Magnetism in Solids

10.1 Types of Magnetism

• Diamagnetism: Weak, negative susceptibility ($\chi \lt 0$). Present in all materials. Arises from the orbital response to an applied field (Lenz’s law). $\chi_d \sim -10^{-5}$.
• Paramagnetism: Positive susceptibility ($\chi \gt 0$). Unpaired spins align with the field. Curie law: $\chi = C/T$ where $C = N\mu_0\mu_{\mathrm{eff}^2/(3k_B)}$.
• Ferromagnetism: Large positive susceptibility. Spontaneous magnetisation below the Curie temperature $T_C$.
• Antiferromagnetism: Adjacent spins antiparallel. Susceptibility peaks at the Néel temperature $T_N$.
• Ferrimagnetism: Antiparallel spins of unequal magnitude. Net magnetisation (e.g., magnetite).

10.2 Diamagnetism

Diamagnetism is the universal tendency of matter to weakly oppose an applied magnetic field.

Langevin diamagnetism. For an atom with $Z$ electrons, each in a circular orbit of radius $\langle r^2 \rangle$A field $B$ along $z$ modifies the angular velocity by $\Delta\omega = eB/(2m_e)$. The induced magnetic moment per atom:

$\mu_{\mathrm{dia} = -\frac{e^2 B}{6m_e}\sum_{i=1}^{Z}\langle r_i^2 \rangle = -\frac{e^2 Z B}{6m_e}\langle r^2 \rangle}$

The susceptibility (per unit volume, with $n$ atoms per unit volume):

$\chi_L = -\frac{\mu_0 n e^2 Z \langle r^2 \rangle}{6m_e}$

This is independent of temperature and very small: $\chi_L \sim -10^{-5}$.

Landau diamagnetism. Free electrons also exhibit diamagnetism. The quantisation of electron Orbits into Landau levels modifies the ground-state energy in an applied field:

$\chi_{\mathrm{Landau} = -\frac{1}{3}\chi_P = -\frac{\mu_0 \mu_B^2 g(\varepsilon_F)}{3}}$

Where $\chi_P$ is the Pauli paramagnetic susceptibility. The total susceptibility of a free electron Gas is $\chi = \chi_P + \chi_{\mathrm{Landau} = \frac{2}{3}\chi_P}$ (still paramagnetic, but Reduced by one-third).

10.3 Paramagnetism

Langevin paramagnetism (classical). For $N$ non-interacting magnetic moments $\boldsymbol{\mu}$ Of magnitude $\mu$ in a field $B$:

$M = N\mu\, L(\mu B/k_B T)$

Where $L(x) = \coth x - 1/x$ is the Langevin function. At high temperature ($\mu B \ll k_B T$):

$L(x) \approx \frac{x}{3} \implies M \approx \frac{N\mu^2 B}{3k_B T}$

Giving the Curie law $\chi = C/T$ with $C = N\mu_0\mu^2/(3k_B)$.

Quantum treatment (Brillouin function). For angular momentum $J$ with $g_J$ the Landé g-factor, The magnetisation is:

$M = Ng_J\mu_B J\, B_J(x)$

Where $x = g_J\mu_B J B/(k_B T)$ and $B_J(x) = \frac{2J+1}{2J}\coth\left(\frac{2J+1}{2J}x\right) - \frac{1}{2J}\coth\left(\frac{x}{2J}\right)$ Is the Brillouin function. For $J = 1/2$ (spin-1/2), $B_{1/2}(x) = \tanh x$.

Pauli paramagnetism. In a metal, the conduction electrons form a degenerate Fermi gas. Only Electrons near $\varepsilon_F$ can flip their spins in response to a field:

$\chi_P = \mu_0\mu_B^2\,g(\varepsilon_F) = \frac{3\mu_0\mu_B^2 N}{2\varepsilon_F}$

This is temperature-independent (up to corrections of order $(T/T_F)^2$), in contrast to the Curie Law. The ratio $\chi_P/\chi_{\mathrm{Curie} \sim T/T_F \sim 10^{-2}}$ at room temperature, Explaining why metals are only weakly paramagnetic.

10.4 Ferromagnetism and the Mean-Field Theory

In the mean-field (Weiss) model, each spin experiences an effective field:

$B_{\mathrm{eff} = B_0 + \lambda M}$

Where $\lambda$ is the molecular field constant and $M$ is the magnetisation.

The spontaneous magnetisation satisfies:

$M = N\mu_B\tanh\left(\frac{\mu_B(B_0 + \lambda M)}{k_B T}\right)$

Setting $B_0 = 0$ and expanding for small $M$ near $T_C$:

$M \approx \frac{N\mu_B^2\lambda M}{k_B T_C}$

Giving the Curie temperature: $T_C = N\mu_B^2\lambda/k_B$.

The critical exponent $\beta = 1/2$ (mean-field value), compared with the experimental value $\beta \approx 1/3$ (3D Ising universality class).

Above $T_C$The susceptibility follows the Curie—Weiss law:

$\chi = \frac{C}{T - T_C}$

10.5 Magnetic Domains

Below $T_C$A ferromagnet divides into domains of uniform magnetisation, separated by domain Walls (Bloch walls). Domain formation reduces the magnetostatic energy.

The domain wall width: $\delta \sim \sqrt{A/K}$ where $A$ is the exchange stiffness and $K$ is the Anisotropy constant. Typical values: $\delta \sim 100$ nm.

The wall energy per unit area: $\sigma_w \sim 4\sqrt{AK}$.

10.6 Magnetic Ordering

Antiferromagnetism. In antiferromagnets (e.g., MnO, NiO), adjacent spins align antiparallel due To negative exchange interaction $J \lt 0$. The Néel temperature is:

$T_N = \frac{\lvert J\rvert z S(S+1)}{3k_B}$

Where $z$ is the number of nearest neighbours. The susceptibility peaks at $T_N$ and decreases at Both higher and lower temperatures.

Ferrimagnetism. In ferrimagnets (e.g., Fe$_3$O$_4$), antiparallel sublattices have different Magnetic moments, giving a net spontaneous magnetisation. The temperature dependence of $M(T)$ is More complex than for simple ferromagnets.

Heisenberg model. The exchange interaction between neighbouring spins is described by:

$\hat{H} = -\sum_{\langle i,j\rangle} J_{ij}\,\hat{\mathbf{S}}_i \cdot \hat{\mathbf{S}}_j$

For $J \gt 0$: ferromagnetic coupling (spins parallel). For $J \lt 0$: antiferromagnetic coupling (spins antiparallel). The exchange integral $J$ arises from the combination of Coulomb repulsion and The Pauli exclusion principle (not from magnetic dipole interactions, which are far too weak).

10.7 Spin Waves (Magnons)

At low temperatures ($T \ll T_C$), the reduction in magnetisation below saturation is carried by Collective excitations called spin waves or magnons.

Linear spin wave theory. For a 1D chain of spins with nearest-neighbour exchange $J$ and Lattice constant $a$The magnon dispersion is:

$\hbar\omega(q) = 2JS[1 - \cos(qa)] = 4JS\sin^2\left(\frac{qa}{2}\right)$

For small $q$ (long wavelength): $\hbar\omega \approx JSa^2 q^2$ (quadratic dispersion, unlike Phonons which are linear).

The magnetisation at low $T$:

$M(T) = M(0)\left[1 - \zeta(3/2)\left(\frac{k_B T}{4\pi JS}\right)^{3/2}\right]$

In 3D, where $\zeta(3/2) \approx 2.612$ is the Riemann zeta function. The $T^{3/2}$ dependence (Bloch $T^{3/2}$ law) is well confirmed experimentally and contrasts with the exponential Freeze-out of a classical paramagnet.

Magnons are bosons and obey Bose—Einstein …/4-statistics-and-probability/2*statistics. They contribute to the low-temperature Specific heat of ferromagnets: $C*{\mathrm{mag} \propto T^{3/2}}$.

10.8 The de Haas—van Alphen Effect

In a magnetic field, the electron orbits are quantised into Landau levels:

$\varepsilon_n = \left(n + \frac{1}{2}\right)\hbar\omega_c, \quad \omega_c = \frac{eB}{m^*}$

The density of states oscillates with $1/B$ (de Haas—van Alphen oscillations). The oscillation period Gives the extremal cross-sectional area of the Fermi surface:

$\Delta\left(\frac{1}{B}\right) = \frac{2\pi e}{\hbar A_{\mathrm{ext}}}$

This is the primary experimental technique for mapping Fermi surfaces.

:::caution Common Pitfall The exchange interaction $J$ in the Heisenberg model is not the magnetic dipole interaction. The dipole energy between two spins is $\sim \mu_0\mu_B^2/a^3 \sim 10^{-4}$ eV, far too small to Explain Curie temperatures of $\sim 10^3$ K ($\sim 0.1$ eV). The exchange interaction is a Consequence of the Coulomb repulsion combined with the antisymmetry of the electron wave function (Pauli principle), and is $10$—$100$ meV.

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