Nearly-free electron model

The nearly-free electron model treats a weak periodic potential by degenerate perturbation theory. At a Bragg crossing, a Fourier coefficient $V_n$ couples the two degenerate plane waves and opens an energy gap of width $2|V_n|$.

For a divalent square Bravais lattice, let $E_X={\hslash }^2{\pi }^2/(2m_e{a}^2)$ and real periodic potential coefficients $V_{10}<V_{11}<0$. The leading two-wave edge levels are $E_X\pm V_{10}$. The four-wave corner levels are $2E_X+V_{11}\pm 2V_{10}$ and a doublet at $2E_X-V_{11}$. Comparing the lowest corner level with the upper edge level gives the displayed indirect band gap. The first spin-degenerate energy band can hold the two electrons per primitive unit cell; it becomes isolated when $-E_X-V_{11}-3V_{10}>0$ in this truncation. This follows from resonant degenerate perturbation theory, and higher plane waves can shift the threshold.

For rectangular real-space periods $a>b$, the free first-zone corner has energy ${\hslash }^2{\pi }^2(a^{-2}+b^{-2})/(2m)$, whereas the free second-zone minimum is ${\hslash }^2{\pi }^2/(2m{a}^2)$. Their difference is the displayed overlap. A divalent spin-degenerate system is a band insulator only when the actual first-band maximum lies below the second-band minimum. A rigid symmetric splitting $\Delta$ removes the overlap if $\Delta >{\Delta }_{\mathrm{overlap}}$, but an arbitrary periodic potential does not shift every corner and face by the same amount. The exact criterion uses the indirect band gap, not a geometry-only local direct-gap width.

For two electrons per conventional monovalent BCC cell, $k_F=(6{\pi }^2{)}^{1/3}/a$ lies inside the first zone. The shortest reciprocal vector is $G=2\sqrt{2}\pi /a$. The free folded-branch difference ${\hslash }^2(G^2-2Gk)/(2m_e)$ is smallest on the occupied Fermi-sphere point toward G, giving the displayed ratio. The destination has magnitude $G-k_F>k_F$ and is empty. A weak periodic potential supplies band mixing and optical matrix elements; its finite corrections and absorption peak shape are not fixed by this leading threshold.

For $-V_0\cos Qx$, the Fourier coupling between the two free states at $k=\pm Q/2$ is $-V_0/2$. The split energies are $E_Q/4\mp V_0/2$, with $E_Q={\hslash }^2{Q}^2/(2m)$. The direct face gap is $V_0$, not $2V_0$.

Consider a spin-degenerate two-dimensional free electron model with rectangular periods $a>b$ and two electrons per unit cell. Write $K=\pi /a$, $L=\pi /b$, and $k_F^2=4\pi /(ab)$. If $K<k_F<L$, the circular Fermi contour crosses only the vertical boundaries of the first Brillouin zone. Folding translates its protruding caps back by a reciprocal lattice vector $(2K,0)$. A weak periodic potential mixes the intersecting plane waves, reconnecting the contours into two open first-band sheets and a closed second-band pocket. The pocket is centred on a zone boundary: its halves on opposite sides of a reduced-zone plot are identified periodically and belong to one closed sheet.

Put $C={\hslash }^2/(2m)$, $v=|V_G|$, $\mathbf{G}=(2K,0)$, and $k_x=K+q$. The upper eigenvalue of the two coupled plane waves isAt fixed Fermi energy $C{k}_F^2$, its closed Fermi contour hasThus $A(v)=4{\int }_0^{q_m}y(q)dq$ can be inverted numerically to estimate the Bragg scattering coupling. The pocket disappears at $v=C(k_F^2-K^2)$. The estimate assumes a fixed chemical potential and neglects coupling to other plane waves; a measured pocket area alone does not determine all Fourier coefficients of an arbitrary lattice potential.

In rectangular-lattice Fermi-surface reconstruction, the zero-coupling closed orbit is the overlap lens of two equal circles of radius $k_F$ with centres separated by $2K$. Two circular sectors minus their triangular portions giveAt fixed chemical potential, the upper hybridized energy band rises when the magnitude of the Fourier coefficient increases. Its occupied pocket therefore shrinks, so this lens is an upper bound in the two-wave model. It is not a universal bound for arbitrary changes of chemical potential or additional energy bands.

For a scalar one-dimensional dispersion $E(k)=\hslash v|k|$ and a potential of period $L$, a reciprocal vector $nG$, $G=2\pi /L$, connects degenerate right- and left-moving states at $k=nG/2$. Their unperturbed energy is $E_n=n\pi \hslash v/L$. Thus consecutive allowed Bragg thresholds have spacing $\pi \hslash v/L$, whereas shifting momentum by a full reciprocal period changes a linear branch's energy by $2\pi \hslash v/L$. The potential's Fourier coefficients and higher-order couplings determine which gaps actually open and their widths, so a geometric period alone does not imply an exactly energy-periodic spectrum. A graphene pseudospin selection rule can further suppress scattering; the scalar-chain model omits that internal structure.

Forwrite $k=n\pi /a+\kappa$. Near the $n$th Bragg crossing,At $\kappa =0$, the two branches differ by $2|V_n|$.