The physics of electronic bands can be rigorously approached from two diametrically opposite yet complementary frontiers: the Nearly Free Electron (NFE) model, which treats the periodic lattice as a weak perturbation on plane waves, and the Tight-Binding (LCAO) approximation, which builds crystal wavefunctions from the overlap of isolated atomic orbitals.
1. The Nearly Free Electron (NFE) Model and Degenerate Perturbation Theory
In simple metals and wide conduction bands, valence electrons experience substantial electrostatic screening, making the effective crystal potential (V(mathbf{r})) weak compared to the kinetic energy. Away from the Brillouin zone boundaries, non-degenerate perturbation theory yields parabolic energy dispersion:
[
E(mathbf{k}) approx rac{hbar^2 k^2}{2m_0} + V_0 + sum_{mathbf{G}
eq 0} rac{|V_{mathbf{G}}|^2}{rac{hbar^2}{2m_0}(k^2 – |mathbf{k} – mathbf{G}|^2)}
]
However, when (mathbf{k}) terminates precisely on a Bragg reflection plane where (|mathbf{k}| approx |mathbf{k} – mathbf{G}|), the energy denominator vanishes, and degenerate perturbation theory must be applied. Considering the two-state subspace spanned by (|mathbf{k}
angle) and (|mathbf{k} – mathbf{G}
angle), the Hamiltonian matrix takes the (2 imes 2) secular form:
[
begin{pmatrix}
E_{mathbf{k}}^0 – E & V_{mathbf{G}}
V_{mathbf{G}}^* & E_{mathbf{k}-mathbf{G}}^0 – E
nd{pmatrix}
begin{pmatrix} C_{mathbf{k}} C_{mathbf{k}-mathbf{G}} nd{pmatrix} = 0
]
Solving the secular determinant ((E_{mathbf{k}}^0 – E)(E_{mathbf{k}-mathbf{G}}^0 – E) – |V_{mathbf{G}}|^2 = 0) at the exact zone boundary ((E_{mathbf{k}}^0 = E_{mathbf{k}-mathbf{G}}^0)) yields two split eigenvalues:
[
E_{pm} = E_{mathbf{k}}^0 pm |V_{mathbf{G}}|
]
Thus, the periodic potential lifts the two-fold degeneracy at the zone boundary, opening an energy bandgap of magnitude:
[
E_g = 2 |V_{mathbf{G}}|
]
The two standing wave solutions correspond to electron probability densities peaked either directly at the positively charged ionic cores ((psi_+ sim cos(Gx/2)), lowering potential energy) or midway between them ((psi_- sim sin(Gx/2)), raising potential energy).
2. The Kronig-Penney 1D Solvable Model
The Kronig-Penney model idealizes the 1D lattice potential as a periodic array of rectangular barriers of height (V_0) and width (b), spaced by wells of width (a). Taking the delta-function limit ((V_0 o infty, b o 0) with barrier strength (P = lim rac{m_0 V_0 b a}{hbar^2}) held constant), matching boundary conditions for Bloch functions yields the celebrated dispersion relation:
[
P rac{sin(alpha a)}{alpha a} + cos(alpha a) = cos(ka), quad ext{where } alpha = rac{sqrt{2m_0 E}}{hbar}
]
Because (|cos(ka)| le 1), solutions exist only when the left-hand side falls within ([-1, +1]). Values exceeding unity define forbidden energy gaps, demonstrating mathematically how quantum tunneling through periodic barriers creates discrete energy bands.
3. The Tight-Binding (LCAO) Approximation
At the opposite extreme, inner-shell core electrons and tightly bound valence electrons (e.g., (d)-orbitals in transition metal dichalcogenides or (sp^3) hybrids in diamond/silicon) are strongly localized near individual atomic nuclei. The tight-binding model constructs the crystal Bloch state (psi_{mathbf{k}}(mathbf{r})) as a Linear Combination of Atomic Orbitals (LCAO) (phi_m(mathbf{r})):
[
psi_{mathbf{k}}(mathbf{r}) = rac{1}{sqrt{N}} sum_{j=1}^N e^{i mathbf{k} cdot mathbf{R}_j} phi(mathbf{r} – mathbf{R}_j)
]
Evaluating the expectation value of the crystal Hamiltonian (hat{H} = hat{H}_{ ext{atom}} + Delta U(mathbf{r})) under the nearest-neighbor approximation yields the energy dispersion:
[
E(mathbf{k}) = psilon_0 – alpha – sum_{mathbf{delta}} gamma(mathbf{delta}) e^{i mathbf{k} cdot mathbf{delta}}
]
where (psilon_0) is the atomic orbital energy, (alpha) is the on-site crystal-field shift, (mathbf{delta}) represents vectors pointing to nearest neighbors, and (gamma(mathbf{delta})) is the nearest-neighbor transfer (hopping) integral:
[
gamma(mathbf{delta}) = -int phi^*(mathbf{r}) Delta U(mathbf{r}) phi(mathbf{r} – mathbf{delta}) , d^3r
]
For a 1D chain with lattice constant (a) and hopping parameter (t), this yields the classic cosine band:
[
E(k) = psilon_0 – 2t cos(ka)
]
with total bandwidth (W = 4t). In graphene, a 2D honeycomb lattice tight-binding Hamiltonian produces linear conical dispersions at the (K) and (K’) points ((E(mathbf{q}) = pm hbar v_F |mathbf{q}|)), manifesting massless Dirac fermions.
