Magnetism in solid state physics

It's all about the interactions.
Single ion magnetism

Atomic Magnetism

  • Transition metals: Fe(3d), Pd(4d), Pt(5d)
  • Rare-earth metals (4f) and actinides (5f)
  • Competition between Hund's coupling, spin-orbit coupling and crystal field
  • Ions with non-zero magnetic moment in ground state
Single ion magnetism

Single ion anisotropy

  • General formula:

    • $$ E_{ani} = \mathbf{S}^T A \mathbf{S} $$
  • For example - Easy axis anisotropy:

    • $$ A = \begin{pmatrix}0 & 0 & 0\\0 & 0 & 0 \\0 & 0 &-A_{zz}\end{pmatrix} $$
  • Depends on type of magnetic ion and crystal field
  • Anisotropy in 3d transition metals often weak, not always, e.g. trigonal Co$^{3+}$ in Ca$_3$Co$_2$O$_6$ with $S = 2$ and $M_L = 1.57/mu B$, $A_{zz} = −7.20(2)$ meV
  • There is also single ion anisotropy in Oh symmetry!

$$ \begin{align} E_{ani} & = -A_{cub} (S_x^4 + S_y^4 + S_z^4)\\ E_{eff} & = -A_{eff} S_x^2 \\ A_{eff} & = 2A_{cub}S^2 \end{align} $$
Magnetic interactions in insulators

Dipole-dipole interaction

$$ E_{DD} \approx \begin{Bmatrix} 3(\mathbf{S}_1\cdot \hat{r})(\mathbf{S}_2\cdot \hat{r}) - \mathbf{S}_1 \cdot \mathbf{S}_2 \end{Bmatrix}$$

For electron spins:

$$ E_{DD} \approx 0.3 \frac{meV}{r^3_A} \rightarrow 1K $$


Interatomic Exchange

  • No hopping: Coulomb repulsion favors parallel spin alignment (FM)
  • Hopping: favors AFM alignment (lower kinetic energy)


Kanamori-Goodenough rules

  • Phenomenological description of superexchange interaction
  • Interaction between overlapping orbitals: strong AF
  • Interaction between non-overlapping orbitals: weak FM
Magnetic interactions in insulators

Spin-orbit coupling (SOC)

  • In 3d transition metals weak, gives weak anisotropy and Dzyaloshinskii-Moriya interaction
  • Crystal field quenches orbital angular momentum
  • In rare earth, SOC is stronger $\rightarrow$ strong anisotropy


Magnetic exchange

  • General formula:

$$ E_{exc} = \sum_{i,j} \mathbf{S}^T_i J_{i,j} \mathbf{S}_j $$
  • Symmetric and asymmetric:

$$ J_S = \begin{pmatrix}J_x & 0 & 0\\0 & J_y & 0\\0 & 0 & J_z\end{pmatrix} \quad J_A = \begin{pmatrix}0 & D_z & -D_y\\-D_z & 0 & D_x\\D_y & -D_x & 0\end{pmatrix} $$