned; whereas if it is lower energy

when spins are anti-aligned

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Spontaneous magnetic order in solids

The Hamiltonian doesn’t indicate a preferred spin direction.

In a real system, atoms are often in an asymmetric environment due to the

lattice and will be directions that the spin would rather point.

Add term to the Heisenberg Hamiltonian:

called anisotropy energy as gives system a preferred direction, here in the

or directions.

Or, for spin pointing along the orthogonal axis directions:

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Spontaneous magnetic order in solids

If the anisotropy term is very large in

It will force the spin to be either or

This gives the Ising Model

where only (and reintroducing the magnetic field B)

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Experiments: MExFM

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(a) Atomic-resolution image of an antiferromagnetic NiO(001) surface

obtained by Non-Contact Atomic Force Microscopy (NC-AFM). The line

section reveals an apparent height difference of 4.5 pm between nickel

(dark) and oxygen (bright) sites.

(b) Spin-resolved image of NiO(001) with atomic resolution as obtained

by Magnetic Exchange Force Microscopy (MExFM)

Imaging & Microscopy, Jun. 01, 2008 R. Wiesendanger,

Experiments: Spin-polarized STM

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Differential conductance asymmetry A dI/dV. a Two magnetic configurations in spin-STM measurements, AP and P,

corresponding to two distinct magnetic states of a bilayer Co nanoisland, pointing up and down, respectively. b, c dI/dV

images of the Co nanoisland ‘A’ in Fig. 5 a measured at μ0 H ext = −1 T and V b = + 0.03 V for AP (b) and P (c) states. d A

dI/dV map calculated from the dI/dV images of b and c. e Two relative magnetization configurations of spin-STM

measurements, corresponding to two distinct magnetic states of a bilayer Fe nanoisland, α and β. f, g dI/dV images of a Fe

nanoisland, measured at external fields of (b) 0 T and (c) a value ≥ H sat. h A dI/dV map calculated from the dI/dV images of f

and g.b–d

Nano Convergence 2017 4:8 Soo-hyon Pharj and Dirk Sander,

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Domains and Hysteresis

In real materials there are regions “domains” with different spin orientation.

Reduces the dipolar energy (resulting from the sum of the individual dipole-

dipole interactions on the atoms).

Can understand like since if view as magnets; two like ends (North/South) will

repel; lower energy by flipping one.

Boundary between domains,

call “domain wall”

Applying magnetic field

increases domain size

of that pointing in same

direction

Ising type ferromagnet

moments only up or down

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Domains and Hysteresis

Another way of understanding why magnetic domains are energetically

preferred is to consider the magnetic field they induce:

The magnetic field will be much lower if they are anti-aligned as can be seen.

The magnetic field has associated energy .

Thus, minimizing the field lowers the energy of the “two dipoles”

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Ferromagnetic domains: Disorder Pinning

Above: both domain walls (red) start and end at same place.

But, one on right, passes through vacancy. It therefore

has one less anti-aligned spins, so overall energy lower

(more favourable) – say, domain wall is “pinned” to the disorder

An Ising ferromagnet

The length of the domain wall

depends on balance between J and

If large, small wall

If small, wide wall

Consider scaling of wall: if length is then each spin twists and angle:

. Then the first term in Eq(1) can be written as:

The spin do not need, however, to only point up or down, corresponding to

large in

The domain wall may be more like a gradual rotation of up pointing spins

to down pointing spins like below:

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Ferromagnetic domains: Bloch/Neel Wall

Called a Bloch Wall or Neel Wall

(1)

Small angle expansion

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Ferromagnetic domains: Bloch/Neel Wall

Can see has lowest energy if so can think of the second term being

an “energy cost”

Then for the N unit cells in the domain wall (the “energy stiffness”), given

per unit area A (per lattice constant a):

Recall from Eq(1)

We also have 2nd term. When spins not exactly up or down, will be energy

cost proportional to per spin, so for the N unit cells in the domain wall:

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Ferromagnetic domains: Bloch/Neel Wall

Total energy cost due to anisotropy:

So with these two energy costs (penalties) we have the total energy cost:

Minimizing this energy with respect to length L we find:

and therefore

Energy balance between cost

of domain wall formation versus

gain due to having domains

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Hysteresis curve

coercive

field

remanent

magnetization

saturation

magnetisation

note that here

We know from electromagnetism that ferromagnets

exhibit a hysteresis loop with applied external field.

When field is returned to zero after being applied, there

is a remanent magnetization

This is because

there is a large

activation energy for

changing the

magnetization

How to understand the activation energy barrier – consider

small crystallite with all spins aligned. The energy per volume

in an external field is:

Where M is the magnetization and is the component in the

-direction.

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Single Domain Crystallites

= Zeeman energy per

unit volume

Number of spins per unit volume

angle of magnetization with respect to axis

Plotting Eq(1) vs gives

parabola -

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Single Domain Crystallites

- minimum of energy when magnetization in plus or minus z-direction,

corresponding to , and energy barrier in between. For increasing

B field, there are stable and metastable states. If B field large enough, spins

will flip – this behaviour can result in the observed hysteresis