Magnetic Ordering and Spin

Learning Objectives

Magnetism is the place where the abstract ideas of Chapter 4 (Pauli exclusion, exchange–correlation) finally show up as a number you can measure with a SQUID. By the end of this section you should be able to:

1. Explain in one sentence why magnetism is fundamentally quantum-mechanical — and why the Bohr–van Leeuwen theorem says no classical solid can be magnetic.
2. Translate between three vocabularies that describe the same physics: atomic spin $\mathbf{S}_i$, magnetic moment $\mathbf{m}_i = -g\mu_B\,\mathbf{S}_i/\hbar$, and the spin density $m(\mathbf{r}) = n_{\uparrow}(\mathbf{r}) - n_{\downarrow}(\mathbf{r})$.
3. Recognise the four canonical magnetic orderings — paramagnetic, ferromagnetic (FM), antiferromagnetic (AFM), and ferrimagnetic (FiM) — by inspection of a unit cell, and predict which one a Heisenberg Hamiltonian $\hat{H} = -\sum_{\langle ij\rangle} J_{ij}\,\mathbf{S}_i\!\cdot\!\mathbf{S}_j$ will favour from the sign of $J_{ij}$.
4. Read a spin-polarised band structure or DOS plot fluently: identify the exchange splitting $\Delta_{\text{ex}}$, compute the net moment as $\mu = \int_{-\infty}^{E_F}\!\bigl[n_{\uparrow}(E) - n_{\downarrow}(E)\bigr]\,dE$, and apply the Stoner criterion $I\,g(E_F) > 1$ to predict ferromagnetism in a metal.
5. Set up a spin-polarised VASP calculation correctly: the meaning of ISPIN, MAGMOM, NUPDOWN, LORBIT, and LMAXMIX; and how to seed an AFM unit cell so the SCF does not collapse to FM.

One-line preview: a magnetic crystal is what happens when the Pauli exclusion principle, the Coulomb interaction, and the translation symmetry of a lattice conspire to make it cheaper for some electrons to align their spins than to antialign them. Everything else — Heisenberg $J$, exchange splitting, Stoner enhancement, MAGMOM in VASP — is bookkeeping on top of that one idea.

Where Magnetism Comes From

It is worth pausing on a deep and counterintuitive fact: classical physics has no room for permanent magnetism in a solid. The Bohr–van Leeuwen theorem (1911, 1921) shows that any ensemble of classical charged particles in thermal equilibrium has zero average magnetic moment, no matter what magnetic field you apply. Lorentz forces do no work, the partition function is independent of the vector potential, and the magnetic susceptibility is identically zero. Yet a refrigerator magnet sticks. Where does the moment come from?

It comes from quantum mechanics, in two distinct steps. First, every electron carries an intrinsic magnetic moment $\boldsymbol{\mu}_e = -g_s\,\mu_B\,\mathbf{S}/\hbar$ with $g_s \approx 2.0023$ and $\mu_B = e\hbar/(2 m_e) \approx 9.27\times10^{-24}\,\text{J/T}$. That is a relativistic effect; it does not exist in the classical world. Second, when you put many electrons into the same crystal, the Pauli exclusion principle forbids two of them from occupying the same spatial orbital with the same spin. This forces the wavefunction to be antisymmetric, which in turn lowers the Coulomb repulsion between same-spin electrons (they cannot get close to each other). Aligning spins becomes energetically favourable — even though the magnetic dipole–dipole interaction is far too weak to do that on its own.

This is why the Heisenberg Hamiltonian we will write down in a moment looks like a spin-spin interaction even though there is no such fundamental force. The $J\,\mathbf{S}_i\!\cdot\!\mathbf{S}_j$ term is an effective description of the spin-dependent part of the Coulomb energy, integrated out from the underlying many-electron wavefunction. The number $J$ is what survives.

Spin as a Second Quantum Label

Section 5.1 introduced the band index $n$ and the Brillouin-zone wavevector $\mathbf{k}$ as the two labels of a Bloch state. There is a third label, hidden in plain sight: spin. A complete Bloch state is $|n,\mathbf{k},\sigma\rangle$ with $\sigma \in \{\uparrow, \downarrow\}$. Inside a non-magnetic, non-relativistic crystal the two spin channels are degenerate by time-reversal symmetry — every band carries two electrons, one up and one down, and we can ignore the label. The moment we break that degeneracy, by either spontaneous magnetism or an external field, the third label comes alive.

For collinear magnetism (every spin pointing along a single global axis, conventionally $\hat{z}$) the bookkeeping simplifies dramatically: the Hamiltonian decomposes into two independent blocks, one for each spin channel, and we have two Kohn–Sham equations instead of one,

with two distinct effective potentials $v_{\uparrow}^{\text{KS}}$ and $v_{\downarrow}^{\text{KS}}$. The whole magnetic story is contained in the fact that, in a magnetic ground state, those two potentials are not equal. They differ by precisely the exchange–correlation field $B_{\text{xc}}(\mathbf{r})$:

That is the entire content of spin-density-functional theory: promote the density $n(\mathbf{r})$ to a pair $(n_{\uparrow}(\mathbf{r}), n_{\downarrow}(\mathbf{r}))$, let the exchange–correlation functional depend on both, and you automatically get spin-split Kohn–Sham potentials. The local magnetisation is then the difference,

and the total moment per unit cell is the integral

where $N_{\sigma}$ is the total number of electrons in spin channel $\sigma$ per unit cell. The headline number a VASP calculation prints — "total moment = 2.20 μ_B/cell" — is exactly this integral.

The Exchange Interaction and the Heisenberg Model

Take two electrons, one in orbital $\phi_a$ on site A and one in $\phi_b$ on site B. Pauli forbids them from occupying the same spin-orbital, so the two-electron wavefunction must be antisymmetric. There are two ways to achieve that:

• Singlet (antiparallel spins). Symmetric spatial part, antisymmetric spin part. The electrons can sit on the same atom for an instant (virtual hopping is allowed), which costs a Coulomb-repulsion penalty $U$ but lowers the kinetic energy by $\sim t^2/U$ through second-order perturbation theory.
• Triplet (parallel spins). Antisymmetric spatial part, symmetric spin part. The two electrons cannot occupy the same orbital — the wavefunction vanishes when $\mathbf{r}_1 = \mathbf{r}_2$ (the "Fermi hole"). No virtual hopping, no kinetic-energy gain from $U$, but also no Coulomb penalty — parallel spins effectively avoid each other, which lowers the direct Coulomb integral by an amount called the exchange integral $K_{ab}$.

The energy difference between the two configurations defines the exchange coupling $J_{ab} = E_{\text{singlet}} - E_{\text{triplet}}$. Through the Heitler–London or Hubbard analyses one finds the compact result

which is a beautiful little equation. The first term $2K_{ab} > 0$ is the direct exchange — it favours parallel spins (FM tendency). The second term $-4t^2/U$ is the kinetic exchange — it favours antiparallel spins (AFM tendency). For most insulating magnets the kinetic term wins and the ground state is AFM; for many metallic 3d ferromagnets (Fe, Co, Ni) the direct term wins. The sign and magnitude of $J$ tell you which ordering will appear.

Once you have computed the per-bond $J_{ij}$ (in DFT this is done by mapping total-energy differences between magnetic configurations onto a model — see "Energy Comparison" below), the magnetic crystal is described by the Heisenberg Hamiltonian:

with a sign convention where $J_{ij} > 0$ favours parallel spins (FM) and $J_{ij} < 0$ favours antiparallel (AFM). The sum runs over neighbour pairs $\langle ij\rangle$. Despite its appearance, this is not a fundamental Hamiltonian — it is the effective low-energy description of the same Coulomb-plus-Pauli physics we just derived.

Interactive — The Four (and a Half) Magnetic Orderings

Below is a 3×3×3 simple-cubic lattice of magnetic atoms. Each atom carries a spin arrow. Click the buttons across the top to switch between paramagnetic (PM), ferromagnetic (FM), G-type and A-type antiferromagnetic (AFM-G, AFM-A), ferrimagnetic (FiM), and a non-collinear (NC) helical state. The footer reports the live net moment $\boldsymbol{\Sigma}\,\mathbf{m} = \sum_i \mathbf{m}_i$ so you can verify by hand that AFM cancels and FM saturates.

Three things to do with the controls, in order:

1. Switch to AFM-G. Every atom is anti-aligned with all six of its nearest neighbours along x, y, and z — a 3-D checkerboard. The net moment ΣM is essentially zero. This is the ground state of MnO, NiO, and most of the antiferromagnetic rock-salt oxides. Note that the magnetic unit cell (period 2 along each axis) is twice as large as the chemical unit cell — a fact that will bite us when we set up VASP supercells.
2. Switch to AFM-A. Within each (xy)-layer the spins are FM; between layers they alternate. This shows up in the perovskite manganites and in CrI₃-type 2D magnets. The magnetic unit cell is doubled along z but not along x/y.
3. Switch to FiM with the amplitude slider. The two sublattices have unequal magnitudes and opposite signs. The net moment is the difference, not zero. This is the structure of magnetite Fe₃O₄ (the original ‘lodestone’): two Fe sites with opposite spins, one with twice the moment of the other, giving a residual macroscopic magnetisation.

Spin-Polarized DFT — Two Densities, One Crystal

We are now ready to plug magnetism into the Kohn–Sham machinery from Section 4.6. The promotion is mechanical: instead of one density-functional Lagrangian we have two, one for each spin channel, coupled only through the exchange–correlation functional. The total energy becomes a functional of the two densities,

with $n = n_{\uparrow} + n_{\downarrow}$. Functional differentiation gives one Kohn–Sham equation per spin channel, with potentials

The first two terms (external potential and Hartree term) are spin-blind — they depend only on the total density. The exchange–correlation term is where spin polarisation enters. In the local spin-density approximation (LSDA), we evaluate the spin-polarised homogeneous electron-gas functional at the local densities,

where $\zeta = (n_{\uparrow}-n_{\downarrow})/n \in [-1, 1]$ is the relative spin polarisation. GGA functionals (PBE, PW91) generalise this by also depending on $\nabla n_{\sigma}$. The point is that $\varepsilon_{\text{xc}}$ is a different function of $\zeta$; for $\zeta \neq 0$ the energy density is lower than the spin-balanced result. That difference is the driving force for magnetism in DFT.

The price for magnetism is computational: ISPIN = 2 doubles the size of the eigenvalue problem (two channels of bands) and roughly doubles the wall time. The pay-off is that every electronic property now comes with a spin label: bands split into majority and minority, DOS becomes two curves, and the per-atom moment is a number you can read off OUTCAR.

Interactive — Exchange Splitting and the Net Moment

The cleanest way to see magnetism in DFT output is the spin-resolved DOS. We plot $n_{\uparrow}(E)$ on one side of the energy axis and $n_{\downarrow}(E)$ on the other, with a horizontal Fermi-level line. The net moment per cell is the integrated difference of occupied states,

which is reported live in the bottom strip of the figure. Drag Δ_ex to control how strongly the two channels split apart — physically this is set by the exchange–correlation field — and drag E_F to control the band filling.

Two limits to feel out:

1. Δ_ex = 0. The two curves coincide. Every state below E_F is doubly occupied (one ↑, one ↓), so $N_{\uparrow} = N_{\downarrow}$ and the moment is zero. This is the non-magnetic, paramagnetic ISPIN = 1 picture.
2. Δ_ex ≈ 1.5 eV, E_F in the gap between the two ↓ peaks. The majority channel is mostly filled, the minority channel is partly emptied, and the moment opens up to ~1–2 μ_B per atom. This is the band-theory picture of the BCC-Fe ground state — a so-called weak ferromagnet, where both channels still cross E_F. Push Δ_ex higher and the minority channel can be pushed entirely above E_F: this is a half-metal (NiMnSb, CrO₂) — gapped in one channel and metallic in the other, the ideal source for spintronic devices.

When Does a Metal Become Ferromagnetic? The Stoner Criterion

Why do Fe, Co, and Ni order ferromagnetically while their neighbours Cu, Pd, and Pt do not? The exchange parameter $I$ is similar across the 3d row, so it must be something about the band structure itself. Stoner's 1938 argument is one of the most elegant criteria in solid-state physics.

Imagine starting from a non-magnetic metal and infinitesimally moving a tiny number of electrons $\delta n$ from the ↓ channel just below E_F to the ↑ channel just above. The cost in kinetic energy is $\delta T = \delta n / g(E_F)$ per electron moved (the inverse density of states sets the energy gap you have to span). The reward in exchange energy is $\delta E_{\text{ex}} = -I\,\delta n$ (each spin flip lowers the energy by $I$ times the imbalance you just created). The total energy change is therefore

and ferromagnetism is the spontaneous winner whenever the bracket is negative — that is, the Stoner criterion:

A high density of states at the Fermi level is the lever; the exchange parameter $I$ is the prize.

Pd is the famous near-miss: $I\,g(E_F) \approx 1.1$, almost satisfying the criterion but not quite. Its susceptibility is enhanced by an order of magnitude over the Pauli value, and a tiny amount of Co or Ni doping pushes it over the threshold. Cu, with its full d-shell, has tiny $g(E_F)$ from the s-band only and the criterion fails by an order of magnitude.

Picking the Ground State by Total-Energy Comparison

For most magnetic materials there is no analytical shortcut for deciding which ordering is the ground state. The standard recipe is embarrassingly direct: compute the total energy of every candidate ordering and pick the smallest. For a binary transition-metal oxide like NiO this means at minimum:

1. A non-magnetic (ISPIN = 1) reference run.
2. A ferromagnetic (FM) run on the chemical unit cell.
3. A G-type AFM run on a doubled magnetic unit cell (e.g. a 2×2×2 supercell of the rock-salt cell).
4. Optionally, A-type, C-type, and other AFM patterns if the lattice allows them.

Order the resulting energies. The lowest is the predicted ground state. The differences also let you fit the Heisenberg $J_{ij}$: writing

gives you $J$ from a single energy difference, with $z$ the coordination number and $S$ the local spin. Repeat with second- and third-neighbour AFM patterns to extract $J_2$, $J_3$, etc. This is the workhorse "four-state mapping" method used in modern magnetic first-principles studies.

VASP — Spin-Polarized Calculations from INCAR to OUTCAR

Setting up a magnetic VASP calculation requires four INCAR tags (ISPIN, MAGMOM, LORBIT, LMAXMIX) and a discipline about reading the output. Here is the canonical template for BCC iron — the simplest interesting ferromagnet — followed by a line-by-line walkthrough.

1bcc Fe (a = 2.866 Å, conventional cell of 2 atoms)
21.0
3   2.866   0.000   0.000
4   0.000   2.866   0.000
5   0.000   0.000   2.866
6Fe
72
8Direct
9   0.0   0.0   0.0
10   0.5   0.5   0.5

1Automatic mesh
20
3Gamma
415 15 15
5 0  0  0

The INCAR — annotated below

1# INCAR — BCC Fe, ferromagnetic ground state
2SYSTEM  = bcc Fe (ferromagnetic)
3PREC    = Accurate
4ENCUT   = 400         # plane-wave cutoff (eV)
5EDIFF   = 1E-6        # SCF tolerance per cell (eV)
6ISMEAR  = 1           # Methfessel-Paxton — best for metals
7SIGMA   = 0.2         # smearing width (eV)
8
9ISPIN   = 2           # turn on spin polarisation (2 channels)
10MAGMOM  = 2*2.5       # initial moment: +2.5 mu_B on each of 2 atoms
11LORBIT  = 11          # write per-atom moments to OUTCAR/PROCAR
12LMAXMIX = 4           # higher l-mixing — needed for transition metals
13
14LREAL   = .FALSE.     # exact projectors (small cell)
15LWAVE   = .FALSE.
16LCHARG  = .TRUE.

Code Walkthrough — Reading the Magnetic Output

After the SCF converges, all the magnetic information you need is scattered through OUTCAR and (for plotting) vasprun.xml. The following pymatgen snippet pulls it out in a few lines.

1# read_magnetic_state.py — extract moments and DOS from a VASP run
2from pymatgen.io.vasp import Outcar, Vasprun
3import numpy as np
4
5oc = Outcar("OUTCAR")
6vr = Vasprun("vasprun.xml")
7
8mags    = [s["tot"] for s in oc.magnetization]   # per-atom total moments
9mu_cell = sum(mags)                              # total moment per cell
10
11dos = vr.complete_dos
12spin_up   = dos.densities[1]                     # n_up(E) on dos.energies grid
13spin_down = dos.densities[-1]                    # n_down(E) — note negative sign
14
15E_F = vr.efermi                                  # Fermi energy in eV
16mask = dos.energies <= E_F
17N_up_occ   = np.trapz(spin_up[mask],   dos.energies[mask])
18N_down_occ = np.trapz(spin_down[mask], dos.energies[mask])
19
20print(f"per-atom moments: {mags}")
21print(f"total moment: {mu_cell:.3f} mu_B / cell")
22print(f"moment from DOS integration: {N_up_occ - N_down_occ:.3f} mu_B / cell")

Pitfalls You Will Hit On Your First Try

Spin-polarised calculations have a small zoo of failure modes. Each of these will eat half a day from a new student; recognising them in advance saves the day.

Looking Ahead — Spin-Orbit Coupling

Everything in this section assumed the spin axis is global — every moment is along $\hat{z}$ (collinear). That is exact in a non-relativistic Hamiltonian, where the spin-rotation symmetry of the kinetic and Coulomb operators makes the choice of axis arbitrary. The world is not non-relativistic. The spin-orbit interaction

couples the spin to the spatial wavefunction and breaks the rotational invariance of the spin space. The consequences for a magnetic crystal are dramatic: the moments now point in a definite crystallographic direction (magnetocrystalline anisotropy), bands split by a few hundred meV in heavy elements, and entirely new phenomena emerge — Rashba splitting, Dirac semimetals, topological insulators. This is the subject of Section 5.7 — Spin–Orbit Coupling, where we will turn on the LSORBIT tag in VASP and see exactly how the band structure of CdSe (a near-textbook material for our chapter-6 case study) splits at the valence-band maximum.

Summary

• Magnetism is quantum. The Bohr–van Leeuwen theorem rules out classical magnetism; permanent moments come from electron spin combined with the Pauli exclusion principle, which turns Coulomb repulsion into a spin-dependent exchange interaction.
• Spin is a third Bloch label. A complete state is $|n,\mathbf{k},\sigma\rangle$. In a magnetic crystal the two spin channels have different Kohn–Sham potentials, differing by the exchange–correlation field $B_{\text{xc}}$.
• The four canonical orderings — PM, FM, AFM, FiM — together with non-collinear states cover essentially every magnetic crystal you will meet. $J_{ij} > 0$ in a Heisenberg model favours FM, $J_{ij} < 0$ favours AFM.
• Spin-polarised DFT (LSDA / GGA with $E_{\text{xc}}[n_{\uparrow}, n_{\downarrow}]$) gives every electronic property a spin label. The exchange splitting $\Delta_{\text{ex}}$ separates majority and minority bands; the integrated occupied difference is the moment.
• The Stoner criterion $I\,g(E_F) > 1$ says ferromagnetism appears in a metal whenever the band exchange parameter times the Fermi-level density of states exceeds unity. Fe/Co/Ni satisfy it; Cu/Pd do not.
• In VASP: ISPIN = 2 turns on spin polarisation, MAGMOM seeds the SCF (must be set explicitly for AFM cells), LORBIT = 11 writes per-atom orbital-resolved moments, and LMAXMIX = 4 stabilises the SCF for 3d transition metals. The ground-state ordering is decided by total-energy comparison across candidate magnetic configurations.

Coming next: Section 5.7 — Spin–Orbit Coupling — where we let the spin axis tilt off $\hat{z}$, watch new band splittings open, and turn on the LSORBIT tag in VASP for the CdSe case study.