Spin–Orbit Coupling in Solids

Spin–orbit coupling in a solid is the relativistic coupling that entangles an electron’s spin with its orbital motion through the crystal potential. It begins with the same local physics that produces atomic fine structure, but a crystal adds orbital hybridization, crystal fields, Bloch momentum, surfaces, interfaces, and space-group symmetry. Those ingredients decide whether spin–orbit coupling merely shifts local levels, splits a band, changes magnetic anisotropy, generates a transverse spin response, or reorganizes the topology of the occupied states.

The phrase “strong spin–orbit coupling” is incomplete without a comparison scale. A matrix element $\lambda$ can be large relative to a magnetic exchange energy and small relative to a crystal-field gap; it can strongly alter a nearly degenerate band crossing while barely changing remote bands. The useful question is therefore not whether spin–orbit coupling is present—it is—but which projected Hilbert space it acts in, which symmetries constrain it, and which competing energy scales it exceeds.

Spin–Orbit Coupling derives the angular-momentum-addition spectrum of $\mathbf L\cdot\mathbf S$. Fine Structure develops its atomic spectroscopic role, and the Pauli Equation gives its nonrelativistic origin. This article starts from those results and develops their consequences in crystalline materials.

Required background. Spin–Orbit Coupling supplies angular-momentum algebra, while Symmetry of Bloch States supplies little-group, double-group, and degeneracy labels.

Helpful background. Tight-Binding Models supplies basis projections, Time Reversal for Spin-1/2 supplies Kramers constraints, and Berry Curvature supplies the geometric response object.

Spin–orbit interactions and crystal energy scales

The local and band models below use these definitions:

• $m_0>0$ is the free-electron mass, while $m^\ast$ is a signed band effective mass when a one-band approximation is valid.
• $V(\mathbf r)$ is the electron’s potential energy, not the electrostatic potential.
• $\boldsymbol{\sigma}$ denotes Pauli matrices in an explicitly stated two-state subspace. It represents physical spin only when that projection is justified.
• $\mathbf k$ is the Bloch wavevector; $\hbar\mathbf k$ is crystal momentum modulo a reciprocal-lattice momentum. A continuum expansion is local to a specified high-symmetry point and is not automatically periodic over the Brillouin zone.
• Time reversal for a physical spin-$1/2$ doublet is $\mathcal T=i\sigma_yK$, up to an overall phase, and $\mathcal T^2=-1$.
• Inversion sends $\mathbf k\mapsto-\mathbf k$ but leaves spin unchanged because spin is an axial vector.
• The signs of Rashba and Dresselhaus coefficients depend on axis, basis, interface-normal, and Pauli-matrix conventions. Spectra determine magnitudes more directly than signs.

Several energy scales must be kept separate:

$$\begin{gathered} \lambda_{\mathrm{at}},\quad\Delta_{\mathrm{CF}},\quad t,\\ \Delta_{\mathrm{ex}},\quad\Delta_{\mathrm{SO}}(\mathbf k),\quad\hbar/\tau. \end{gathered}$$

Here $\lambda_{\mathrm{at}}$ is an atomic or local spin–orbit scale, $\Delta_{\mathrm{CF}}$ a crystal-field splitting, $t$ a hopping or bandwidth scale, $\Delta_{\mathrm{ex}}$ an exchange splitting, $\Delta_{\mathrm{SO}}(\mathbf k)$ a band splitting or avoided-crossing gap, and $\hbar/\tau$ a disorder broadening. There is no universal ordering among them.

Relativistic origin

For a static scalar potential, the Foldy–Wouthuysen expansion of the Dirac Hamiltonian contains the Pauli spin–orbit term

$$H_{\mathrm{SO}} = \frac{\hbar}{4m_0^2c^2} \boldsymbol{\sigma}\cdot \left( \boldsymbol{\nabla}V \times \mathbf p \right).$$

The coefficient includes the Thomas-precession factor. If one instead writes an electrostatic potential $\phi$ for an electron of charge $-e$, then $V=-e\phi$; replacing $V$ by $\phi$ without the charge factor changes both the sign and the units.

For a central potential $V(r)$,

$$\boldsymbol{\nabla}V \times \mathbf p = \frac{1}{r} \frac{\mathrm dV}{\mathrm dr} \mathbf L ,$$

so that

$$H_{\mathrm{SO}} = \frac{1}{2m_0^2c^2r} \frac{\mathrm dV}{\mathrm dr} \mathbf L\cdot\mathbf S .$$

The large gradient near a heavy nucleus is why atomic spin–orbit scales generally increase strongly with atomic number. In a solid, however, a band state is a coherent mixture of atomic orbitals. The relevant matrix element depends on which atomic regions it samples, the orbital characters and covalency, and the projection used to construct a low-energy model. “Made from a heavy element” is useful chemical guidance, not a quantitative answer.

In the central-potential equation, $\mathbf L$ and $\mathbf S=\hbar\boldsymbol\sigma/2$ are physical angular momenta. For the local-shell model, introduce dimensionless operators $\boldsymbol\ell=\mathbf L/\hbar$ and $\mathbf s_{\mathrm{loc}}=\mathbf S/\hbar$, so that the coefficient $\lambda$ is an energy:

$$H_{\mathrm{loc}} = H_{\mathrm{CF}} + \lambda\,\boldsymbol\ell\cdot\mathbf s_{\mathrm{loc}} + H_{\mathrm{cov}} + H_{\mathrm{int}},$$

Here $H_{\mathrm{CF}}$ is the crystal field, $H_{\mathrm{cov}}$ represents hybridization with ligands, and $H_{\mathrm{int}}$ contains local interactions. Equivalently, the spin–orbit term is $(\lambda/\hbar^2)\mathbf L\cdot\mathbf S$ in physical angular momenta. The fitted $\lambda$ is an effective energy scale; it need not equal a free-ion constant.

Crystal fields, orbital quenching, and pseudospin

A crystal field removes some or all of an atomic multiplet’s orbital degeneracy. If an orbital eigenstate is nondegenerate and can be chosen real, its first-order expectation value of $\mathbf L$ vanishes. This is the usual sense in which the orbital moment is quenched. It does not mean that $\mathbf L$ has disappeared as an operator. Virtual admixture of excited crystal-field levels produces corrections of order

$$\delta E_{\mathrm{SO}} \sim \frac{\lambda^2}{\Delta_{\mathrm{CF}}},$$

as well as anisotropic $g$ factors and magnetocrystalline anisotropy. Near an orbital degeneracy, spin–orbit coupling can instead act in first order.

Cubic d orbitals

In an ideal octahedral environment, the five $d$ orbitals split into a lower $t_{2g}$ triplet and an upper $e_g$ doublet for the usual ligand-field sign. Within the $t_{2g}$ manifold, projected dimensionless orbital matrices can be represented by an effective $\ell_{\mathrm{eff}}=1$ algebra with the opposite sign:

$$P_{t_{2g}}\,\boldsymbol\ell\,P_{t_{2g}} = -\boldsymbol\ell_{\mathrm{eff}}.$$

Consequently,

$$P_{t_{2g}}H_{\mathrm{SO}}P_{t_{2g}} = -\lambda\, \boldsymbol\ell_{\mathrm{eff}}\cdot\mathbf s_{\mathrm{loc}} .$$

The statement is about matrices in a specified orbital basis, not a claim that the electron’s microscopic orbital angular momentum has literally changed sign. Electron versus hole descriptions, distortions, covalency, and the sign convention for $\lambda$ must all be declared before assigning level order.

For one electron, positive $\lambda$ and an isolated ideal $t_{2g}$ manifold, adding $\ell_{\mathrm{eff}}=1$ to $s=1/2$ gives

$$E_j=-\frac{\lambda}{2}[j(j+1)-2-3/4].$$

The $j_{\mathrm{eff}}=3/2$ quartet lies at $-\lambda/2$ and the $j_{\mathrm{eff}}=1/2$ doublet at $+\lambda$, with gap $3\lambda/2$. These are electron levels; a hole filling or a many-electron multiplet must be treated with its own occupancy and interaction conventions. The ideal quartet can split into Kramers doublets under lower-symmetry crystal fields.

In the ideal $e_g$ subspace, the direct projection $P_{e_g}\mathbf LP_{e_g}$ vanishes. Spin–orbit effects nevertheless survive through mixing with $t_{2g}$ states, ligand orbitals, and lower-symmetry fields. Thus “orbital quenching” and “negligible spin–orbit coupling” are not synonyms.

Three useful coupling regimes

The order in which one diagonalizes local terms should follow the scale hierarchy:

1. If $\Delta_{\mathrm{CF}}\gg|\lambda|$, first diagonalize the crystal field and project $\boldsymbol\ell\cdot\mathbf s_{\mathrm{loc}}$ into its low-energy manifold.
2. If $|\lambda|\gg\Delta_{\mathrm{CF}}$, first form atomic $j$ multiplets and then apply the crystal field within them.
3. If the scales are comparable, neither limiting label is exact; diagonalize the full local Hamiltonian and report wave-function composition rather than forcing a pure $LS$- or $jj$-coupling name.

These choices describe a one-electron shell or an already selected many-electron multiplet. For a many-electron ion, Coulomb and Hund splittings also determine whether $LS$, $jj$ or an intermediate-coupling description is appropriate. Comparing only $\lambda$ and $\Delta_{\mathrm{CF}}$ does not decide that question.

The low-energy two-state object is often a Kramers pseudospin rather than a bare spin. Its magnetic moment has the general form

$$\boldsymbol{\mu} = -\mu_{\mathrm B}\, \mathbf g\cdot \frac{\boldsymbol{\sigma}}{2},$$

where $\mathbf g$ is a tensor. Neutron scattering, magnetic resonance, and Zeeman coupling probe matrix elements of the physical moment, not the abstract Pauli labels alone.

Top: for $\lambda>0$, the isolated one-electron $t_{2g}$ model gives a quartet below a doublet, separated by $3\lambda/2$. Horizontal marks at the same height denote degeneracy. Bottom: for positive coefficients and physical-spin Pauli matrices, arrows show $\langle\boldsymbol\sigma\rangle_+$ for the linear Rashba and [001] Dresselhaus models. The lower branch reverses every arrow; the circles sample directions and do not assert a particular Fermi energy.

Bloch Hamiltonians and symmetry

Near a chosen point, an isolated two-state band subspace can be written

$$H(\mathbf k) = \varepsilon_0(\mathbf k)I + \mathbf g(\mathbf k)\cdot\boldsymbol{\sigma}.$$

Its eigenvalues are

$$E_\pm(\mathbf k) = \varepsilon_0(\mathbf k) \pm |\mathbf g(\mathbf k)|.$$

For $\boldsymbol{\sigma}$ representing physical spin, the two eigenstates have

$$\langle\boldsymbol{\sigma}\rangle_\pm = \pm \frac{\mathbf g}{|\mathbf g|}.$$

For a spin–orbital pseudospin, this equation describes the pseudospin texture; the physical spin texture follows only after projecting the spin operator into the same subspace.

Time reversal and inversion

Choose a local Kramers basis in which time reversal is represented by $i\sigma_yK$ at the paired momenta. Time-reversal invariance requires

$$\mathcal T H(\mathbf k) \mathcal T^{-1} = H(-\mathbf k),$$

which gives

$$\varepsilon_0(\mathbf k) = \varepsilon_0(-\mathbf k), \qquad \mathbf g(\mathbf k) = -\mathbf g(-\mathbf k).$$

A general momentum-dependent projected basis uses a sewing matrix instead; the simple even/odd relations are statements in the declared basis, not gauge-independent component equations. At a time-reversal-invariant momentum $\boldsymbol{\Lambda}$ satisfying

$$\boldsymbol{\Lambda} = -\boldsymbol{\Lambda} + \mathbf G,$$

with reciprocal vector $\mathbf G$, the odd field vanishes and the pair is Kramers degenerate.

If inversion is also present, then

$$\mathcal P H(\mathbf k) \mathcal P^{-1} = H(-\mathbf k).$$

Within a same-parity spin doublet this makes $\mathbf g$ even. Time reversal makes it odd, so $\mathbf g=0$. More generally, the antiunitary operation $\mathcal P\mathcal T$ leaves every $\mathbf k$ fixed and satisfies

$$(\mathcal P\mathcal T)^2=-1$$

for spinful electrons when inversion and time reversal commute. Every Bloch level is then at least twofold degenerate at every $\mathbf k$, not only at the time-reversal-invariant momenta. Kramers Degeneracy develops the theorem and its hypotheses.

Breaking inversion while preserving time reversal allows spin-split bands away from the invariant momenta. Breaking time reversal allows a Zeeman or exchange term even at those momenta. A measured band splitting therefore does not identify Rashba coupling until the relevant crystal, surface, magnetic, and little-group symmetries have been established.

Rashba coupling

For an approximately isotropic two-dimensional band with a polar axis $\hat{\mathbf z}$, the leading structural-inversion-asymmetry term is

$$H_{\mathrm R} = \alpha_{\mathrm R} \left( k_y\sigma_x-k_x\sigma_y \right) = \alpha_{\mathrm R} \left( \mathbf k\times\hat{\mathbf z} \right)\cdot \boldsymbol{\sigma}.$$

This convention gives

$$\mathbf g_{\mathrm R} = \alpha_{\mathrm R} \left( k_y,-k_x,0 \right).$$

The sign changes if the surface normal, coordinate orientation, or Pauli basis is reversed. For a parabolic band,

$$H = \frac{\hbar^2k^2}{2m^\ast}I + H_{\mathrm R},$$

and

$$E_\pm(k) = \frac{\hbar^2k^2}{2m^\ast} \pm |\alpha_{\mathrm R}|k .$$

For $m^\ast>0$, useful scales are

$$k_{\mathrm R} = \frac{m^\ast|\alpha_{\mathrm R}|}{\hbar^2}, \qquad E_{\mathrm R} = \frac{m^\ast\alpha_{\mathrm R}^2}{2\hbar^2}.$$

The lower branch has a ring minimum at $k=k_{\mathrm R}$ lying $E_{\mathrm R}$ below the unsplit crossing energy in this ideal model. The two pseudospins wind tangentially in opposite directions around the crossing.

It is tempting to identify $\alpha_{\mathrm R}$ with a bare electric field in the vacuum Pauli term. In actual solids, $\alpha_{\mathrm R}$ is a band parameter produced by inversion-asymmetric orbital hybridization together with atomic spin–orbit matrix elements. Wave-function weight near heavy atoms, avoided crossings, confinement, electrostatic gating, and structural relaxation can all matter. A large potential gradient without suitable orbital mixing need not produce a large band splitting.

Crystal symmetry also permits higher harmonics. Hexagonal warping, for example, can generate an out-of-plane spin component and a noncircular contour. The linear isotropic Hamiltonian is a controlled low-$k$ model, not a universal definition of every surface splitting.

Dresselhaus coupling

Bulk inversion asymmetry in a zinc-blende crystal permits the cubic Dresselhaus invariant

$$H_{\mathrm D}^{(3\mathrm D)} =\gamma\big[ k_x(k_y^2-k_z^2)\sigma_x +k_y(k_z^2-k_x^2)\sigma_y +k_z(k_x^2-k_y^2)\sigma_z \big].$$

The angular dependence follows the crystal axes and is qualitatively different from the rotationally symmetric Rashba field. Along selected high-symmetry directions, the leading splitting vanishes.

Confinement in a [001] quantum well replaces $k_z^2$ by a subband expectation value and produces a linear in-plane term. Absorbing projection signs into $\beta$, we write

$$H_{\mathrm D}^{(1)} = \beta \left( k_x\sigma_x-k_y\sigma_y \right),$$

while residual cubic terms become important as the in-plane Fermi momentum grows. The fitted $\beta$ depends on well width, strain, composition, subband occupation, and convention; it is not simply a universal bulk $\gamma$ multiplied by one geometric number.

Rashba and Dresselhaus together

For a [001] two-dimensional system with both linear terms,

$$H_{\mathrm{RD}} = \alpha \left( k_y\sigma_x-k_x\sigma_y \right) + \beta \left( k_x\sigma_x-k_y\sigma_y \right),$$

so

$$\mathbf g_{\mathrm{RD}} = \left( \alpha k_y+\beta k_x,\, -\alpha k_x-\beta k_y,\, 0 \right).$$

The splitting is anisotropic:

$$|\mathbf g_{\mathrm{RD}}|^2 = (\alpha^2+\beta^2) (k_x^2+k_y^2) + 4\alpha\beta k_xk_y .$$

At $\alpha=\beta$, the field points along the fixed spin axis $[1\bar 10]$:

$$H_{\mathrm{RD}} = 2\alpha k_+\sigma_-,$$

where

$$k_+ = \frac{k_x+k_y}{\sqrt 2}, \qquad \sigma_- = \frac{\sigma_x-\sigma_y}{\sqrt 2}.$$

The component $\sigma_-$ is conserved in the ideal parabolic, linear-coupling model. For spatially uniform coefficients, the associated exact SU(2) structure supports a persistent spin helix with wavevector magnitude

$$Q = \frac{4m^\ast|\alpha|}{\hbar^2}.$$

Spin-independent scalar disorder preserves the exact SU(2) symmetry of this ideal model, as shown by Bernevig, Orenstein, and Zhang (2006). Cubic Dresselhaus terms, unequal coefficients, symmetry-breaking spatial variations of the spin–orbit coefficients, spin-dependent disorder, additional subbands, and other spin-relaxation channels can limit the lifetime in real samples. The symmetry point is robust physics, not a promise of infinite experimental lifetime.

Magnetic consequences

Spin–orbit coupling lets the lattice distinguish spin directions. After high-energy orbital states are eliminated, it can generate:

• anisotropic $g$ tensors and single-ion anisotropy;
• symmetric anisotropic exchange and antisymmetric Dzyaloshinskii–Moriya exchange;
• magnetocrystalline anisotropy and easy axes or easy planes;
• spin–momentum locking and current-induced nonequilibrium spin polarization;
• gaps at magnon or electronic band crossings;
• orbital moments and mixed spin–orbital collective modes.

Exchange Interactions derives the exchange-tensor decomposition. Magnetic Anisotropy develops the resulting easy directions, shape and interface competition, texture length scales, and material inference. The important power counting is that a local spin–orbit term can enter an effective spin Hamiltonian only after it is combined with hopping, hybridization, exchange, or crystal-field processes. A Dzyaloshinskii–Moriya vector is therefore constrained by bond symmetry and microscopic virtual paths; it is not obtained by attaching an arbitrary vector to every bond.

In an itinerant ferromagnet, exchange already breaks time reversal and separates spin-mixed bands. Spin–orbit coupling then changes their avoided crossings, orbital moments, Berry curvature, and orientation-dependent total energy. Describing such states as pure “spin up” and “spin down” is generally an approximation.

Spin Hall response

The spin Hall effect is a transverse spin response to a longitudinal electric field. A commonly used conventional spin-current operator is

$$J_i^a = \frac{1}{2} \left\{ v_i,s_a \right\},$$

where $i$ is the flow direction and $a$ is the spin-polarization axis. Here $s_a$ is a physical spin angular-momentum operator, rather than the dimensionless local-shell $s_{\mathrm{loc},a}$ above. Because spin–orbit coupling produces a torque,

$$\frac{\mathrm ds_a}{\mathrm dt} = \frac{i}{\hbar} \left[ H,s_a \right] \ne 0,$$

this current is generally not associated with a strictly conserved bulk density. Alternative conserved-current definitions shift torque-dipole terms between “current” and “source.” A quoted spin Hall conductivity is meaningful only with the operator, axes, units, frequency, and boundary protocol stated.

Intrinsic contribution

For clean, independent quasiparticle bands with nondegenerate energies at the momentum in question, an intrinsic dc contribution can be written as $j_i^a=\sigma_{ij}^{a,\mathrm{int}}E_j$. With electron charge $-e$ and $e>0$, the Kubo expression is

$$\sigma_{ij}^{a,\mathrm{int}} = e\hbar \sum_n \int_{\mathrm{BZ}} \frac{\mathrm d^dk}{(2\pi)^d} f_{n\mathbf k}\, \Omega_{n,ij}^{a}(\mathbf k),$$

with the spin Berry curvature

$$\Omega_{n,ij}^{a} = -2\,\operatorname{Im} \sum_{m\ne n} \frac{ \langle n|J_i^a|m\rangle \langle m|v_j|n\rangle }{ (\varepsilon_n-\varepsilon_m)^2 }.$$

All states and energies here are evaluated at the same $\mathbf k$. The displayed expression is not an instruction to divide by zero within an exactly degenerate multiplet: combine such states using the projector or full matrix Kubo response and take the appropriate static limit. Individual curvatures inside a degenerate pair depend on its basis, while the physical summed response does not. The spin-current and velocity operators must be transformed consistently with any band projection; a momentum-dependent basis generally requires its connection in addition to differentiating matrix entries. Near spin–orbit-induced avoided crossings, the small denominator can produce large hot spots. The total response still requires the occupation, all relevant bands, and symmetry-related cancellations.

Time reversal forbids the antisymmetric intrinsic charge Hall conductivity in a nonmagnetic equilibrium crystal because charge Berry curvature is odd in $\mathbf k$. It does not generally forbid spin Hall response: both velocity and spin reverse under time reversal, so their symmetrized product is even.

Extrinsic contribution

Disorder can generate two distinct leading mechanisms:

• skew scattering, an asymmetric scattering probability whose conductivity often scales with the transport lifetime in a dilute-disorder regime;
• side jump, a coordinate displacement during scattering whose leading conductivity is often independent of the lifetime, even though scattering events are essential.

The decomposition into intrinsic, side-jump, and skew terms can depend on model and representation, especially in multiband systems. Scaling with longitudinal conductivity, impurity control, thickness dependence, and comparison with microscopic calculations are more informative than assigning a mechanism from one transverse voltage.

A widely used conversion is

$$\theta_{\mathrm{SH}} = \frac{2e}{\hbar} \frac{j_s}{j_c},$$

but some authors absorb $\hbar/2e$ into the definition of $j_s$ or reverse an axis. A spin Hall angle without its convention cannot be compared safely across papers.

The ideal linear Rashba two-dimensional electron gas is a useful warning. Its clean-band calculation suggests a simple intrinsic dc value, but vertex corrections cancel that result for a common model of spin-independent short-range disorder in the uniform, zero-frequency limit. The cancellation is not a theorem for real multiorbital crystals, finite frequency, finite size, nonlinear coupling, or arbitrary disorder.

Hall Effect explains charge-tensor signs and measurements. Spintronics connects spin injection, inverse conversion and torques to devices.

Bridge to topological materials

Spin–orbit coupling can gap a band crossing, invert orbital characters, or change the symmetry representations of occupied states. None of those observations alone proves topology. A topological phase is diagnosed by the global occupied-state bundle and the protecting symmetries.

In a two-band avoided crossing,

$$H(\mathbf q) = v_xq_x\sigma_x + v_yq_y\sigma_y + m_{\mathrm{SO}}\sigma_z,$$

Here $\mathbf q$ is a wavevector, so $v_x,v_y$ in this notation have units of energy times length (equal to $\hbar$ times the corresponding Dirac velocities), and $m_{\mathrm{SO}}$ is an energy. The gap is $2|m_{\mathrm{SO}}|$, and the Berry curvature is concentrated near $\mathbf q=0$. Whether the integrated contribution is topologically nontrivial depends on how this valley is completed elsewhere in the Brillouin zone and on the signs of all symmetry-related masses.

For a time-reversal-invariant two-dimensional insulator, Kramers structure permits a $\mathbb Z_2$ classification even when no spin component is conserved. In the spin-conserving limit of the Kane–Mele model, one can picture two opposite-Chern sectors related by time reversal. Rashba coupling mixes those sectors, yet the quantum spin Hall phase can survive as long as time reversal and the bulk gap remain intact. The spin Chern-number picture is then no longer the fundamental definition.

Berry Curvature develops the local geometric response, Chern Numbers develops the integer invariant, and Topological Quantum Numbers compares invariant families. Topological Insulators develops the class-AII $\mathbb Z_2$ construction and its material tests. A material claim additionally needs a bulk invariant, a gap or mobility-gap statement, symmetry control, and boundary evidence consistent with the same phase.

How to infer spin–orbit physics in a material

No single measurement returns “the spin–orbit coupling.” A defensible analysis separates model identification, parameter extraction, and mechanism attribution.

Begin with the bulk and interface symmetries: inversion asymmetry, magnetic order and unresolved orbital bands can produce different kinds of band separation. Symmetry of Bloch States identifies protected degeneracies. Fit symmetry-allowed momentum dependence in a sufficiently large orbital and spin subspace before assigning a Rashba or Dresselhaus coefficient.

Spectroscopy and transport

Angle-resolved photoemission can measure dispersions and, with spin resolution, spin expectation values. Matrix elements, overlapping bands, final-state effects, surface band bending, and finite resolution can mimic or obscure a splitting. Quantum oscillations resolve extremal orbit areas but require care: two frequencies can arise from warping, multiple pockets, domains, or magnetic breakdown rather than spin splitting.

Weak antilocalization, spin resonance, optical orientation, and spin-relaxation anisotropy constrain spin precession and decoherence. Their extracted “spin–orbit field” is model dependent. Fits should state the diffusive or ballistic regime, dimensionality, intervalley and magnetic scattering assumptions, and the hierarchy among $\tau$, the spin-precession time, and the phase-coherence time.

Spin Hall measurements often infer a bulk conversion through edge spin accumulation, inverse spin Hall voltages, spin-torque ferromagnetic resonance, nonlocal transport, or terahertz emission. Interface transparency, current shunting, spin diffusion, thermoelectric backgrounds, and spin-memory loss must be calibrated before assigning a bulk angle.

Comparing band calculations

Comparing scalar-relativistic and fully relativistic bands at the same converged structure can isolate a spin–orbit-induced feature. Orbital projections then help identify the states involved. A spinor Wannier model should reproduce the relevant gaps, projected physical moments and symmetry transformations; Berry-curvature sums often require a much denser momentum mesh than the band plot. Switching spin–orbit coupling off in a calculation is a useful counterfactual, but it does not by itself establish a microscopic path or a topological invariant.

Exercises

1. Degeneracy with inversion and time reversal

For a spinful, nonmagnetic centrosymmetric crystal, show that every Bloch energy is at least doubly degenerate at each $\mathbf k$. Why does this not imply that the eigenstates are pure spin states?

Solution

Inversion maps $\mathbf k$ to $-\mathbf k$, and time reversal does the same. Their product $\mathcal P\mathcal T$ therefore leaves $\mathbf k$ fixed. For ordinary inversion, $\mathcal P^2=1$ and $[\mathcal P,\mathcal T]=0$, while $\mathcal T^2=-1$. Hence

$$(\mathcal P\mathcal T)^2 = \mathcal P^2\mathcal T^2 = -1.$$

If $|u_{n\mathbf k}\rangle$ is an eigenstate, $\mathcal P\mathcal T|u_{n\mathbf k}\rangle$ is an orthogonal eigenstate at the same energy and momentum. Spin–orbit coupling may still entangle spin with orbital and sublattice amplitudes within this doublet. Degeneracy constrains the energy; it does not provide a globally unique spin quantization axis.

2. Rashba momentum and energy scales

For $m^\ast>0$ and

$$E_-(k) = \frac{\hbar^2k^2}{2m^\ast} -|\alpha_{\mathrm R}|k,$$

find the ring minimum and its energy relative to $E_-(0)$. Give the spin direction for $\alpha_{\mathrm R}>0$ on the $+k_x$ axis.

Solution

Differentiation gives

$$\frac{\mathrm dE_-}{\mathrm dk} = \frac{\hbar^2k}{m^\ast} -|\alpha_{\mathrm R}| = 0,$$

so

$$k_{\mathrm R} = \frac{m^\ast|\alpha_{\mathrm R}|}{\hbar^2}.$$

Substitution yields

$$E_-(k_{\mathrm R})-E_-(0) = -\frac{m^\ast\alpha_{\mathrm R}^2}{2\hbar^2} = -E_{\mathrm R}.$$

At $\mathbf k=(k,0)$,

$$\mathbf g_{\mathrm R} = (0,-\alpha_{\mathrm R}k,0).$$

The $+$ helicity has spin along $-\hat{\mathbf y}$ and the $-$ helicity along $+\hat{\mathbf y}$. Which energy branch is called “inner” or “outer” depends on the chosen Fermi energy.

3. Persistent-spin-helix symmetry

Set $\alpha=\beta$ in the linear Rashba–Dresselhaus Hamiltonian. Show that one spin component is conserved and derive the helix wavevector for a parabolic band.

Solution

At equal couplings,

$$\begin{aligned} H_{\mathrm{RD}} &= \alpha(k_x+k_y)(\sigma_x-\sigma_y) \\ &= 2\alpha k_+\sigma_-. \end{aligned}$$

Therefore $[H,\sigma_-]=0$ for

$$\sigma_- = \frac{\sigma_x-\sigma_y}{\sqrt 2}.$$

The kinetic and SOC terms along $k_+$ complete the square:

$$\frac{\hbar^2k_+^2}{2m^\ast}+2\alpha k_+\sigma_- =\frac{\hbar^2}{2m^\ast} \left(k_++\frac{2m^\ast\alpha}{\hbar^2}\sigma_-\right)^2 -\frac{2m^\ast\alpha^2}{\hbar^2}.$$

The two conserved-spin branches are shifted by

$$Q = \frac{4m^\ast|\alpha|}{\hbar^2},$$

which is the spatial helix wavevector. Spin-independent scalar disorder preserves the ideal helical symmetry. Conserving the single component $\sigma_-$ alone is not sufficient for this fixed-wavevector helix: symmetry-breaking spatial variations of the spin–orbit coupling can destroy the momentum-shift relation even if that component remains conserved.

4. Why time reversal permits a spin Hall response

Determine the time-reversal parity of the charge current $j_i=-ev_i$ and of the conventional spin current $J_i^a=\{v_i,s_a\}/2$. Use it to explain why a nonmagnetic crystal can have zero charge Hall conductivity and nonzero spin Hall conductivity.

Solution

Time reversal changes both velocity and spin:

$$\mathcal T v_i\mathcal T^{-1} = -v_i, \qquad \mathcal T s_a\mathcal T^{-1} = -s_a.$$

Thus charge current is odd, whereas

$$\mathcal T J_i^a\mathcal T^{-1} = \frac{1}{2} \left\{ -v_i,-s_a \right\} = J_i^a$$

is even. In a time-reversal-invariant equilibrium crystal, symmetry-related momenta cancel the antisymmetric charge Hall response. A symmetric off-diagonal conductivity from anisotropy can still produce a transverse voltage. The extra time-reversal-odd spin insertion reverses that cancellation pattern, so a transverse spin current is symmetry allowed. Its actual value can still vanish because of additional crystal symmetries or dynamical cancellations.

5. A gap is not yet a topological invariant

A time-reversal-invariant insulator has two valleys with equal gap magnitudes. Must their total charge Chern number vanish? Does the gap measurement determine the $\mathbb Z_2$ index?

Solution

Time reversal relates the occupied-band curvature at opposite momenta by a sign reversal, so the total charge Chern number vanishes. The local Dirac contribution depends on the signed product $m v_xv_y$, not merely on the gap $2|m|$, but the full time-reversal partners must cancel in the charge sum. This cancellation occurs in both trivial and quantum spin Hall insulators. In a spin-conserving description, the two valleys can add within one spin sector while its time-reversed sector contributes the opposite Chern number. With spin mixing, the global Kramers-pair structure supplies the $\mathbb Z_2$ invariant. Equal local gaps alone do not decide it.

References

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