A periodic crystal potential organizes energies by wavevector and band.
Key knowledge
For a periodic one-electron Hamiltonian, Bloch states have ψ_nk(r) = e^(ik·r)u_nk(r), with u_nk(r+R) = u_nk(r) for lattice vectors R. Their energies E_n(k) form bands.
Values and equations
ψ_nk(r) = e^(ik·r)u_nk(r); u_nk(r+R) = u_nk(r); energy E_n(k)
ψ_nk, u_nk
State and lattice-periodic part in a chosen normalization
k
Crystal wavevector m⁻¹
r, R
Position and lattice translation m
n
Band index
E_n(k)
Band energy J
Conditions and scope
Ideal periodic crystal and independent-electron or effective one-electron model; boundaries and disorder need additional treatment.
Crystal wavevector is not generally the free-particle mechanical momentum; strong correlations can invalidate the simple band description.
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References
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For a periodic one-electron Hamiltonian, Bloch states have ψ_nk(r) = e^(ik·r)u_nk(r), with u_nk(r+R) = u_nk(r) for lattice vectors R. Their energies E_n(k) form bands.
The periodic effective one-electron Hamiltonian selects Bloch states and bands.
In a semiconductor band model, conduction-band electrons and holes in an otherwise filled valence band carry charge. Donor and acceptor doping can produce electron- or hole-dominated conduction in the stated operating regime.
Carrier descriptions use conduction/valence bands and a material-specific gap.
A Bravais lattice consists of translations R=Σ_i n_i a_i with integer n_i and independent primitive vectors. A crystal is a lattice with a repeated atomic basis at positions R+d_α; a primitive cell contains one lattice point, whereas conventional cells may contain several.
The lattice supplies translations of a periodic model; a basis is distinct from the Bravais translation lattice.
For a wave packet in one smooth band E(k), neglecting anomalous velocity, v=(1/ħ)∇_kE and ħk̇=F. Band curvature defines (m*⁻¹)_ij=(1/ħ²)∂²E/(∂k_i∂k_j), so a_i=Σ_j(m*⁻¹)_ijF_j. An isotropic quadratic minimum gives E≈E_c+ħ²|k−k_c|²/(2m*) with positive electron mass; hole excitation mass at a valence maximum is defined from the opposite curvature.
An isolated smooth indexed band supplies packet group velocity and curvature, with interband/Berry exclusions.
For a real periodic one-dimensional potential of period a, define V_G=a⁻¹∫₀ᵃV(x)e^(−iGx)dx, G=2π/a, and ε_k=ħ²k²/(2m). Near k=G/2, the coupled plane waves |k⟩ and |k−G⟩ have the leading two-state Hamiltonian with diagonal ε_k+V₀, ε_(k−G)+V₀ and off-diagonal V_G,V_G*. Its energies are V₀+(ε_k+ε_(k−G))/2 ± √[((ε_k−ε_(k−G))/2)²+|V_G|²]. At the boundary the leading gap is 2|V_G|.
A representative weak-potential method computes the leading one-dimensional boundary gap behind the broader Bloch band description.
For a periodic chain of N orthonormal localized orbitals, one per site with spacing a, take H_nm=ε₀δ_nm−t(δ_(n+1,m)+δ_(n−1,m)). The normalized eigenvectors φ_n=e^(ikna)/√N, k=2πj/(Na), have E(k)=ε₀−2t cos(ka). In the infinite-chain limit this single band spans ε₀−2|t| to ε₀+2|t|; zero hopping gives the degenerate atomic limit.
The localized-orbital model provides a concrete single-band dispersion, with material filling and extra interactions left to additional claims.
In correlated superconductors, spin-fluctuation-mediated pairing, electron–phonon contributions and other electronic correlations must be tested against gap symmetry, spectroscopy and thermodynamics for the particular material. Scalapino's spin-fluctuation account is a model-supported proposal connecting several classes; correlated evidence alone does not uniquely establish a pairing kernel.
The electronic states constrain the candidate pairing interaction.
For a gapped noninteracting two-dimensional band system, the occupied-band Berry curvature integrates to an integer C=(2π)⁻¹∫_BZTrF_xy d²k. Under the applicable insulating zero-temperature conditions, σ_xy=C e²/h. Interfaces carry net chiral edge modes set by the change in C; the invariant requires the separating gap to remain open.
Spinful time-reversal symmetry with T²=−1 permits a Z₂ band distinction even when total Chern number vanishes. An odd boundary Kramers-pair structure cannot be removed by symmetry-preserving band deformation without closing the relevant bulk gap. Photoemission and transport probe different surface/bulk properties; protected does not mean every scattering process is absent.
Boundary and bulk bands must be distinguished in measured spectra.