Nearly Free Electrons

The nearly-free-electron model is controlled perturbation theory around free plane waves in a weak periodic potential. Away from free-electron degeneracies, the potential produces small energy shifts and weak reciprocal-lattice satellites. At a Bragg plane, however, two or more plane waves are degenerate, ordinary perturbation theory fails, and even an arbitrarily weak Fourier component can split the degeneracy at first order.

For an isolated two-wave crossing, the leading direct gap is

$$\Delta_{\mathbf G} = 2|V_{\mathbf G}|,$$

where $V_{\mathbf G}$ is the Fourier coefficient of the periodic potential at the reciprocal vector connecting the two waves. This compact result explains why zone boundaries are special, but it must be read with its assumptions: the crossing must be isolated from other free branches, omitted couplings must be perturbative, and a local avoided crossing is not automatically a global insulating gap.

Bloch’s Theorem is exact and supplies the crystal-momentum sectors. We now approximate the Hamiltonian within those sectors, keeping the waves that become resonant at a Bragg plane and estimating the effects of the others.

Required background. Bloch’s Theorem supplies the exact crystal-momentum sectors, Brillouin Zones supplies zone-boundary geometry and folding conventions, and Degenerate Perturbation Theory supplies the avoided-crossing calculation at a Bragg plane.

Electrons in a weak periodic potential

Consider

$$H = H_0+V, \qquad H_0 = \frac{\mathbf p^2}{2m},$$

with a real lattice-periodic potential,

$$V(\mathbf r+\mathbf R) = V(\mathbf r).$$

Expand it in reciprocal vectors:

$$V(\mathbf r) = \sum_{\mathbf G} V_{\mathbf G} e^{i\mathbf G\cdot\mathbf r},$$

where

$$V_{\mathbf G} = \frac{1}{\Omega_c} \int_{\mathrm{cell}} d^dr\, V(\mathbf r) e^{-i\mathbf G\cdot\mathbf r}.$$

Reality implies

$$V_{-\mathbf G} = V_{\mathbf G}^{\ast}.$$

The average $V_{\mathbf0}$ shifts every energy equally and can be absorbed into the energy origin. The nonzero coefficients scatter a plane wave with wavevector $\mathbf q$ into waves at

$$\mathbf q-\mathbf G.$$

“Nearly free” means that the relevant $V_{\mathbf G}$ are small compared with the energy separations to plane-wave branches excluded from a chosen perturbative subspace. It does not mean that every microscopic ionic potential is weak at every point. In practical electronic-structure reasoning, screening and pseudopotentials can make the effective potential seen by valence electrons much smoother than the bare core potential.

Free Electrons in a Periodic Box

For $N_c$ primitive cells with Born–von Karman boundary conditions, free eigenstates are normalized plane waves,

$$\langle\mathbf r|\mathbf q\rangle = \frac{1}{\sqrt V} e^{i\mathbf q\cdot\mathbf r}, \qquad V=N_c\Omega_c,$$

with energies

$$\varepsilon_{\mathbf q} = \frac{\hbar^2|\mathbf q|^2}{2m}.$$

Choose $\mathbf k$ in one Brillouin zone. Every allowed plane wave can be written uniquely as

$$\mathbf q = \mathbf k+\mathbf G.$$

At fixed $\mathbf k$, the free basis is therefore

$$\left\{ |\mathbf k+\mathbf G\rangle \right\}_{\mathbf G\in\Lambda^\ast},$$

with folded branches

$$\varepsilon_{\mathbf G}(\mathbf k) = \frac{\hbar^2}{2m} |\mathbf k+\mathbf G|^2.$$

Folding these parabolas into one zone is only a relabeling of the free spectrum. It creates branch crossings on a plot but does not open a gap. Gaps appear only after the periodic potential couples crossing branches.

The Central Equation

A Bloch state at fixed $\mathbf k$ has the plane-wave expansion

$$\psi_{n\mathbf k}(\mathbf r) = \sum_{\mathbf G} c_{n\mathbf k}(\mathbf G) e^{i(\mathbf k+\mathbf G)\cdot\mathbf r}.$$

The matrix element of the potential is

$$\langle \mathbf k+\mathbf G | V | \mathbf k+\mathbf G' \rangle = V_{\mathbf G-\mathbf G'}.$$

Substitution into the Schrödinger equation gives the central equation

$$\sum_{\mathbf G'} \left[ \varepsilon_{\mathbf G}(\mathbf k) \delta_{\mathbf G\mathbf G'} + V_{\mathbf G-\mathbf G'} \right] c_{n\mathbf k}(\mathbf G') = E_n(\mathbf k) c_{n\mathbf k}(\mathbf G).$$

Equivalently,

$$\left[ \varepsilon_{\mathbf G}(\mathbf k) -E_n(\mathbf k) \right] c_{n\mathbf k}(\mathbf G) + \sum_{\mathbf G'} V_{\mathbf G-\mathbf G'} c_{n\mathbf k}(\mathbf G') = 0.$$

This is an infinite matrix eigenproblem within one Bloch sector. A plane-wave calculation truncates the reciprocal basis at a kinetic-energy cutoff; the nearly-free approximation instead keeps only the resonant branches analytically and treats the rest perturbatively.

Away from Degeneracy

Suppose the free plane wave $|\mathbf q\rangle$ is separated in energy from every $|\mathbf q-\mathbf G\rangle$ coupled appreciably by the potential. Nondegenerate perturbation theory gives

$$E(\mathbf q) = \varepsilon_{\mathbf q} + V_{\mathbf0} + \sum_{\mathbf G\ne\mathbf0} \frac{|V_{\mathbf G}|^2} {\varepsilon_{\mathbf q}-\varepsilon_{\mathbf q-\mathbf G}} + O(V^3).$$

The first-order correction is only the spatial average $V_{\mathbf0}$. The nonuniform potential changes a nondegenerate free energy first at second order, although it changes the eigenvector at first order.

The expansion fails when a denominator becomes small:

$$\varepsilon_{\mathbf q} \approx \varepsilon_{\mathbf q-\mathbf G}.$$

This is not a technical nuisance to be ignored. It identifies the Bragg planes where a qualitatively new two-wave or multiwave problem must be solved.

Bragg Planes

Free waves $\mathbf k$ and $\mathbf k-\mathbf G$ are degenerate when

$$\frac{\hbar^2|\mathbf k|^2}{2m} = \frac{\hbar^2|\mathbf k-\mathbf G|^2}{2m}.$$

Canceling common terms yields

$$\mathbf k\cdot\mathbf G = \frac{|\mathbf G|^2}{2}.$$

This is the perpendicular-bisector equation for a reciprocal-lattice vector. The nearest such planes are exactly the faces used to construct the first Brillouin zone.

The same condition is the elastic Bragg-scattering condition. The periodic potential can transfer reciprocal momentum $\hbar\mathbf G$ without changing the free kinetic energy, so the incident and Bragg-reflected waves mix coherently.

Two-Wave Degenerate Perturbation Theory

Near an isolated Bragg plane, retain

$$|\mathbf k\rangle \quad\text{and}\quad |\mathbf k-\mathbf G\rangle.$$

To first order in the weak potential, the effective Hamiltonian is

$$H_{\mathrm{eff}} = \begin{pmatrix} \varepsilon_{\mathbf k}+V_{\mathbf0} & V_{\mathbf G} \\ V_{\mathbf G}^{\ast} & \varepsilon_{\mathbf k-\mathbf G}+V_{\mathbf0} \end{pmatrix}.$$

Its eigenvalues are

$$E_{\pm}(\mathbf k) = V_{\mathbf0} + \frac{ \varepsilon_{\mathbf k} + \varepsilon_{\mathbf k-\mathbf G} }{2} \pm \sqrt{ \left( \frac{ \varepsilon_{\mathbf k} - \varepsilon_{\mathbf k-\mathbf G} }{2} \right)^2 + |V_{\mathbf G}|^2 }.$$

At exact degeneracy,

$$E_{\pm} = \varepsilon_{\mathrm B} + V_{\mathbf0} \pm |V_{\mathbf G}|,$$

so the leading avoided-crossing gap is

$$\Delta_{\mathbf G} = E_+-E_- = 2|V_{\mathbf G}|.$$

Corrections from omitted plane waves begin at higher order when those waves remain well separated. If several free branches are degenerate, the two-state formula is insufficient and the full degenerate block must be diagonalized.

Folding alone produces a crossing between $\varepsilon_{\mathbf k}$ and $\varepsilon_{\mathbf k-\mathbf G}$. The Fourier coefficient $V_{\mathbf G}$ couples the degenerate waves and replaces the crossing by two smooth branches separated by $2|V_{\mathbf G}|$ at the Bragg plane. Here $\kappa$ measures displacement normal to the plane.

One-Dimensional Zone Boundary

For a lattice spacing $a$, take

$$G = \frac{2\pi}{a}, \qquad k = \frac{G}{2}+\kappa.$$

The coupled wavevectors are

$$k = \frac{G}{2}+\kappa, \qquad k-G = -\frac{G}{2}+\kappa.$$

Their average free energy and half-difference are

$$\frac{ \varepsilon_k+\varepsilon_{k-G} }{2} = \frac{\hbar^2G^2}{8m} + \frac{\hbar^2\kappa^2}{2m},$$ $$\frac{ \varepsilon_k-\varepsilon_{k-G} }{2} = \frac{\hbar^2G\kappa}{2m}.$$

The two-wave dispersion is therefore

$$E_{\pm}(\kappa) = V_0 + \frac{\hbar^2G^2}{8m} + \frac{\hbar^2\kappa^2}{2m} \pm \sqrt{ \left( \frac{\hbar^2G\kappa}{2m} \right)^2 + |V_G|^2 }.$$

Far enough from the boundary that

$$\left| \frac{\hbar^2G\kappa}{2m} \right| \gg |V_G|,$$

the branches approach the folded free parabolas. Inside the mixing region, the eigenstates contain comparable right- and left-moving components and the free crossing is rounded.

Standing waves at the boundary

Write

$$V_G = |V_G|e^{i\phi}.$$

At $k=G/2$, the coefficient vectors have unit norm. The corresponding functions, with unit cell-average density, may be chosen as

$$\psi_{\pm}(x) = \frac{1}{\sqrt2} \left[ e^{iGx/2} \pm e^{-i\phi}e^{-iGx/2} \right].$$

Their densities are

$$|\psi_{\pm}(x)|^2 = 1 \pm \cos(Gx+\phi).$$

The resonant part of the potential is

$$V_Ge^{iGx} + V_{-G}e^{-iGx} = 2|V_G| \cos(Gx+\phi).$$

For a unit-normalized wavefunction in a finite chain of length $L$, divide the displayed function by $\sqrt L$. The upper state concentrates density near maxima of this harmonic, while the lower state concentrates near its minima. Translating the coordinate origin changes $\phi$ and shifts both patterns, but it cannot change the energies or the gap $2|V_G|$.

The Factor-of-Two Convention

The safest rule is to compute the Fourier coefficient before quoting a gap. If

$$V(x) = 2U\cos(Gx),$$

then

$$V_G = V_{-G} = U,$$

and

$$\Delta_G = 2|U|.$$

If another author writes

$$V(x) = U\cos(Gx),$$

then $V_G=U/2$ and the gap is $|U|$. Both statements are the same physics with different definitions of the cosine amplitude. Writing “the gap equals twice the potential” without defining the Fourier convention is ambiguous.

Following the coupled bands

The reduced, extended and repeated zone schemes display the same states in different ways. Folding labels the uncoupled branches; a nonzero matrix element changes their eigenvalues. Near the avoided crossing, the eigenvectors continuously exchange their dominant plane-wave character. Consequently, following the most intense plane-wave component can jump between energy-ordered bands.

Higher Dimensions

On a generic face of a Brillouin zone, the two-wave derivation applies to motion normal to that face. Momentum parallel to the face remains continuous, so the avoided crossing forms a gapped sheet rather than an isolated point.

At an edge or corner, several reciprocal vectors can generate simultaneous degeneracies. If the resonant plane waves are

$$\left\{ |\mathbf k+\mathbf G_1\rangle, \ldots, |\mathbf k+\mathbf G_M\rangle \right\},$$

one must diagonalize the $M\times M$ matrix

$$\left[ H_{\mathrm{deg}}(\mathbf k) \right]_{ij} = \varepsilon_{\mathbf k+\mathbf G_i} \delta_{ij} + V_{\mathbf G_i-\mathbf G_j}.$$

Point-group symmetry constrains this matrix and can protect residual degeneracies. A geometrical free-electron crossing does not guarantee a first-order gap: the required $V_{\mathbf G}$ may vanish by symmetry or motif interference.

Local gap versus global gap

$2|V_{\mathbf G}|$ is a local direct splitting between two branches at a particular Bragg plane. A band insulator at zero temperature requires a global gap at the chosen integer filling: all occupied bands must be separated in energy from all unoccupied bands throughout the zone. This criterion concerns clean independent-electron band insulation; disorder and interactions can produce other insulating mechanisms. In two and three dimensions, different folded free branches often overlap in energy even though each individual avoided crossing is nonzero.

Worked Examples

Single cosine

Let

$$V(x) = 2U\cos\left(\frac{2\pi x}{a}\right).$$

Only

$$V_{\pm 2\pi/a} = U$$

are nonzero apart from the average. At the first-zone boundaries $k=\pm\pi/a$, the leading gap is $2|U|$. The same harmonic also couples other pairs of plane waves separated by $2\pi/a$, but only resonant pairs acquire a first-order splitting.

Shifted cosine

For

$$V(x) = 2U \cos \left[ G(x-x_0) \right],$$

the coefficient is

$$V_G = Ue^{-iGx_0}.$$

The shift changes the phase of the coupling and the position of the standing-wave nodes. Since

$$|V_G| = |U|,$$

the band energies and gap are independent of the arbitrary origin.

Rectangular lattice

Take

$$V(x,y) = 2U_x\cos(G_xx) + 2U_y\cos(G_yy).$$

At the Bragg plane $k_x=G_x/2$ away from a zone corner, the relevant pair differs by $G_x\hat{\mathbf x}$ and the leading splitting is $2|U_x|$. Similarly, the $k_y=G_y/2$ face has splitting $2|U_y|$.

At their intersection, four free plane waves are degenerate. For this separable potential the leading block is

$$H_{\rm corner}=\varepsilon_{\rm B}I_4 +U_x\sigma_x\otimes I_2+U_y I_2\otimes\sigma_x,$$

with energies $\varepsilon_{\rm B}\pm U_x\pm U_y$ for real $U_x,U_y$. If $U_x=U_y=U$, the shifts are $-2U,0,0,2U$: a doublet remains in the middle. A Fourier harmonic coupling both directions generally destroys this separability and requires the full four-state calculation.

Validity and Limitations

Controlled truncation

Let $P$ project onto the retained resonant plane waves and $Q=1-P$. The approximation is controlled when matrix elements coupling $P$ to $Q$ are small compared with the corresponding free-energy separations:

$$\frac{ |\langle p|V|q\rangle| }{ |E_p^{(0)}-E_q^{(0)}| } \ll 1$$

for the omitted states that matter. Many individually small terms can nevertheless accumulate. A useful state-specific estimate of omitted weight is

$$w_Q\simeq\sum_{q\in Q} \frac{|\langle q|V|p\rangle|^2}{|E_p^{(0)}-E_q^{(0)}|^2}\ll1.$$

For a retained block, the corresponding condition involves the norm of the coupling times the omitted-space resolvent. Convergence with a larger plane-wave cutoff tests the cumulative effect. Near a multiwave degeneracy, enlarge $P$ rather than forcing a two-state treatment.

Strong potentials and localized orbitals

When many Fourier components mix strongly, a low-order nearly-free expansion loses quantitative accuracy. A larger plane-wave diagonalization remains valid as a numerical representation, while a tight-binding or Wannier description may be more economical for localized bands. These are different approximations to the same Bloch problem, not competing versions of Bloch’s theorem.

Core electrons and pseudopotentials

Bare ionic potentials are singular and core wavefunctions oscillate strongly. Nearly-free reasoning is most natural for delocalized valence states after core physics has been incorporated into an effective potential. Orthogonality to core states and nonlocal pseudopotentials can be essential even when the resulting valence dispersion looks free-electron-like.

Interactions, disorder, and lifetimes

The elementary model is a one-particle Hamiltonian. Electron–electron interactions can renormalize dispersions, transfer spectral weight, and produce finite lifetimes or entirely new phases. Disorder broadens crystal momentum and can obscure small gaps. A fitted nearly-free band should therefore be identified as a one-particle or quasiparticle description, not as the exact many-body spectrum.

Exercises

Exercise 1: standing-wave energy ordering

For real $V_G>0$ at $k=G/2$, show that the symmetric standing wave is the upper state and the antisymmetric standing wave is the lower state. How does the answer change when $V_G<0$?

Solution

The two-state matrix at the boundary is

$$H_{\mathrm{eff}} = \begin{pmatrix} \varepsilon_{\mathrm B} & V_G \\ V_G & \varepsilon_{\mathrm B} \end{pmatrix}.$$

For $V_G>0$, the vector $(1,1)/\sqrt2$ has energy $\varepsilon_{\mathrm B}+V_G$ and produces

$$\psi_+(x) \propto \cos(Gx/2).$$

The vector $(1,-1)/\sqrt2$ has energy $\varepsilon_{\mathrm B}-V_G$ and produces

$$\psi_-(x) \propto \sin(Gx/2)$$

up to an overall phase. When $V_G<0$, the eigenvectors are the same but their energy ordering reverses. The lower state always places more density near minima of the resonant potential harmonic.

Exercise 2: recover the free branches

Use the one-dimensional formula for $E_{\pm}(\kappa)$ to show that far from the mixing region the two eigenvalues approach $\varepsilon_k+V_0$ and $\varepsilon_{k-G}+V_0$.

Solution

Define

$$\delta(\kappa) = \frac{ \varepsilon_k-\varepsilon_{k-G} }{2} = \frac{\hbar^2G\kappa}{2m}.$$

When $|\delta|\gg|V_G|$,

$$\sqrt{\delta^2+|V_G|^2} = |\delta| + O\left( \frac{|V_G|^2}{|\delta|} \right).$$

The two eigenvalues become the average free energy plus or minus $|\delta|$. Those are the larger and smaller of $\varepsilon_k$ and $\varepsilon_{k-G}$, with only second-order corrections. Their ordering swaps as $\kappa$ crosses zero, which is why the coupled branches avoid rather than pass through one another.

Exercise 3: origin dependence

Translate the coordinate origin by $\mathbf r_0$. Show how $V_{\mathbf G}$ transforms and prove that the two-wave energies are unchanged.

Solution

With the new coordinate $\mathbf r'=\mathbf r-\mathbf r_0$,

$$V(\mathbf r'+\mathbf r_0) = \sum_{\mathbf G} V_{\mathbf G} e^{i\mathbf G\cdot\mathbf r_0} e^{i\mathbf G\cdot\mathbf r'}.$$

Thus

$$V_{\mathbf G}' = e^{i\mathbf G\cdot\mathbf r_0} V_{\mathbf G}.$$

The phase changes but

$$|V_{\mathbf G}'| = |V_{\mathbf G}|.$$

Since the eigenvalues depend on the coupling through $|V_{\mathbf G}|^2$, the dispersion and gap are origin independent. The eigenvector phases transform so that the real-space state is described consistently.

Exercise 4: a gap without the direct Fourier harmonic

Take $V(x)=2U\cos(Gx)$ with real $|U|\ll E_G=\hbar^2G^2/(2m)$. At reduced momentum $k=0$, the free waves $|+G\rangle$ and $|-G\rangle$ are degenerate. Although $V_{2G}=0$, show that a gap opens at second order. Include the common diagonal energy shift.

Solution

The path $+G\to0\to-G$ has two matrix elements $U$ and denominator $E_G-0$. It produces the off-diagonal element $U^2/E_G$. For each diagonal entry, intermediate waves at 0 and $\pm2G$ give

$$\delta E=\frac{U^2}{E_G}+\frac{U^2}{E_G-4E_G} =\frac{2U^2}{3E_G}.$$

Thus

$$H_{\rm eff}=\left(E_G+\frac{2U^2}{3E_G}\right)I_2 +\frac{U^2}{E_G}\sigma_x+O(U^4/E_G^3),$$

and the leading gap is $2U^2/E_G$. Each scattering event changes the reciprocal index by one, so a process connecting these even-separated indices has even order; no third-order term occurs for this single harmonic. The absent direct coefficient removes the first-order gap, not all possible virtual couplings. Exact symmetry protection would have to forbid the effective coupling at every order.

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976), Chapter 9.
• P. A. Lee, “Band Structure,” lecture notes for Theory of Solids I, MIT OpenCourseWare (2004), course materials.
• S. H. Simon, The Oxford Solid State Basics (Oxford University Press, 2013), Chapter 15, Sec. 15.1.