    from matplotlib import pyplot

import numpy as np
from scipy.optimize import curve_fit
from scipy.integrate import quad

from common import draw_classic_axes, configure_plotting

configure_plotting()

Lecture 8 – Magnetism¶

(based on chapters 19-20 of the book)
Relevant exercises 19.3, 19.6, 20.1, 20.2, 20.4.

In this lecture we will:

discuss common forms of magnetism
introduce magnetic susceptibility
discuss Hund's rules
discuss magnetic order

Magnetic field¶

To start off confusing, there are two different quantities that are often referred to as the magnetic field: the ${\bf B}$-field (in units of Tesla) and the ${\bf H}$-field (in A/m). In vacuum, they are proportional:

\[ {\bf B}=\mu_0 {\bf H} \]

with $\mu_0$ the permeability of free space. Technically, inside a material they differ. Here, ${\bf B}$ is the total magnetic field that one can measure locally, while ${\bf H}$ is the same, but corrected for the magnetisation:

\[ {\bf B}=\mu_0({\bf H}+{\bf M}),\ {\rm or}\ \ {\bf H}=\frac{\bf B}{\mu_0}-{\bf M} \]

where ${\bf M}$ is the magnetisation. In other words, ${\bf B}$ is the field due to free currents and 'bound currents' (i.e. local magnetised moments) while ${\bf H}$ is the field due only to free currents. But in practice, ${\bf M}$ is small compared to ${\bf H}$, so we always assume ${\bf B}=\mu_0 {\bf H}$.

Magnetic susceptibility¶

The susceptibility $\chi$ is the extent to which a material is prone to magnetisation:

\[ {\bf M}=\chi{\bf H}=\chi\frac{\bf B}{\mu_0} \]

Depending on the sign, you can get different forms of magnetism:

$\chi>0\Rightarrow$ paramagnetism: the material tends to magnetise along the local field.

$\chi<0\Rightarrow$ diamagnetism: the material tends to magnetise opposite to the local field.

Further on, we will discuss two forms of spontaneous magnetisation: ferromagnetism and anti-ferromagnetism. Unlike the ones mentioned above, these forms persist even in the absense of a magnetic field.

Hund's rules¶

We discussed before that the magnetic configuration of electrons in an atomic orbital can be described with four quantum numbers: $n$, $l$, $l_z$ and $\sigma_z$. Question is: how do these spin and orbital momental align inside the atom? There are various mechanism governing this, but the results can be summarized in three Hund's rules:

Rule 1: Maximise $S$.
Rule 2: Maximise $L$ (provided that rule 1 is satisfied).
Rule 3: For shells less than half filled: $J=|L-S|$. For shells more than half filled: $J=L+S$.

Examples:

Fe = [Ar]4s$^2$3d$^6$

$S=2$, $L=2$, $J=4$ $\Rightarrow$ $^{2S+1}L_J =\ ^5{\rm D}_4$

V = [Ar]4s$^2$3d$^3$

$S=\frac{3}{2}$, $L=3$, $J=\frac{3}{2}$ $\Rightarrow$ $^{2S+1}L_J =\ ^4{\rm F}_{3/2}$

Paramagnetism: free spin ½¶

If single electron spin ${\bf \sigma}$ is situated in a magnetic field ${\bf B}$, it can be described by the Zeeman Hamiltonian (note convention minus sign):

\[ {\mathcal H}=g\mu_{\rm B}{\bf B}\cdot{\bf \sigma} \]

where $\mu_{\rm B}=\frac{e\hbar}{2m_{\rm e}}=10^{-23}$ J/T is the Bohr magneton and $g$ is the g-factor, which for an electron is almost exactly 2. Its two eigenstates are $+\mu_{\rm B}B$ and $-\mu_{\rm B}B$. Now using the Bolzmann weights we compute the expectation value of the magnetisation of $n$ spins:

\[ M = -n \frac{ \mu_{\rm B}{\rm e}^{-\beta\mu_{\rm B}B}-\mu_{\rm B}{\rm e}^{+\beta\mu_{\rm B}B} }{ {\rm e}^{-\beta\mu_{\rm B}B}+{\rm e}^{+\beta\mu_{\rm B}B}} = n\mu_{\rm B}{\rm tanh}(\beta\mu_{\rm B}B) \]

For small $B$ this function is linear, allowing us to extract the susceptibility

\[ \chi=\mu_0\frac{{\rm d}M}{{\rm d}B}\approx\frac{n\mu_0 \mu_{\rm B}^2}{k_{\rm B}T}=\frac{C}{k_{\rm B}T} \]

with $C$ a constant. This is known as the Curie law, and it shows that at high temperature a paramagnetic material becomes less susceptible to magnetisation.

B = np.linspace(-2, 2, 1000)
fig, ax = pyplot.subplots()

ax.plot(B, np.tanh(B * 3), label="low $T$")
ax.plot(B, np.tanh(B), label="high $T$")

ax.legend()
ax.set_ylabel("$M$")
ax.set_xlabel("$B$")
ax.set_xticks([])
ax.set_yticks([])
draw_classic_axes(ax, xlabeloffset=.2)

svg

Paramagnetism: free spin $J$¶

For an atom with magnetic moments $L$ and $S$ the Hamiltonian will become:

\[ {\mathcal H}=\mu_{\rm B}{\bf B}\cdot\left({\bf L}+g{\bf S}\right) \]

This can be written in terms of the total magnetic moment ${\bf J}$ using the Landé factor $\tilde{g}\ $:

\[ {\mathcal H}=\tilde{g}\ \mu_{\rm B}{\bf B}\cdot{\bf J} \]

with

\[ \tilde{g}\ =\frac{1}{2}(g+1)+\frac{1}{2}(g-1)\left[\frac{S(S+1)-L(L+1)}{J(J+1)}\right]=\frac{3}{2}-\frac{L(L+1)-S(S+1)}{2J(J+1)} \]

The result is a ladder of equally spaced levels. The bigger the contribution of $S$ inside $J$, the bigger te level spacing. Now, the magnetisation for $n$ atoms becomes:

\[ M=-n\frac{\sum_{m_{J}=-J}^{J}\tilde{g}\ \mu_{\rm B}m_{J}{\rm e}^{-\beta\mu_{\rm B}m_J B}}{\sum_{m_{J}=-J}^{J}{\rm e}^{-\beta\mu_{\rm B}m_J B}} \]

The resulting susceptibility is:

\[ \chi=\frac{n\mu_0(\tilde{g}\ \mu_{\rm B})^2}{3}\frac{J(J+1)}{k_{\rm B}T} \]

Atoms in solids¶

Until now we have considered magnetic atoms in free space. When embedded inside a solid, many things change. Below, we will discuss two effects:

Interaction of magnetic atoms with other magnetic atoms $\Rightarrow$ Heisenberg model
Interaction of spins with non-magnetic atoms $\Rightarrow$ Crystal field

Heisenberg model¶

Consider a one-dimensional chain of spins $\frac{1}{2}$ that are nearest-neighbor coupled with equal coupling strength $J$:

\[ {\mathcal H}=-\frac{1}{2}J\sum_{\langle i,j \rangle} {\bf S}_i\cdot{\bf S}_j+\sum_i g\mu_{\rm B}{\bf B}\cdot{\bf S}_i=\sum_i \left( -J{\bf S}_i\cdot{\bf S}_{i+1}+ g\mu_{\rm B}{\bf B}\cdot{\bf S}_i \right) \]

Ferromagnetism¶

Let's assume ${\bf B}=0$ for now. When $J>0$, the spins tend to align and form a ferromagnet. In this case, $\left|\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\right\rangle$ and $\left|\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\right\rangle$ are both valid eigenstates. The order parameter is the total magnetisation $M=\sum_i S_z^i$.

Antiferromagnetism¶

When $J<0$, the term favors anti-alignment, leading to antiferromagnetism. Antiferromagnets typically have zero magnetization, so we need a different order parameter. We can define the staggered magnetization $\tilde{M} = \sum_i (-1)^i S_z^i$, which will become finite if there is alternating order.

Surprisingly, the so-called Néel states $\left|\uparrow\downarrow\uparrow\downarrow\uparrow\downarrow\right\rangle$ and $\left|\downarrow\uparrow\downarrow\uparrow\downarrow\uparrow\right\rangle$ are not eigenstates of the Heisenberg Hamiltonian. This can be seen as follows:

\[ {\mathcal H}=-J\sum_i{\bf S}_i\cdot{\bf S}_{i+1}=-J\sum_i \left( S_x^i S_x^{i+1}+S_y^i S_y^{i+1}+S_z^i S_z^{i+1} \right) \]

Using $S_x=\frac{S_{+}+S_{-}}{2}$ and $S_y=\frac{S_{+}-S_{-}}{2i}$, we can rewrite this as:

\[ {\mathcal H}=-J\sum_i\left(\frac{S_+S_+ + S_+S_- + S_-S_+ + S_-S_-}{4}-\frac{S_+S_+ - S_+S_- - S_-S_+ + S_-S_-}{4}+S_zS_z\right)\\ =-J\sum_i\left(\frac{S_+S_-}{2}+\frac{S_-S_+}{2}+S_zS_z\right) \]

Frustration¶

Finding the correct behavior of the $J<0$ Heisenberg chain is a challenge, both theoretically and experimentally. As particularly fascinating situation arises if there is frustration, where no configuration of the spins can satisfy all bonds (Fig. 20.2). For just three spins the ground state can still be worked out (see Problem 20.2), but nobody knows what happens in large frustrated spin systems.

Forms of coupling¶

Why do spins sometimes prefer to align and sometimes prefer to anti-align? The most obvious explanation is dipolar interaction. But if we estimate the dipolar interaction between two moments $\mu_{\rm B}$ at a distance of 3 Å, we find:

\[ \Delta E\sim \mu_B B_{\rm dipolar} \sim \frac{\mu_0\mu_{\rm B}^2}{4\pi r^3} \sim 2\ \mu{\rm eV} \]

This corresponds to a thermal energy of only ~20 mK! Therefore, other mechanism must be responsible for magnetic ordering:

Exchange interaction – The total wavefunction of two electrons needs to be antisymmetric upon exchange:

\[ \psi({\bf r}_1,{\bf r}_2,{\bf s}_1,{\bf s}_2)=-\psi({\bf r}_2,{\bf r}_1,{\bf s}_2,{\bf s}_1) \]

Coulomb interaction favors symmetric spatial wavefunction, resulting in a preferred antisymmetric spin wavefunction $\Rightarrow J>0$.

Superexchange interaction – When magnetic atoms are connected via one non-magnetic mutual neighbor, simultaneous exchange of electrons with the neighbor can favor anti-alignment $\Rightarrow J<0$.

Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction - Coupling of magnetic atoms via the itinerant electrons inside a metal can lead to an oscillating behavior of $J$ as a function of separation.

Crystal field¶

For a free atom, the orbitals are spherically symmetric (spherical harmonics). Inside a crystal, it can happen that, due to the Coulomb interaction with neighboring non-magnetic atoms, the degeneracy between orbitals is broken. As a result, electrons can no longer complete a full circular orbit around an atom, causing the orbital angular momentum to be quenched: ${\bf L}\Rightarrow 0$.

The remaining spin ${\bf S}$ can favor certain directions for magnetisation (due to spin-orbit coupling). This can be expressed in the Heisenberg Hamiltonian as:

\[ {\mathcal H}=\sum_i \left( -J{\bf S}_i\cdot{\bf S}_{i+1}+ g\mu_{\rm B}{\bf B}\cdot{\bf S}_i -\kappa(S_i^z)^2 \right) \]

If $S>\frac{1}{2}$, e.g. $S=2$, in this case the $S_z = \pm 2$ states will be favored. This is called crystal field anisotropy. If the anisotropy becomes very large, it makes sense to reduce the system to just the $S_z = \pm S$ subspace. This is called the Ising model:

\[ {\mathcal H}=\sum_i \left( -J \sigma_i \sigma_{i+1}+ g\mu_{\rm B}B\sigma_i\right) \]

where $\sigma_i=\pm S$.